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| {{Hatnote|For a more general introduction to the topic, please see [[Introduction to quantum mechanics#Schrödinger wave equation|Introduction to quantum mechanics]].}}
| | Name: Lillian Monds<br>My age: 30<br>Country: Austria<br>Town: Kainratsdorf <br>ZIP: 3231<br>Address: Simmeringer Hauptstrasse 10<br><br>My blog post; [http://bestregistrycleanerfix.com/fix-it-utilities fix it utilities] |
| {{Use dmy dates|date=October 2012}}
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| [[File:Erwin Schrodinger2.jpg|right|thumb|150px|[[Erwin Schrödinger]]]]
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| {{Quantum mechanics|cTopic=Equations}}
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| In [[quantum mechanics]], the '''Schrödinger equation''' is a [[partial differential equation]] that describes how the [[quantum state]] of some [[physical system]] changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist [[Erwin Schrödinger]].<ref name = sch>
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| {{cite journal
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| | last = Schrödinger | first = E.
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| | title = An Undulatory Theory of the Mechanics of Atoms and Molecules
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| | url = http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
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| | archiveurl = http://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
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| | archivedate = 17 December 2008
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| | journal = [[Physical Review]]
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| | volume = 28 | issue = 6 | pages = 1049–1070
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| | year = 1926
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| | doi = 10.1103/PhysRev.28.1049
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| |bibcode = 1926PhRv...28.1049S }}</ref>
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| In [[classical mechanics]], the [[equation of motion]] is [[Newton's second law]], and equivalent formulations are the [[Euler–Lagrange equations]] and [[Hamilton's equations]]. All of these formulations are used to solve for the motion of a mechanical system and mathematically predict what the system will do at any time beyond the initial settings and configuration of the system.
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| In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but (in general) a [[linear differential equation|linear]] [[partial differential equation]]. The differential equation describes the [[wave function]] of the system, also called the [[quantum state]] or state vector.<ref name=Griffiths2004>{{citation| author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn= 0-13-111892-7}}</ref>{{rp|1–2}}
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| The concept of a state vector is a fundamental [[Mathematical formulation of quantum mechanics|postulate of quantum mechanics]]. Although often presented as a postulate, Schrödinger's equation can in fact be derived from symmetry principles.<ref name=Ballentine>{{citation|last=Ballentine|first=Leslie|title=Quantum Mechanics: A Modern Development | publisher=World Scientific Publishing Co.|year=1998| isbn = 9810241054}}</ref>{{rp|Chapter 3}}
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| In the [[Copenhagen interpretation|standard interpretation of quantum mechanics]], the wave function is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only [[molecule|molecular]], [[atom]]ic, and [[subatomic particle|subatomic]] systems, but also [[macroscopic scale|macroscopic system]]s, possibly even the whole [[universe]].<ref name=Laloe>{{citation|last=Laloe|first=Franck|title=Do We Really Understand Quantum Mechanics| publisher=Cambridge University Press|year=2012| isbn = 978-1-107-02501-1}}</ref>{{rp|292ff}}
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| Like Newton's second law (''F = ma''), the Schrödinger equation can be mathematically transformed into other formulations such as [[Werner Heisenberg]]'s [[matrix mechanics]], and [[Richard Feynman]]'s [[path integral formulation]]. Also like Newton's second law, the Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem that is not as severe in matrix mechanics and completely absent in the path integral formulation.
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| {{TOC limit|3}}
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| ==Equation==
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| ===Time-dependent equation===
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| The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the [[Schrödinger equation#Time dependent|time-dependent Schrödinger equation]], which gives a description of a system evolving with time:<ref name=Shankar1994>
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| {{cite book
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| |last=Shankar |first=R.
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| |year=1994
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| |title=Principles of Quantum Mechanics
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| |edition=2nd
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| |publisher=[[Kluwer Academic]]/[[Plenum Publishers]]
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| |isbn=978-0-306-44790-7
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| }}</ref>{{rp|143}}
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| {{Equation box 1
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| |indent=:
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| |title='''Time-dependent Schrödinger equation''' ''(general)''
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| |equation=<math>i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi</math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where ''i'' is the [[imaginary unit]], the symbol "''∂/∂t''" indicates a [[partial derivative]] with respect to time ''t'', ''ħ'' is [[Planck's constant]] divided by 2''π'', ''[[Ψ]]'' is the [[wave function]] of the quantum system, and <math>\hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] [[operator (physics)|operator]] (which characterizes the total energy of any given wave function and takes different forms depending on the situation).
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| [[File:Wave packet (dispersion).gif|thumb|200px|A [[wave function]] that satisfies the non-relativistic Schrödinger equation with ''V''=0. In other words, this corresponds to a particle traveling freely through empty space. The [[real part]] of the [[wave function]] is plotted here.]]
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| The most famous example is the [[relativistic quantum mechanics|non-relativistic]] Schrödinger equation for a single particle moving in an [[electric field]] (but not a [[magnetic field]]; see the [[Pauli equation]]):
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| {{Equation box 1
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| |indent=:
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| |title='''Time-dependent Schrödinger equation''' ''(single [[relativistic quantum mechanics|non-relativistic]] particle)''
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| |equation=<math>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)</math>
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| where ''m'' is the particle's mass, ''V'' is its [[potential energy]], ∇<sup>2</sup> is the [[Laplacian]], and ''Ψ'' is the wave function (more precisely, in this context, it is called the "position-space wave function"). In plain language, it means "total energy equals [[kinetic energy]] plus [[potential energy]]", but the terms take unfamiliar forms for reasons explained below.
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| Given the particular differential operators involved, this is a [[linear differential equation|linear]] [[partial differential equation]]. It is also a [[diffusion equation]], but unlike the [[heat equation]], this one is also a wave equation given the [[imaginary unit]] present in the transient term.
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| The term ''"Schrödinger equation"'' can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the [[Dirac equation]] to [[quantum field theory]], by plugging in various complicated expressions for the Hamiltonian. The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see [[relativistic quantum mechanics]] and [[relativistic quantum field theory]]).
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| To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.
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| ==={{anchor|Time independent equation}}Time-independent equation===
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| [[File:StationaryStatesAnimation.gif|300px|thumb|right|Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a [[quantum harmonic oscillator|harmonic oscillator]]. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The [[probability distribution]] of finding the particle with this wave function at a given position. The top two rows are examples of '''[[stationary state]]s''', which correspond to [[standing wave]]s. The bottom row an example of a state which is ''not'' a stationary state. The right column illustrates why stationary states are called "stationary".]]
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| The time-independent Schrödinger equation predicts that wave functions can form [[standing wave]]s, called [[stationary state]]s (also called "orbitals", as in [[atomic orbital]]s or [[molecular orbital]]s). These states are important in their own right, and moreover if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for ''any'' state. The ''time-independent Schrödinger equation'' is the equation describing stationary states. (It is only used when the [[Hamiltonian (quantum mechanics)|Hamiltonian]] itself is not dependent on time. In general, the wave function still has a time dependency! )
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| {{Equation box 1
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| |indent=:
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| |title='''Time-independent Schrödinger equation''' (''general'')
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| |equation=<math>E\Psi=\hat H \Psi</math>
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| In words, the equation states:
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| ::''When the Hamiltonian operator acts on a certain wave function Ψ, and the result is proportional to the same wave function Ψ, then Ψ is a [[stationary state]], and the proportionality constant, E, is the energy of the state Ψ.''
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| The time-independent Schrödinger equation is discussed further [[#Time_independent|below]]. In [[linear algebra]] terminology, this equation is an [[Eigenvalues and eigenvectors|eigenvalue equation]].
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| As before, the most famous manifestation is the [[relativistic quantum mechanics|non-relativistic]] Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):
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| {{Equation box 1
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| |indent=:
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| |title='''Time-independent Schrödinger equation''' (''single non-[[relativistic quantum mechanics|relativistic]] particle'')
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| |equation=<math>E \Psi(\mathbf{r}) = \left[ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r})</math>
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| with definitions as above.
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| ==Implications==
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| The Schrödinger equation, and its solutions, introduced a breakthrough in thinking about physics. Schrödinger's equation was the first of its type, and solutions led to very unusual and unexpected consequences for the time.
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| ===Total, kinetic, and potential energy===
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| The ''overall'' form of the equation is ''not'' unusual or unexpected as it uses the principle of the [[conservation of energy]]. The terms of the nonrelativistic Schrödinger equation can be interpreted as total energy of the system, equal to the system [[kinetic energy]] plus the system [[potential energy]]. In this respect, it is just the same as in classical physics.
