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| {{distinguish|Wave equation}}
| | When it comes to losing fat fast, most dieters, plus people all together, seem to feel that this involves either starving oneself or taking diet pills. Unfortunately, if you've ever tried those methods, we recognize all too perfectly which they don't function. If getting slim was just because convenient as taking a pill!<br><br>The Cabbage Diet. lose weight fast with all the cabbage diet. This diet consists of making a soup with cabbage, that you will eat throughout the 7-day length of the diet. The diet is low inside fat, plus significant in fiber which may help we lose fat swiftly. During this diet you'll turn between eating cabbage soup and fruit with eating greens plus meat on each additional day. This diet will enable we lose fat because it is actually low inside calories. Some persons find it hard to maintain this diet due to the limited food choices it allows.<br><br>Trick a body. A mistake folks often create is striving to learn how to lose weight solely by dieting. This really is tough, because as the calories lessen, the body will try plus keep itself secure inside case the calories dry up altogether in the future - plus it does this by turning the fat storage as much as maximum - making more food than ever be turned into fat. Counteract this by exercising first, that may get the body into the mode of burning fat, not storing it.<br><br>Have breakfast. Skipping breakfast leads to low blood sugar mid morning that results in cravings for sugary foods to pump the blood glucose back up. A vicious cycle results plus continues throughout the day. Your breakfast should comprise of entire grains, a protein source plus fruit. It doesn't have to be fancy. A cut of complete wheat toast spread with a tablespoon of low fat peanut butter and an apple can do really fine. Or we can blend 1/2 cup of nonfat cottage cheese with chopped fresh peaches plus a couple of entire wheat crackers. Another possibility is a scrambled egg served on an English muffin with a glass of orange juice.<br><br>Once we are burning more calories then you'll start to lose [http://safedietplansforwomen.com/how-to-lose-weight-fast how to lose weight fast for women] the weight. And how do you receive you body to burn calories - with exercise plus exercising regularly.<br><br>Vegan diet is nothing but a diet which involves only vegetarian foods and no foods from dead animals. You can even frame a diet by oneself by deciding the amount of calories we want to consume inside a day. Given below is one such sample diet which allows you to consume 1400 calories in a day.<br><br>If I wish To lose 10lbs extremely immediately, I usually add 30 minutes of strolling or bike riding to my daily schedule. That alone might cause you to lose at least 1 pound of fat a week, causing we to get rid of weight even quicker. |
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| [[File:QuantumHarmonicOscillatorAnimation.gif|thumb|300px|right|Some trajectories of a [[harmonic oscillator]] (a ball attached to a [[Hooke's law|spring]]) in [[classical mechanics]] (A–B) and [[quantum mechanics]] (C–H). In quantum mechanics (C–H), the ball has a '''wave function''', which is shown with [[real part]] in blue and [[imaginary part]] in red. The trajectories C, D, E, F, (but not G or H) are examples of [[standing wave]]s, (or "[[stationary state]]s"). Each standing-wave frequency is proportional to a possible [[energy level]] of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]]
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| A '''wave function''' or '''wavefunction''' (also named a state function) in [[quantum mechanics]] describes the [[quantum state]] of a particle and how it behaves. Typically, its values are [[complex number]]s and, for a single particle, it is a [[function (mathematics)|function]] of space and time. The [[Schrödinger equation]] describes how the wave function evolves over time. The wave function behaves qualitatively like other [[wave]]s, like [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]].
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| The most common symbols for a wave function are {{math|''ψ''}} or {{math|Ψ}} (lower-case and capital [[psi (letter)|psi]]).
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| Although values of {{math|''ψ''}} are complex numbers, {{math|{{abs|''ψ''}}<sup>2</sup>}} is [[real number|real]] corresponding by [[Max Born]]'s proposal to the [[probability density function|probability density]] of finding a particle in a given place at a given time, if the particle's position is to be [[measurement in quantum mechanics|measured]]. Louis de Broglie in his later years 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 [[unit of measurement]] for {{math|''ψ''}} depends on the system. For one particle in three dimensions, its units are [length]<sup>−3/2</sup>. These unusual units are required so that an integral of {{math|{{abs|''ψ''}}<sup>2</sup>}} over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different and can be found by [[dimensional analysis]].<ref>{{cite book|pages=1223–1229| author=R.G. Lerner, G.L. Trigg| title=Encyclopaedia of Physics| publisher=VHC Publishers|edition=2nd| year=1991| isbn=0-89573-752-3}}</ref>
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| The wave function is central to quantum mechanics as the most direct way to describe the [[motion (physics)|motion]] of a [[particle]].