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| ===Quantization===
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| The Schrödinger equation predicts that if certain properties of a system are measured, the result may be ''quantized'', meaning that only specific discrete values can occur. One example is ''energy quantization'': the energy of an electron in an atom is always one of the [[Energy level|quantized energy levels]], a fact discovered via [[atomic spectroscopy]]. (Energy quantization is discussed [[#Time_independent|below]].) Another example is [[angular momentum operator|quantization of angular momentum]]. This was an ''assumption'' in the earlier [[Bohr model|Bohr model of the atom]], but it is a ''prediction'' of the Schrödinger equation.
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| Another result of the Schrödinger equation is that not every measurement gives a quantized result in quantum mechanics. For example, position, momentum, time, and (in some situations) energy can have any value across a continuous range.{{citation needed|date=August 2013}}
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| ===Measurement and uncertainty===
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| {{main|Measurement in quantum mechanics|Heisenberg uncertainty principle|Interpretations of quantum mechanics}}
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| In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change [[determinism|deterministically]] as the particle moves according to [[Newton's laws]]. In quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a [[probability distribution]]. The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.
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| The [[Heisenberg uncertainty principle]] is the statement of the inherent measurement uncertainty in quantum mechanics. It states that the more precisely a particle's position is known, the less precisely its momentum is known, and vice versa.
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| The Schrödinger equation describes the (deterministic) evolution of the [[wave function]] of a particle. However, even if the wave function is known exactly, the result of a specific measurement on the wave function is uncertain.
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| ===Quantum tunneling===
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| {{main|Quantum tunneling}}
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| [[File:TunnelEffektKling1.png|width = 300px|thumb|[[Quantum tunneling]] through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side. ]]
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| In classical physics, when a ball is rolled slowly up a large hill, it will come to a stop and roll back, because it doesn't have enough [[energy]] to get over the top of the hill to the other side. However, the Schrödinger equation predicts that there is a small probability that the ball will get to the other side of the hill, even if it has too little energy to reach the top. This is called [[quantum tunneling]]. It is related to the uncertainty principle: Although the ball seems to be on one side of the hill, its position is uncertain so there is a chance of finding it on the other side.
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| ===Particles as waves===
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| {{main|Matter wave|Wave–particle duality|Double-slit experiment}}
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| [[File:Double-slit experiment results Tanamura 2.jpg|175px|thumb|A double slit experiment showing the accumulation of electrons on a screen as time passes.]]
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| The nonrelativistic Schrödinger equation is a type of [[partial differential equation]] called a [[wave equation]]. Therefore it is often said particles can exhibit behavior usually attributed to waves. In most modern interpretations this description is reversed – the quantum state, i.e. wave, is the only genuine physical reality, and under the appropriate conditions it can show features of particle-like behavior.
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| [[Double-slit experiment|Two-slit diffraction]] is a famous example of the strange behaviors that waves regularly display, that are not intuitively associated with particles. The overlapping waves from the two slits cancel each other out in some locations, and reinforce each other in other locations, causing a complex pattern to emerge. Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.
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| However, since the Schrödinger equation is a [[wave equation]], a single particle fired through a double-slit ''does'' show this same pattern (figure on left). Note: The experiment must be repeated many times for the complex pattern to emerge. The appearance of the pattern proves that each electron passes through ''both'' slits simultaneously.<ref>O Donati G F Missiroli G Pozzi May 1973 An Experiment on Electron Interference ''American Journal of Physics'' '''41''' 639–644</ref><ref>Brian Greene, ''The Elegant Universe,'' p. 110</ref><ref>Feynman Lectures on Physics (Vol. 3), R. Feynman, R.B. Leighton, M. Sands, Addison-Wesley, 1965, ISBN 0-201-02118-8</ref> Although this is counterintuitive, the prediction is correct; in particular, [[electron diffraction]] and [[neutron diffraction]] are well understood and widely used in science and engineering.
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| Related to [[diffraction]], particles also display [[Superposition principle#Application to waves|superposition]] and [[Interference (wave propagation)|interference]].
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| The superposition property allows the particle to be in a [[quantum superposition]] of two or more states with different classical properties at the same time. For example, a particle can have several different energies at the same time, and can be in several different locations at the same time. In the above example, a particle can pass through two slits at the same time. This superposition is still a single quantum state, as shown by the interference effects, even though that conflicts with classical intuition.
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| ==Interpretation of the wave function==
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| {{main|Interpretations of quantum mechanics}}
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| The Schrödinger equation provides a way to calculate the possible wave functions of a system and how they dynamically change in time. However, the Schrödinger equation does not directly say ''what'', exactly, the wave function is. [[Interpretations of quantum mechanics]] address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.
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| An important aspect is the relationship between the Schrödinger equation and [[wavefunction collapse]]. In the oldest [[Copenhagen interpretation]], particles follow the Schrödinger equation ''except'' during wavefunction collapse, during which they behave entirely differently. The advent of [[quantum decoherence|quantum decoherence theory]] allowed alternative approaches (such as the [[Everett many-worlds interpretation]] and [[consistent histories]]), wherein the Schrödinger equation is ''always'' satisfied, and wavefunction collapse should be explained as a consequence of the Schrödinger equation.
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| ==Historical background and development==
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| {{Main|Theoretical and experimental justification for the Schrödinger equation}}
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| Following [[Max Planck]]'s quantization of light (see [[black body radiation]]), [[Albert Einstein]] interpreted Planck's [[quantum|quanta]] to be [[photon]]s, [[corpuscular theory of light|particles of light]], and proposed that the [[Planck relation|energy of a photon is proportional to its frequency]], one of the first signs of [[wave–particle duality]]. Since [[energy]] and [[momentum]] are related in the same way as [[frequency]] and [[wavenumber]] in [[special relativity]], it followed that the momentum ''p'' of a photon is inversely proportional to its [[wavelength]] ''λ'', or proportional to its [[wavenumber]] ''k''.
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| :<math>p = \frac{h}{\lambda} = \hbar k</math> | |
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| where ''h'' is [[Planck's constant]]. [[Louis de Broglie]] hypothesized that this is true for all particles, even particles such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form [[standing wave]]s, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.<ref>
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| {{Cite journal
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| |last=de Broglie |first=L.
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| |year=1925
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| |title=Recherches sur la théorie des quanta
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| |trans_title=On the Theory of Quanta
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| |url=http://tel.archives-ouvertes.fr/docs/00/04/70/78/PDF/tel-00006807.pdf
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| |journal=[[Annales de Physique]]
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| |volume=10 |issue=3 |pages=22–128
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| |doi=
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| }} [http://web.archive.org/web/20090509012910/http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.pdf Translated version].</ref>
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| These quantized orbits correspond to discrete [[energy level]]s, and de Broglie reproduced the [[Bohr model]] formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum ''L'' according to:
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| :<math> L = n{h \over 2\pi} = n\hbar.</math>
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| According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:
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| :<math>n \lambda = 2 \pi r.\,</math>
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| This approach essentially confined the electron wave in one dimension, along a circular orbit of radius ''r''.
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| In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum 4-vector to derive what we now call the de Broglie relation<ref>{{cite journal|last=Weissman|first=M.B.|coauthors=V. V. Iliev and I. Gutman|title=A pioneer remembered: biographical notes about Arthur Constant Lunn|journal=Communications in Mathematical and in Computer Chemistry|year=2008|volume=59|issue=3|pages=687–708}}</ref> Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation, and solve for its energy eigenvalues for the hydrogen atom. Unfortunately the paper was rejected by the Physical Review, as recounted by Kamen.<ref>{{cite book|last=Kamen|first=Martin D.|title=Radiant Science, Dark Politics|year=1985|publisher=University of California Press|location=Berkeley and Los Angeles, CA|isbn=0-520-04929-2|pages=29–32}}</ref>
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| Following up on de Broglie's ideas, physicist [[Peter Debye]] made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by [[William Rowan Hamilton|William R. Hamilton]]'s analogy between [[mechanics]] and [[optics]], encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system — the trajectories of light rays become sharp tracks that obey [[Fermat's principle]], an analog of the [[principle of least action]].<ref>
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| {{Cite book
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| |last=Schrodinger |first=E.
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| |year=1984
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| |title=Collected papers
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| |publisher=[[Friedrich Vieweg und Sohn]]
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| |isbn=3-7001-0573-8
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| }} See introduction to first 1926 paper.</ref><!--
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| --- Is the non-formulation by Hamilton really relevant?
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| Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves.<ref>
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| {{cite web
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| |last=Michon |first=G.P.
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| |year=2009
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| |title=Hamilton's Analogy: Paths to the Schrödinger Equation
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| |url=http://home.att.net/~numericana/answer/schrodinger.htm#1928
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| |work=Final Answers: The Schrödinger Equation
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| |accessdate=28 February 2010
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| }}</ref>
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| ----> A modern version of his reasoning is reproduced below. The equation he found is:<ref name="verlagsgesellschaft1991">Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) ISBN 0-89573-752-3</ref>
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| :<math>i\hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t).</math>
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| However, by that time, [[Arnold Sommerfeld]] had [[Sommerfeld–Wilson quantization|refined the Bohr model]] with [[fine structure|relativistic corrections]].<ref>
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| {{Cite book
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| |last=Sommerfeld |first=A.
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| |year=1919
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| |title=Atombau und Spektrallinien
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| |publisher=[[Friedrich Vieweg und Sohn]]
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| |location=Braunschweig
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| |isbn=3-87144-484-7
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| }}</ref><ref>For an English source, see {{Cite journal
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| |last=Haar |first=T.
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| |year=
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| |title=The Old Quantum Theory
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| |url=
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| |publisher=
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| |pages=
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| |isbn=
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| }}</ref> Schrödinger used the relativistic energy momentum relation to find what is now known as the [[Klein–Gordon equation]] in a [[Coulomb potential]] (in [[natural units]]):
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| :<math>\left(E + {e^2\over r} \right)^2 \psi(x) = - \nabla^2\psi(x) + m^2 \psi(x).</math>
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| <!-- <math>\frac{1}{c^2}\left(E + {e^2\over 4 \pi \epsilon_0 r} \right)^2 \psi(x) = -\hbar^2 \nabla^2\psi(x) + \frac{m^2c^2}{\hbar^2} \psi(x).</math> ?-->
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| He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin with a lover, in December 1925.<ref>
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| {{cite book
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| |last=Rhodes |first=R.
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| |year=1986
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| |title=Making of the Atomic Bomb
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| |publisher=[[Touchstone Books|Touchstone]]
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| |pages=
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| |isbn=0-671-44133-7
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| }}</ref>
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| While at the cabin, Schrödinger decided that his earlier non-relativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite difficulties solving the differential equation for hydrogen (he had later help from his friend the mathematician [[Hermann Weyl]]<ref name="Schrödinger1982"/>{{rp|3}}) Schrödinger showed that his non-relativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.<ref name="Schrödinger1982">{{cite book|author=Erwin Schrödinger|title=Collected Papers on Wave Mechanics: Third Edition|year=1982|publisher=American Mathematical Soc.|isbn=978-0-8218-3524-1}}</ref>{{rp|1}}<ref>{{cite journal
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| |last=Schrödinger |first=E.
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| |year=1926
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| |title=Quantisierung als Eigenwertproblem; von Erwin Schrödinger
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| |url=http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination
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| |journal=[[Annalen der Physik]]
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| |volume= |issue= |pages=361–377
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| |doi=
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| }}</ref> In the equation, Schrödinger computed the [[hydrogen spectral series]] by treating a [[hydrogen atom]]'s [[electron]] as a wave ''Ψ''(''x'', ''t''), moving in a [[potential well]] ''V'', created by the [[proton]]. This computation accurately reproduced the energy levels of the [[Bohr model]]. In a paper, Schrödinger himself explained this equation as follows:
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| {{cquote|The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog. |20px|20px|Erwin Schrödinger<ref>Erwin Schrödinger, "The Present situation in Quantum Mechanics," p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in '''Proceedings of the American Philosophical Society''', 124, 323–38. It later appeared as Section I.11 of Part I of '''Quantum Theory and Measurement''' by J.A. Wheeler and W.H. Zurek, eds., Princeton University Press, New Jersey 1983).</ref>}}
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| This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's [[matrix mechanics]], which he considered overly formal.<ref>
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| {{Cite journal
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| |last=Einstein |first=A.
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| |coauthors=''et. al''.
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| |year=
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| |title=Letters on Wave Mechanics: Schrodinger–Planck–Einstein–Lorentz
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| |url=
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| }}</ref>
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| The Schrödinger equation details the behavior of ''ψ'' but says nothing of its ''nature''. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful.<ref name=Moore1992>
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| {{cite book
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| |last=Moore |first=W.J.
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| |year=1992
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| |title=Schrödinger: Life and Thought
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| |publisher=[[Cambridge University Press]]
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| |isbn=0-521-43767-9
| |
| }}</ref>{{rp|219}} In 1926, just a few days after Schrödinger's fourth and final paper was published, [[Max Born]] successfully interpreted ''ψ'' as the [[probability amplitude]], whose absolute square is equal to [[Probability density function|probability density]].<ref name=Moore1992/>{{rp|220}} Schrödinger, though, always opposed a [[statistical]] or probabilistic approach, with its associated [[wavefunction collapse|discontinuities]]—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying [[determinism|deterministic theory]]— and never reconciled with the [[Copenhagen interpretation]].{{#tag:ref |It is clear that even in his last year of life, as shown in a letter to Max Born, that Schrödinger never accepted the Copenhagen interpretation.<ref name=Moore1992/>{{rp|220}}}}
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| Louis de Broglie in his later years has proposed a real valued [[wave function]] connected to the complex wave function by a proportionality constant and developed the [[De Broglie–Bohm theory]].
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| ==The wave equation for particles==
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| {{main|wave–particle duality}}
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| The Schrödinger equation is mathematically a ''[[wave equation]]'', since the ''solutions'' are functions which describe wave-like motions. Normally wave equations in physics can be derived from other physical laws – the wave equation for [[Vibration|mechanical vibrations]] on strings and in matter can be derived from [[Newton's laws]] – where the analogue wave function is the [[Displacement (vector)|displacement]] of matter, and [[electromagnetic waves]] from [[Maxwell's equations]], where the wave functions are [[electric field|electric]] and [[magnetic field|magnetic]] fields. On the contrary, the basis for Schrödinger's equation is the energy of the ''particle'', and a separate [[Mathematical formulation of quantum mechanics#Postulates of quantum mechanics|postulate of quantum mechanics]]: the wave function is a description of the system.<ref name="Quantum Chemistry 1977">Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> The SE is therefore a ''new concept in itself''; as Feynman put it:
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| {{cquote|Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. |20px|20px|Richard Feynman<ref>The New Quantum Universe, T.Hey, P.Walters, Cambridge University Press, 2009, ISBN 978-0-521-56457-1</ref>}}
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| The equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function ''Ψ'', which contains all the information that can be known about the system. In the [[Copenhagen interpretation]], the modulus of ''Ψ'' is related to the [[probability]] the particles are in some spatial configuration at some instant of time. Solving the equation for ''Ψ'' can be used to predict how the particles will behave under the influence of the specified potential and with each other.
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| The Schrödinger equation was developed principally from the [[De Broglie hypothesis]], a wave equation that would describe particles,<ref name="Quanta 1974">Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1</ref> and can be constructed in the following way.<ref name="Molecules, B.H. Bransden 1983">Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2</ref> For a more rigorous description of Schrödinger's equation, see also.<ref name="Atoms, Molecules 1985"/>
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| === Consistency with energy conservation ===
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| The total [[energy]] ''E'' of a particle is the sum of kinetic energy ''T'' and potential energy ''V'', this sum is also the frequent expression for the [[Hamiltonian mechanics#Basic physical interpretation|Hamiltonian]] ''H'' in classical mechanics:
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| :<math>E = T + V =H \,\!</math>
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| Explicitly, for a particle in one dimension with position ''x'', [[mass]] ''m'' and [[momentum]] ''p'', and potential energy ''V'' which generally [[Harmonic function|varies with position]] and [[time]] ''t'':
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| :<math> E = \frac{p^2}{2m}+V(x,t)=H.</math>
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| For three dimensions, the [[position vector]] '''r''' and momentum vector '''p''' must be used:
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| :<math>E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H</math>
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| This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian. However, there can be [[interactions#Physics|interactions]] between the particles (an [[many body problem|''N''-body problem]]), so the potential energy ''V'' can change as the spatial configuration of particles changes, and possibly with time. The potential energy, in general, is ''not'' the sum of the separate potential energies for each particle, it is a function of all the spatial positions of the particles. Explicitly:
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| :<math>E=\sum_{n=1}^N \frac{\mathbf{p}_n\cdot\mathbf{p}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2\cdots\mathbf{r}_N,t) = H \,\!</math>
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| | |
| === Linearity ===
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| The simplest wavefunction is a [[plane wave]] of the form:
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| :<math> \Psi(\mathbf{r},t) = A e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} \,\!</math>
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| where the ''A'' is the amplitude, '''k''' the wavevector, and ''ω'' the angular frequency, of the plane wave. In general, physical situations are not purely described by plane waves, so for generality the [[superposition principle]] is required; any wave can be made by superposition of sinusoidal plane waves. So if the equation is linear, a [[linear combination]] of plane waves is also an allowed solution. Hence a necessary and separate requirement is that the Schrödinger equation is a [[linear differential equation]].
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| For discrete '''k''' the sum is a [[superposition principle#Application to waves|superposition]] of plane waves:
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| :<math> \Psi(\mathbf{r},t) = \sum_{n=1}^\infty A_n e^{i(\mathbf{k}_n\cdot\mathbf{r}-\omega_n t)} \,\!</math>
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| for some real amplitude coefficients ''A<sub>n</sub>'', and for continuous '''k''' the sum becomes an integral, the [[Fourier transform]] of a momentum space wavefunction:<ref name="Quantum Mechanics Demystified 2006">Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>
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| :<math> \Psi(\mathbf{r},t) = \frac{1}{(\sqrt{2\pi})^3}\int\Phi(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}d^3\mathbf{k} \,\!</math>
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| where {{nowrap|1=''d''<sup>3</sup>'''k''' = ''dk<sub>x</sub>dk<sub>y</sub>dk<sub>z</sub>''}} is the differential volume element in [[momentum space|'''k'''-space]], and the integrals are taken over all '''k'''-space. The momentum wavefunction Φ('''k''') arises in the integrand since the position and momentum space wavefunctions are Fourier transforms of each other.
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| === Consistency with the De Broglie relations ===
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| [[File:Wavefunction values.svg|300px|"300px"|thumb|Diagrammatic summary of the quantities related to the wavefunction, as used in De broglie's hypothesis and development of the Schrödinger equation.<ref name="Quanta 1974"/>]]
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| [[Photoelectric effect|Einstein's light quanta hypothesis]] (1905) states that the energy ''E'' of a photon is proportional to the [[frequency]] ''ν'' (or [[angular frequency]], ''ω'' = 2π''ν'') of the corresponding quantum wavepacket of light:
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| :<math>E = h\nu = \hbar \omega \,\!</math>
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| Likewise [[Matter wave|De Broglie's hypothesis]] (1924) states that any particle can be associated with a wave, and that the momentum ''p'' of the particle is inversely proportional to the [[wavelength]] ''λ'' of such a wave (or proportional to the [[wavenumber]], ''k'' = 2π/''λ''), in one dimension, by:
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| :<math>p = \frac{h}{\lambda} = \hbar k\;,</math>
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| while in three dimensions, wavelength ''λ'' is related to the magnitude of the [[wavevector]] '''k''':
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| :<math>\mathbf{p} = \hbar \mathbf{k}\,,\quad |\mathbf{k}| = \frac{2\pi}{\lambda} \,. </math>
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| The Planck–Einstein and de Broglie relations illuminate the deep connections between energy with time, and space with momentum, and express [[wave–particle duality]]. In practice, [[natural units]] comprising {{nowrap|1=''ħ'' = 1}} are used, as the De Broglie ''equations'' reduce to ''identities'': allowing momentum, wavenumber, energy and frequency to be used interchangeably, to prevent duplication of quantities, and reduce the number of dimensions of related quantities. For familiarity SI units are still used in this article.
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| Schrödinger's insight,{{citation needed|date=January 2014}} late in 1925, was to express the [[Phase (waves)|phase]] of a [[plane wave]] as a [[complex number|complex]] [[phase factor]] using these relations:
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| :<math>\Psi = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} </math>
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| and to realize that the first order [[partial derivatives]] with respect to space
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| :<math> \nabla\Psi = \dfrac{i}{\hbar}\mathbf{p}Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = \dfrac{i}{\hbar}\mathbf{p}\Psi </math>
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| and time
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| :<math> \dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = -\dfrac{i E}{\hbar} \Psi </math>
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| Another postulate of quantum mechanics is that all observables are represented by [[linear operator|linear]] [[Self-adjoint operator|Hermitian operators]] which act on the wavefunction, and the eigenvalues of the operator are the values the observable takes. The previous derivatives are consistent with the [[energy operator]], corresponding to the time derivative,
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| :<math>\hat{E} \Psi = i\hbar\dfrac{\partial}{\partial t}\Psi = E\Psi </math>
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| where ''E'' are the energy [[eigenvalue]]s, and the [[momentum operator]], corresponding to the spatial derivatives (the [[del|gradient]] ∇),
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| :<math>\hat{\mathbf{p}} \Psi = -i\hbar\nabla \Psi = \mathbf{p} \Psi </math>
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| where '''p''' is a vector of the momentum eigenvalues. In the above, the "[[circumflex|hats]]" ( ^ ) indicate these observables are operators, not simply ordinary numbers or vectors. The energy and momentum operators are ''[[differential operators]]'', while the potential energy function ''V'' is just a multiplicative factor.
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| Substituting the energy and momentum operators into the classical energy conservation equation obtains the operator:
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| :<math>E= \dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V \quad \rightarrow \quad \hat{E} = \dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V </math>
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| so in terms of derivatives with respect to time and space, acting this operator on the wavefunction ''Ψ'', immediately led Schrödinger to his equation:{{citation needed|date=January 2014}}
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| :<math>i\hbar\dfrac{\partial \Psi}{\partial t}= -\dfrac{\hbar^2}{2m}\nabla^2\Psi +V\Psi</math>
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| Wave–particle duality can be assessed from these equations as follows. The kinetic energy ''T'' is related to the square of momentum '''p'''. As the particle's momentum increases, the kinetic energy increases more rapidly, but since the wavenumber |'''k'''| increases the wavelength ''λ'' decreases. In terms of ordinary scalar and vector quantities (not operators):
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| :<math> \mathbf{p}\cdot\mathbf{p} \propto \mathbf{k}\cdot\mathbf{k} \propto T \propto \dfrac{1}{\lambda^2}</math>
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| The kinetic energy is also proportional to the second spatial derivatives, so it is also proportional to the magnitude of the ''[[curvature]]'' of the wave, in terms of operators:
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| :<math> \hat{T} \Psi = \frac{-\hbar^2}{2m}\nabla\cdot\nabla \Psi \, \propto \, \nabla^2 \Psi \,.</math>
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| As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.<ref name="Quanta 1974"/>
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| ===Wave and particle motion===
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| {{multiple image
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| | align = right
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| | direction = horizontal
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| | image1 = Quantum mechanics travelling wavefunctions.svg
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| | caption1 = Increasing levels of [[wavepacket]] localization, meaning the particle has a more localized position.
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| | width1 = 400
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| | image2 = Perfect localization.svg
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| | caption2 = In the limit ''ħ'' → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.
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| | width2 = 150
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| }}
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| Schrödinger required that a [[wave packet]] solution near position '''r''' with wavevector near '''k''' will move along the trajectory determined by classical mechanics for times short enough for the spread in '''k''' (and hence in velocity) not to substantially increase the spread in '''r''' . Since, for a given spread in '''k''', the spread in velocity is proportional to Planck's constant ''ħ'', it is sometimes said that in the limit as ''ħ'' approaches zero, the equations of classical mechanics are restored from quantum mechanics.<ref name="Analytical Mechanics 2008">Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0</ref> Great care is required in how that limit is taken, and in what cases.
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| The limiting short-wavelength is equivalent to ''ħ'' tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the [[Heisenberg uncertainty principle]] for position and momentum, the products of uncertainty in position and momentum become zero as {{nowrap|''ħ'' → 0}}:
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| :<math> \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!</math>
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| | |
| where ''σ'' denotes the (root mean square) [[measurement uncertainty]] in ''x'' and ''p<sub>x</sub>'' (and similarly for the ''y'' and ''z'' directions) which implies the position and momentum can only be known to arbitrary precision in this limit.
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| The Schrödinger equation in its general form
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| :<math> i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right) \,\!</math>
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| is closely related to the [[Hamilton–Jacobi equation]] (HJE)
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| :<math> \frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\!</math>
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| where ''S'' is [[action (physics)|action]] and ''H'' is the [[Hamiltonian mechanics|Hamiltonian function]] (not operator). Here the [[generalized coordinates]] ''q<sub>i</sub>'' for {{nowrap|1=''i'' = 1, 2, 3}} (used in the context of the HJE) can be set to the position in Cartesian coordinates as {{nowrap|1='''r''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>, ''q''<sub>3</sub>) = (''x'', ''y'', ''z'')}}.<ref name="Analytical Mechanics 2008"/>
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| Substituting
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| :<math> \Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}\,\!</math>
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| where ρ is the probability density, into the Schrödinger equation and then taking the limit {{nowrap|''ħ'' → 0}} in the resulting equation, yields the Hamilton–Jacobi equation.
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| The implications are:
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| * The motion of a particle, described by a (short-wavelength) wave packet solution to the Schrödinger equation, is also described by the Hamilton–Jacobi equation of motion.
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| * The Schrödinger equation includes the wavefunction, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.
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| ==Non-relativistic quantum mechanics==
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| The quantum mechanics of particles without accounting for the effects of [[special relativity]], for example particles propagating at speeds much less than [[speed of light|light]], is known as '''non-relativistic quantum mechanics'''. Following are several forms of Schrödinger's equation in this context for different situations: time independence and dependence, one and three spatial dimensions, and one and ''N'' particles.
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| In actuality, the particles constituting the system do not have the numerical labels used in theory. The language of mathematics forces us to label the positions of particles one way or another, otherwise there would be confusion between symbols representing which variables are for which particle.<ref name="Atoms, Molecules 1985">Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0</ref>
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| ===Time independent===
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| If the Hamiltonian is not an explicit function of time, the equation is [[Separation of variables|separable]] into a product of spatial and temporal parts. In general, the wavefunction takes the form:
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| :<math>\Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,.</math>
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| where ''ψ''(space coords) is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and ''τ''(''t'') is a function of time only.
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| Substituting for ''Ψ'' into the Schrödinger equation for the relevant number of particles in the relevant number of dimensions, solving by [[separation of variables]] implies the general solution of the time-dependent equation has the form:<ref name="verlagsgesellschaft1991"/>
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| :<math> \Psi(\text{space coords},t) = \psi(\text{space coords}) e^{-i{E t/\hbar}} \,.</math>
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| Since the time dependent phase factor is always the same, only the spatial part needs to be solved for in time independent problems. Additionally, the energy operator <math> \hat{E} = i \hbar \partial / \partial t \,\!</math> can always be replaced by the energy eigenvalue ''E'', thus the time independent Schrödinger equation is an [[eigenvalue]] equation for the Hamiltonian operator:<ref name=Shankar1994/>{{rp|143ff}}
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| :<math>\hat{H} \psi = E \psi </math>
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| This is true for any number of particles in any number of dimensions (in a time independent potential). This case describes the [[standing wave]] solutions of the time-dependent equation, which are the states with definite energy (instead of a probability distribution of different energies). In physics, these standing waves are called "[[stationary state]]s" or "energy eigenstates"; in chemistry they are called "[[atomic orbital]]s" or "[[molecular orbital]]s". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels.
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| The energy eigenvalues from this equation form a discrete [[Spectrum (functional analysis)|spectrum]] of values, so mathematically energy must be quantized. More specifically, the energy eigenstates form a basis – any wavefunction may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is the [[spectral theorem]] in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a [[Hermitian matrix]].
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| ===One-dimensional examples===
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| For a particle in one dimension, the Hamiltonian is:
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|
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| :<math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x) \,, \quad \hat{p} = -i\hbar \frac{d}{d x} </math>
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| and substituting this into the general Schrödinger equation gives:
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| :<math> -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi(x) + V(x)\psi(x) = E\psi(x) </math>
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| This is the only case the Schrödinger equation is an [[ordinary derivative|ordinary]] differential equation, rather than a [[partial derivative|partial]] differential equation. The general solutions are always of the form:
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| :<math> \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . </math>
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| For ''N'' particles in one dimension, the Hamiltonian is:
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| :<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n}</math>
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| where the position of particle ''n'' is ''x<sub>n</sub>''. The corresponding Schrödinger equation is:
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| :<math> -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\psi(x_1,x_2,\cdots x_N) + V(x_1,x_2,\cdots x_N)\psi(x_1,x_2,\cdots x_N) = E\psi(x_1,x_2,\cdots x_N) \, .</math>
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| so the general solutions have the form:
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| :<math> \Psi(x_1,x_2,\cdots x_N,t) = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N) </math>
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| For non-interacting distinguishable particles,<ref>{{cite book|author=N. Zettili|title=Quantum Mechanics: Concepts and Applications|edition=2nd|page=458|isbn=978-0-470-02679-3}}</ref> the potential of the system only influences each particle separately, so the total potential energy is the sum of potential energies for each particle:
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| :<math> V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .</math>
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| and the wavefunction can be written as a product of the wavefunctions for each particle:
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| :<math> \Psi(x_1,x_2,\cdots x_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, ,</math>
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| For non-interacting [[identical particles]], the potential is still a sum, but wavefunction is a bit more complicated - it is a sum over the permutations of products of the separate wavefunctions to account for particle exchange. In general for interacting particles, the above decompositions are ''not'' possible.
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| ====Free particle====
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| For no potential, {{nowrap|1=''V'' = 0}}, so the particle is free and the equation reads:<ref name=Shankar1994/>{{rp|151ff}}
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| :<math> - E \psi = \frac{\hbar^2}{2m}{d^2 \psi \over d x^2}\,</math>
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| which has oscillatory solutions for {{nowrap|''E'' > 0}} (the ''C''<sub>n</sub> are arbitrary constants):
| |
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| :<math>\psi_E(x) = C_1 e^{i\sqrt{2mE/\hbar^2}\,x} + C_2 e^{-i\sqrt{2mE/\hbar^2}\,x}\,</math>
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| and exponential solutions for {{nowrap|''E'' < 0}}
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| :<math>\psi_{-|E|}(x) = C_1 e^{\sqrt{2m|E|/\hbar^2}\,x} + C_2 e^{-\sqrt{2m|E|/\hbar^2}\,x}.\,</math>
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| The exponentially growing solutions have an infinite norm, and are not physical. They are not allowed in a finite volume with periodic or fixed boundary conditions.
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| ====Constant potential====
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| [[File:EffetTunnel.gif|left|Animation of a de Broglie wave incident on a barrier.]]
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| For a constant potential, {{nowrap|1=''V'' = ''V''<sub>0</sub>}}, the solution is oscillatory for {{nowrap|''E'' > ''V''<sub>0</sub>}} and exponential for {{nowrap|''E'' < ''V''<sub>0</sub>}}, corresponding to energies that are allowed or disallowed in classical mechanics. Oscillatory solutions have a classically allowed energy and correspond to actual classical motions, while the exponential solutions have a disallowed energy and describe a small amount of quantum bleeding into the classically disallowed region, due to [[quantum tunneling]]. If the potential ''V''<sub>0</sub> grows at infinity, the motion is classically confined to a finite region, which means that in quantum mechanics every solution becomes an exponential far enough away. The condition that the exponential is decreasing restricts the energy levels to a discrete set, called the allowed energies.<ref name="Quantum Mechanics Demystified 2006"/>
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| {{-}}
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| ====Harmonic oscillator====
| |
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| [[File:QuantumHarmonicOscillatorAnimation.gif|300px|thumb|right|A [[harmonic oscillator]] in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a [[Hooke's law|spring]], oscillates back and forth. (C–H) are six solutions to the Schrödinger Equation for this situation. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the [[wavefunction]]. [[Stationary state]]s, or energy eigenstates, which are solutions to the time-independent Schrödinger Equation, are shown in C,D,E,F, but not G &H.]]
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| {{Main|Quantum harmonic oscillator}}
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| The Schrödinger equation for this situation is
| |
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| :<math> E\psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi + \frac{1}{2}m\omega^2x^2\psi </math>
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| | |
| It is a notable quantum system to solve for; since the solutions are exact (but complicated – in terms of [[Hermite polynomials]]), and it can describe or at least approximate a wide variety of other systems, including [[Molecular vibration|vibrating atoms, molecules]],<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7</ref> and atoms or ions in lattices,<ref>Solid State Physics (2nd Edition), J.R. Hook, H.E. Hall, Manchester Physics Series, John Wiley & Sons, 2010, ISBN 978-0-471-92804-1</ref> and approximating other potentials near equilibrium points. It is also the [[Perturbation theory (quantum mechanics)#Applications of perturbation theory|basis of perturbation methods]] in quantum mechanics.
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| There is a family of solutions – in the position basis they are
| |
| | |
| :<math> \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{
| |
| - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right) </math>
| |
| | |
| where ''n'' = 0,1,2..., and the functions ''H<sub>n</sub>'' are the [[Hermite polynomials]].
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| ===Three-dimensional examples===
| |
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| The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the [[gradient]] operator.
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| The Hamiltonian for one particle in three dimensions is:
| |
| | |
| :<math> \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \,, \quad \hat{\mathbf{p}} = -i\hbar \nabla </math>
| |
| | |
| generating the equation:
| |
| | |
| :<math> -\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r}) </math>
| |
| | |
| with stationary state solutions of the form:
| |
| | |
| :<math> \Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-iEt/\hbar} </math>
| |
| | |
| where the position of the particle is '''r'''. Two useful coordinate systems for solving the Schrödinger equation are [[Cartesian coordinates]] so that {{nowrap|1='''r''' = (''x'', ''y'', ''z'')}} and [[spherical polar coordinates]] so that {{nowrap|1='''r''' = (''r'', ''θ'', ''φ'')}}, although other [[orthogonal coordinates]] are useful for solving the equation for systems with certain geometric symmetries.
| |
| | |
| For ''N'' particles in three dimensions, the Hamiltonian is:
| |
| | |
| :<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n </math>
| |
| | |
| where the position of particle ''n'' is '''r'''<sub>''n''</sub> and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle ''n'', the position vector is {{nowrap|1='''r'''''<sub>n</sub>'' = (''x<sub>n</sub>'', ''y<sub>n</sub>'', ''z<sub>n</sub>'')}} while the gradient and [[Laplacian operator]] are respectively:
| |
| | |
| :<math>\nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}\,,\quad \nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}</math>
| |
| | |
| The Schrödinger equation is:
| |
| | |
| :<math> -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N)\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) = E\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) </math>
| |
| | |
| with stationary state solutions:
| |
| | |
| :<math> \Psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N,t) = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N) </math>
| |
| | |
| Again, for non-interacting distinguishable particles the potential is the sum of particle potentials
| |
| | |
| :<math> V(\mathbf{r}_1,\mathbf{r}_2,\cdots \mathbf{r}_N) = \sum_{n=1}^N V(\mathbf{r}_n) </math>
| |
| | |
| and the wavefunction is a product of the particle wavefuntions
| |
| | |
| :<math> \Psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(\mathbf{r}_n) \, . </math>
| |
| | |
| For non-interacting identical particles, the potential is a sum but the wavefunction is a sum over permutations of products. The previous two equations do not apply to interacting particles.
| |
| | |
| Following are examples where exact solutions are known. See the main articles for further details.
| |
| | |
| ====Hydrogen atom====
| |
| | |
| This form of the Schrödinger equation can be applied to the [[hydrogen atom]]:<ref name="Quantum Chemistry 1977"/><ref name="Quanta 1974"/>
| |
| | |
| :<math> E \psi = -\frac{\hbar^2}{2\mu}\nabla^2\psi - \frac{e^2}{4\pi\epsilon_0 r}\psi </math>
| |
| | |
| where ''e'' is the electron charge, '''r''' is the position of the electron ({{nowrap|1=''r'' = {{abs|'''r'''}}}} is the magnitude of the position), the potential term is due to the [[Coulomb's law|coloumb interaction]], wherein ''ε''<sub>0</sub> is the [[electric constant]] (permittivity of free space) and
| |
| | |
| :<math> \mu = \frac{m_em_p}{m_e+m_p} </math>
| |
| | |
| is the 2-body [[reduced mass]] of the hydrogen [[Nucleus (atomic structure)|nucleus]] (just a [[proton]]) of mass ''m''<sub>p</sub> and the electron of mass ''m''<sub>e</sub>. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common centre of mass, and constitute a two-body problem to solve. The motion of the electron is of principle interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.
| |
| | |
| The wavefunction for hydrogen is a function of the electron's coordinates, and in fact can be separated into functions of each coordinate.<ref>Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7</ref> Usually this is done in [[spherical polar coordinates]]:
| |
| | |
| :<math> \psi(r,\theta,\phi) = R(r)Y_\ell^m(\theta, \phi) = R(r)\Theta(\theta)\Phi(\phi)</math>
| |
| | |
| where ''R'' are radial functions and <math>\scriptstyle Y_{\ell}^{m}(\theta, \phi ) \,</math> are [[spherical harmonic]]s of degree ''ℓ'' and order ''m''. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximative methods. The family of solutions are:<ref>{{cite book|author=David Griffiths|title=Introduction to elementary particles|url=http://books.google.com/books?id=w9Dz56myXm8C&pg=PA162|accessdate=27 June 2011|year=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|pages=162–}}</ref>
| |
| | |
| :<math> \psi_{n\ell m}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^{\ell} L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^{m}(\theta, \phi ) </math>
| |
| | |
| where:
| |
| *<math> a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2} </math> is the [[Bohr radius]],
| |
| *<math> L_{n-\ell-1}^{2\ell+1}(\cdots) </math> are the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] of degree {{nowrap|''n'' − ''ℓ'' − 1}}.
| |
| *''n, ℓ, m'' are the [[principal quantum number|principal]], [[azimuthal quantum number|azimuthal]], and [[magnetic quantum number|magnetic]] [[quantum numbers]] respectively: which take the values:
| |
| :<math> \begin{align} n & = 1,2,3 \cdots \\
| |
| \ell & = 0,1,2 \cdots n-1 \\
| |
| m & = -\ell\cdots\ell
| |
| \end{align}</math>
| |
| | |
| NB: [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] are defined differently by different authors—see main article on them and the hydrogen atom.
| |
| | |
| ====Two-electron atoms or ions====
| |
| The equation for any two-electron system, such as the neutral [[helium atom]] (He, ''Z'' = 2), the negative [[hydrogen]] [[ion]] (H<sup>–</sup>, ''Z'' = 1), or the positive [[lithium]] ion (Li<sup>+</sup>, ''Z'' = 3) is:<ref name="Molecules, B.H. Bransden 1983"/>
| |
| | |
| :<math> E\psi = -\hbar^2\left[\frac{1}{2\mu}\left(\nabla_1^2 +\nabla_2^2 \right) + \frac{1}{M}\nabla_1\cdot\nabla_2\right] \psi + \frac{e^2}{4\pi\epsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1}+\frac{1}{r_2} \right) \right] \psi </math>
| |
| | |
| where '''r'''<sub>1</sub> is the position of one electron ({{nowrap|1=''r''<sub>1</sub> = {{abs|'''r'''<sub>1</sub>}}}} is its magnitude), '''r'''<sub>2</sub> is the position of the other electron (''r''<sub>2</sub> = |'''r'''<sub>2</sub>| is the magnitude), ''r''<sub>12</sub> = |'''r'''<sub>12</sub>| is the magnitude of the separation between them given by
| |
| | |
| :<math> |\mathbf{r}_{12}| = |\mathbf{r}_2 - \mathbf{r}_1 | \,\!</math>
| |
| | |
| ''μ'' is again the two-body reduced mass of an electron with respect to the nucleus of mass ''M'', so this time
| |
| | |
| :<math> \mu = \frac{m_e M}{m_e+M} \,\!</math>
| |
| | |
| and ''Z'' is the [[atomic number]] for the element (not a [[quantum number]]).
| |
| | |
| The cross-term of two laplacians
| |
| | |
| :<math>\frac{1}{M}\nabla_1\cdot\nabla_2\,\!</math>
| |
| | |
| is known as the ''mass polarization term'', which arises due to the motion of [[Atomic nucleus|atomic nuclei]]. The wavefunction is a function of the two electron's positions:
| |
| | |
| :<math> \psi = \psi(\mathbf{r}_1,\mathbf{r}_2). </math>
| |
| | |
| There is no closed form solution for this equation.
| |
| | |
| ===Time dependent===
| |
| | |
| This is the equation of motion for the quantum state. In the most general form, it is written:<ref name=Shankar1994/>{{rp|143ff}}
| |
| | |
| :<math>i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi.</math>
| |
| | |
| and the solution, the wavefunction, is a function of all the particle coordinates of the system and time. Following are specific cases.
| |
| | |
| For one particle in one dimension, the Hamiltonian
| |
| | |
| :<math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) \,,\quad \hat{p} = -i\hbar \frac{\partial}{\partial x} </math>
| |
| | |
| generates the equation:
| |
| | |
| :<math> i\hbar\frac{\partial}{\partial t}\Psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi(x,t) + V(x,t)\Psi(x,t) </math>
| |
| | |
| For ''N'' particles in one dimension, the Hamiltonian is:
| |
| | |
| :<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N,t) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n} </math>
| |
| | |
| where the position of particle ''n'' is ''x<sub>n</sub>'', generating the equation:
| |
| | |
| :<math> i\hbar\frac{\partial}{\partial t}\Psi(x_1,x_2\cdots x_N,t) = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi(x_1,x_2\cdots x_N,t) + V(x_1,x_2\cdots x_N,t)\Psi(x_1,x_2\cdots x_N,t) \, .</math>
| |
| | |
| For one particle in three dimensions, the Hamiltonian is:
| |
| | |
| :<math> \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r},t) \,,\quad \hat{\mathbf{p}} = -i\hbar \nabla </math>
| |
| | |
| generating the equation:
| |
| | |
| :<math> i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},t) + V(\mathbf{r},t)\Psi(\mathbf{r},t) </math>
| |
| | |
| For ''N'' particles in three dimensions, the Hamiltonian is:
| |
| | |
| :<math> \hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t)\,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n </math>
| |
| | |
| where the position of particle ''n'' is '''r'''<sub>''n''</sub>, generating the equation:<ref name=Shankar1994/>{{rp|141}}
| |
| :<math> i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t)\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) </math>
| |
| | |
| This last equation is in a very high dimension, so the solutions are not easy to visualize.
| |
| | |
| == Solution methods ==
| |
| | |
| {{multicol begin}}
| |
| General techniques:
| |
| * [[Perturbation theory (quantum mechanics)|Perturbation theory]]
| |
| * The [[variational method]]
| |
| * [[Quantum Monte Carlo]] methods
| |
| * [[Density functional theory]]
| |
| * The [[WKB approximation]] and semi-classical expansion
| |
| {{multicol-break}}
| |
| | |
| Methods for special cases:
| |
| * [[List of quantum-mechanical systems with analytical solutions]]
| |
| * [[Hartree–Fock]] method and [[post Hartree–Fock]] methods
| |
| {{multicol-end}}
| |
| | |
| == Properties ==
| |
| | |
| The Schrödinger equation has the following properties: some are useful, but there are shortcomings. Ultimately, these properties arise from the Hamiltonian used, and solutions to the equation.
| |
| | |
| ===Linearity===
| |
| | |
| {{See also|Linear differential equation}}
| |
| | |
| In the development above, the Schrödinger equation was made to be linear for generality, though this has other implications. If two wave functions ''ψ''<sub>1</sub> and ''ψ''<sub>2</sub> are solutions, then so is any [[linear combination]] of the two:
| |
| | |
| :<math>\displaystyle \psi = a\psi_1 + b \psi_2 </math>
| |
| | |
| where ''a'' and ''b'' are any complex numbers (the sum can be extended for any number of wavefunctions). This property allows [[Quantum superposition|superpositions of quantum states]] to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over all single state solutions achievable. For example, consider a wave function <math>\Psi (x,t)</math> such that the wave function is a product of two functions: one time independent, and one time dependent. If states of definite energy found using the time independent Shrödinger equation are given by <math> \psi_E (x)</math> with amplitude <math>A_n</math> and time dependent phase factor is given by
| |
| | |
| :<math>e^{{-iE_n t}/\hbar}, </math>
| |
| | |
| then a valid general solution is
| |
| | |
| :<math>\displaystyle \Psi(x,t) = \sum\limits_{n} A_n \psi_{E_n}(x) e^{{-iE_n t}/\hbar}. </math>
| |
| | |
| Additionally, the ability to scale solutions allows one to solve for a wave function without normalizing it first. If one has a set of normalized solutions <math>\psi_n</math>, then
| |
| | |
| :<math>\displaystyle \Psi = \sum\limits_{n} A_n \psi_n </math>
| |
| | |
| can be normalized by ensuring that
| |
| | |
| :<math>\displaystyle \sum\limits_{n}|A_n|^2 = 1. </math>
| |
| | |
| This is much more convenient than having to verify that
| |
| | |
| :<math>\displaystyle \int\limits_{-\infty}^{\infty}|\Psi(x)|^2\,dx = \int\limits_{-\infty}^{\infty}\Psi(x)\Psi^{*}(x)\,dx = 1. </math>
| |
| | |
| === Real energy eigenstates ===
| |
| For the time-independent equation, an additional feature of linearity follows: if two wave functions ''ψ''<sub>1</sub> and ''ψ''<sub>2</sub> are solutions to the time-independent equation with the same energy ''E'', then so is any linear combination:
| |
| | |
| :<math> \hat H (a\psi_1 + b \psi_2 ) = a \hat H \psi_1 + b \hat H \psi_2 = E (a \psi_1 + b\psi_2). </math>
| |
| | |
| Two different solutions with the same energy are called ''degenerate''.<ref name="Quantum Mechanics Demystified 2006"/>
| |
| | |
| In an arbitrary potential, if a wave function ''ψ'' solves the time-independent equation, so does its complex conjugate ''ψ''*. By taking linear combinations, the real and imaginary parts of ''ψ'' are each solutions (if there is no degeneracy they can only differ by a factor). Thus, the time-independent eigenvalue problem can be restricted to real-valued wave functions.
| |
| | |
| In the time-dependent equation, complex conjugate waves move in opposite directions. If {{nowrap|''Ψ''(''x'', ''t'')}} is one solution, then so is {{nowrap|''Ψ''(''x'', –''t'')}}. The symmetry of complex conjugation is called [[T-symmetry|time-reversal symmetry]].
| |
| | |
| ===Space and time derivatives===
| |
| [[File:Wavefunction continuity space.svg|250px|"250px"|thumb|Continuity of the wavefunction and its first spatial derivative (in the ''x'' direction, ''y'' and ''z'' coordinates not shown), at some time ''t''.]]
| |
| | |
| The Schrödinger equation is first order in time and second in space, which describes the time evolution of a quantum state (meaning it determines the future amplitude from the present).
| |
| | |
| Explicitly for one particle in 3d Cartesian coordinates – the equation is
| |
| | |
| :<math>i\hbar{\partial \Psi \over \partial t} = - {\hbar^2\over 2m} \left ( {\partial^2 \Psi \over \partial x^2} + {\partial^2 \Psi \over \partial y^2} + {\partial^2 \Psi \over \partial z^2} \right ) + V(x,y,z,t)\Psi.\,\!</math>
| |
| | |
| The first time partial derivative implies the initial value (at {{nowrap|1=''t'' = 0}}) of the wavefunction
| |
| | |
| :<math> \Psi(x,y,z,0) \,\!</math>
| |
| | |
| is an arbitrary constant. Likewise – the second order derivatives with respect to space implies the wavefunction ''and'' its first order spatial derivatives
| |
| | |
| :<math> \begin{align}
| |
| & \Psi(x_b,y_b,z_b,t) \\
| |
| & \frac{\partial}{\partial x}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial y}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial z}\Psi(x_b,y_b,z_b,t)
| |
| \end{align} \,\!</math>
| |
| | |
| are all arbitrary constants at a given set of points, where ''x<sub>b</sub>'', ''y<sub>b</sub>'', ''z<sub>b</sub>'' are a set of points describing boundary ''b'' (derivatives are evaluated at the boundaries). Typically there are one or two boundaries, such as the [[step potential]] and [[particle in a box]] respectively.
| |
| | |
| As the first order derivatives are arbitrary, the wavefunction can be a [[Smooth function|continuously differentiable function]] of space, since at any boundary the gradient of the wavefunction can be matched.
| |
| | |
| On the contrary, wave equations in physics are usually ''second order in time'', notable are the family of classical [[wave equation]]s and the quantum [[Klein–Gordon equation]].
| |
| | |
| === Local conservation of probability ===
| |
| | |
| {{Main|Probability current|Continuity equation}}
| |
| | |
| The Schrödinger equation is consistent with [[conservation of probability|probability conservation]]. Multiplying the Schrödinger equation on the right by the complex conjugate wavefunction, and multiplying the wavefunction to the left of the complex conjugate of the Schrödinger equation, and subtracting, gives the [[Continuity equation#Quantum mechanics|continuity equation]] for probability:<ref name="Quantum Mechanics 2004">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0</ref>
| |
| :<math>{ \partial \over \partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0, </math>
| |
| where
| |
| :<math>\rho=|\Psi|^2=\Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t)\,\!</math>
| |
| is the [[probability density function|probability density]] (probability per unit volume, * denotes [[complex conjugate]]), and
| |
| :<math> \mathbf{j} = {1 \over 2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right)\,\!</math>
| |
| is the [[probability current]] (flow per unit area).
| |
| | |
| Hence predictions from the Schrödinger equation do not violate probability conservation.
| |
| | |
| === Positive energy ===
| |
| | |
| If the potential is bounded from below, meaning there is a minimum value of potential energy, the eigenfunctions of the Schrödinger equation have energy which is also bounded from below. This can be seen most easily by using the [[variational principle]], as follows. (See also below).
| |
| | |
| For any linear operator <math> \hat{A} </math> [[Bounded operator|bounded]] from below, the eigenvector with the smallest eigenvalue is the vector ''ψ'' that minimizes the quantity
| |
| | |
| :<math> \langle \psi |\hat{A}|\psi \rangle </math>
| |
| | |
| over all ''ψ'' which are [[normalizable wave function|normalized]].<ref name="Quantum Mechanics 2004"/> In this way, the smallest eigenvalue is expressed through the [[variational principle]]. For the Schrödinger Hamiltonian <math>\hat{H}</math> bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of
| |
| | |
| :<math>\langle \psi|\hat{H}|\psi\rangle = \int \psi^*(\mathbf{r}) \left[ - \frac{\hbar^2}{2m} \nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r})\right] d^3\mathbf{r} = \int \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 + V(\mathbf{r}) |\psi|^2 \right] d^3\mathbf{r} = \langle \hat{H}\rangle </math>
| |
| | |
| (using [[integration by parts]]). Due to the [[Absolute value#Complex numbers|complex modulus]] of ''ψ'' squared (which is positive definite), the right hand side always greater than the lowest value of ''V''(''x''). In particular, the ground state energy is positive when ''V''(''x'') is everywhere positive.
| |
| | |
| For potentials which are bounded below and are not infinite over a region, there is a ground state which minimizes the integral above. This lowest energy wavefunction is real and positive definite – meaning the wavefunction can increase and decrease, but is positive for all positions. It physically cannot be negative: if it were, smoothing out the bends at the sign change (to minimize the wavefunction) rapidly reduces the gradient contribution to the integral and hence the kinetic energy, while the potential energy changes linearly and less quickly. The kinetic and potential energy are both changing at different rates, so the total energy is not constant, which can't happen (conservation). The solutions are consistent with Schrödinger equation if this wavefunction is positive definite.
| |
| | |
| The lack of sign changes also shows that the ground state is nondegenerate, since if there were two ground states with common energy ''E'', not proportional to each other, there would be a linear combination of the two that would also be a ground state resulting in a zero solution.
| |
| | |
| ===Analytic continuation to diffusion===
| |
| | |
| {{see also|path integral formulation#The Schrödinger equation|label 1=Path integral formulation (The Schrödinger equation)}}
| |
| | |
| The above properties (positive definiteness of energy) allow the [[analytic continuation]] of the Schrödinger equation to be identified as a [[stochastic process]]. This can be interpreted as the [[Huygens–Fresnel principle]] applied to De Broglie waves; the spreading wavefronts are diffusive probability amplitudes.<ref name="Quantum Mechanics 2004"/>
| |
| | |
| For a particle in a [[random walk]] (again for which {{nowrap|1=''V'' = 0}}), the continuation is to let:<ref>http://www.stt.msu.edu/~mcubed/Relativistic.pdf</ref>
| |
| | |
| :<math> \tau = it \,, \!</math>
| |
| | |
| substituting into the time-dependent Schrödinger equation gives:
| |
| | |
| :<math> {\partial \over \partial \tau} \Psi(\mathbf{r},\tau) = \frac{\hbar}{2m} \nabla ^2 \Psi(\mathbf{r},\tau), </math>
| |
| | |
| which has the same form as the [[diffusion equation]], with diffusion coefficient ''ħ''/2''m''.
| |
| | |
| ==Relativistic quantum mechanics==
| |
| | |
| [[Relativistic quantum mechanics]] is obtained where quantum mechanics and [[special relativity]] simultaneously apply. The general form of the Schrödinger equation is still applicable, but the Hamiltonian operators are much less obvious, and more complicated.
| |
| | |
| One wishes to build [[relativistic wave equations]] from the relativistic [[energy–momentum relation]]
| |
| | |
| :<math>E^2 = (pc)^2 + (m_0c^2)^2 \, , </math>
| |
| | |
| instead of classical energy equations. The [[Klein–Gordon equation]] and the [[Dirac equation]] are two such equations. The Klein–Gordon equation was the first such equation to be obtained, even before the non-relativistic one, and applies to massive spinless particles. The Dirac equation arose from taking the "square root" of the Klein–Gordon equation by factorizing the entire relativistic wave operator into a product of two operators – one of these is the operator for the entire Dirac equation.
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| The Dirac equation introduces ''spin matrices'', in the particular case of the Dirac equation they are the [[gamma matrices]] for [[spin-1/2]] particles, and the solutions are 4-component [[spinor field]]s with two components corresponding to the particle and the other two for the antiparticle. In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of position and momentum operators, but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin ''s'', are complex-valued {{nowrap|2(2''s'' + 1)}}-component [[spinor field]]s.
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| ==Quantum field theory==
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| The general equation is also valid and used in [[quantum field theory]], both in relativistic and non-relativistic situations. However, the solution ''ψ'' is no longer interpreted as a "wave", but more like a "[[field (physics)|field]]".
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| ==See also==
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| {{div col}}
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| *[[Fractional Schrödinger equation]]
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| *[[Nonlinear Schrödinger equation]]
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| *[[Quantum carpet]]
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| *[[Quantum revival]]
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| *[[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
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| *[[Schrödinger field]]
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| *[[Schrödinger picture]]
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| *[[Schrödinger's cat]]
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| *[[Theoretical and experimental justification for the Schrödinger equation]]
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| {{div col end}}
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| == Notes ==
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| {{Reflist|colwidth=30em}}
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| == References ==
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| * {{cite book | author= P. A. M. Dirac |author-link=Paul Dirac |title= [[Principles of Quantum Mechanics|The Principles of Quantum Mechanics]] | edition = 4th | publisher= Oxford University Press |year=1958 }}
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| * {{cite book | author= B.H. Bransden and C.J. Joachain |title= Quantum Mechanics | edition = 2nd | publisher= Prentice Hall PTR |year=2000 | isbn = 0-582-35691-1 }}
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| * {{cite book | author = [[David J. Griffiths]] | title = Introduction to Quantum Mechanics | edition = 2nd | year = 2004 | publisher = Benjamin Cummings | isbn = 0-13-124405-1 }}
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| * {{cite book | author = [[Richard Liboff]] | title = Introductory Quantum Mechanics | edition = 4th | year = 2002 | publisher = Addison Wesley | isbn = 0-8053-8714-5 }}
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| * {{cite book | author = David Halliday | title = Fundamentals of Physics | edition = 8th | year = 2007 | publisher = Wiley | isbn = 0-471-15950-6 }}
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| * {{cite book | author = Serway, Moses, and Moyer | title = Modern Physics | edition = 3rd | year = 2004 | publisher = Brooks Cole | isbn = 0-534-49340-8}}
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| * {{cite journal | author = Schrödinger, Erwin |date=December 1926 | title = An Undulatory Theory of the Mechanics of Atoms and Molecules | journal = Phys. Rev. | pages = 1049–1070 | doi = 10.1103/PhysRev.28.1049 | volume = 28 |bibcode = 1926PhRv...28.1049S | issue = 6 }}
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| * {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year=2009 |url=http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=978-0-8218-4660-5 }}
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| ==External links==
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| * {{springer|title=Schrödinger equation|id=p/s083410}}
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| * [http://www.lightandmatter.com/html_books/0sn/ch13/ch13.html Quantum Physics] — textbook with a treatment of the time-independent Schrödinger equation
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| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde108.pdf Linear Schrödinger Equation] at EqWorld: The World of Mathematical Equations.
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| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde1403.pdf Nonlinear Schrödinger Equation] at EqWorld: The World of Mathematical Equations.
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| * [http://www.colorado.edu/UCB/AcademicAffairs/ArtsSciences/physics/TZD/PageProofs1/TAYL07-203-247.I.pdf The Schrödinger Equation in One Dimension] as well as the [http://www.colorado.edu/UCB/AcademicAffairs/ArtsSciences/physics/TZD/PageProofs1/ directory of the book].
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| * [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html All about 3D Schrödinger Equation ]
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| *Mathematical aspects of Schrödinger equations are discussed on the [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page Dispersive PDE Wiki].
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| * [http://www.nanotechnology.hu/online/web-schroedinger/index.html Web-Schrödinger: Interactive solution of the 2D time-dependent and stationary Schrödinger equation ]
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| * [http://behindtheguesses.blogspot.com/2009/06/schrodinger-equation-corrections.html An alternate reasoning behind the Schrödinger Equation ]
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| * Online software-[http://nanohub.org/resources/3847 Periodic Potential Lab] Solves the time-independent Schrödinger equation for arbitrary periodic potentials.
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| *[http://www.felderbooks.com/papers/psi.html What Do You Do With a Wavefunction?]
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| *[http://www.felderbooks.com/papers/quantum.html The Young Double-Slit Experiment]
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| {{DEFAULTSORT:Schrodinger Equation}}
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| [[Category:Wave mechanics]]
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| [[Category:Schrödinger equation]]
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| [[Category:Austrian inventions]]
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