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| Although the wavefunction contains information, it is a [[complex number|complex-valued]] quantity; only its relative phase and relative magnitude can be measured. It does not directly tell anything about the magnitudes or directions of measurable observables. An operator extracts this information by acting on the wavefunction {{math|''ψ''}}. For details and examples on how quantum mechanical operators act on the wave function, commutation of operators, and expectation values of operators; see [[Operator (quantum mechanics)|Operator (physics)]].
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| ==Historical background==
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| {{Quantum mechanics|cTopic=Fundamental concepts}}
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| In the 1920s and 1930s, quantum mechanics was developed using [[calculus]] and [[linear algebra]]. Those who used the techniques of calculus included [[Louis-Victor de Broglie|Louis de Broglie]], [[Erwin Schrödinger]], and others, developing "[[wave|wave mechanics]]". Those who applied the methods of linear algebra included [[Werner Heisenberg]], [[Max Born]], and others, developing "matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent.<ref>
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| {{Citation
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| | last = Hanle
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| | first = P.A.
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| | title = Erwin Schrodinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory.
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| | journal = Isis
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| | volume = 68
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| | issue = 4
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| | pages = 606–609
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| | date = December 1977
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| | doi = 10.1086/351880
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| }}
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| </ref> In each case, the wavefunction was at the centre of attention in two forms, giving quantum mechanics its unity.
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| In 1905 Planck postulated the proportionality between the frequency of a photon and its energy, in the [[Planck–Einstein equation]], {{math|''E'' {{=}} ''hf''}}. In 1925, De Broglie published the symmetric relation between [[momentum]] and [[wavelength]], {{math|''p'' {{=}} ''h''/''λ''}}, now called the [[matter wave|De Broglie relation]]. These equations represent [[wave–particle duality]]. In 1926, Schrödinger published the famous wave equation now named after him, indeed the [[Schrödinger equation]], based on [[Classical physics|classical]] [[energy conservation]] using [[operator (physics)|quantum operators]] and the de Broglie relations such that the solutions of the equation are the wavefunctions for the quantum system. Later [[Wolfgang Pauli|Pauli]] invented the [[Pauli equation]] that adds a description of electron's [[spin (physics)|spin]] and [[magnetic dipole]]. However, ''no one'', even Schrödinger and De Broglie, were clear on ''how to interpret it''.<ref>Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7</ref>
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| Around 1924–27, Max Born, Heisenberg, Bohr and others provided the perspective of ''[[probability amplitude]]''.<ref>Sears' and Zemansky's University Physics, Young and Freedman (12th edition), Pearson Ed. & Addison-Wesley Inc., 2008, ISBN 978-0-321-50130-1</ref> This is the ''[[Copenhagen interpretation]]'' of quantum mechanics. There are many other [[interpretations of quantum mechanics]], but this is considered the most important – since quantum ''calculations'' can be understood.
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| In 1927, [[Douglas Hartree|Hartree]] and [[Vladimir Fock|Fock]] made the first step in an attempt to solve the [[Many-body problem|''N''-body]] wave function, and developed the ''self-consistency cycle'': an [[Iteration|iterative]] [[algorithm]] to approximate the solution. Now it is also known as the [[Hartree–Fock method]].<ref name="Quanta 1974">Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1</ref> The [[Slater determinant|Slater]] [[determinant]] and [[permanent]] (of a [[Matrix (mathematics)|matrix]]) was part of the method, provided by [[John C. Slater]].
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| Interestingly, Schrödinger did encounter an equation for which the wave function satisfied [[theory of relativity|relativistic]] energy conservation ''before'' he published the non-relativistic one, but it led to unacceptable consequences; negative [[probability|probabilities]] and negative [[energy|energies]], so he discarded it.<ref>{{cite book
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| | last =McMahon
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| | first =David
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| | authorlink =
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| | title =Quantum Field Theory Demystified
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| | publisher =McGraw Hill Professional
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| | year =2008
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| | isbn =9780071643528
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| }}</ref>{{rp|3}} In 1927, [[Oskar Klein|Klein]], [[Walter Gordon (physicist)|Gordon]] and Fock also found it, but taking a step further: incorporated the [[Electromagnetic force|electromagnetic]] [[interaction]] into it and proved it was [[Lorentz covariance|Lorentz-invariant]]. De Broglie also arrived at exactly the same equation in 1928. This relativistic wave equation is now known most commonly as the [[Klein–Gordon equation]].<ref>Particle Physics (3rd Edition), B.R. Martin, G. Shaw, Manchester Physics Series, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7</ref>
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| In 1927, [[Wolfgang Pauli|Pauli]] phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the [[Pauli equation]]. Pauli found the wavefunction was not a single complex number, but two complex numbers, which correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, [[Paul Dirac|Dirac]] found an equation from the first successful unification of [[special relativity]] and quantum mechanics applied to the [[electron]] – now called the [[Dirac equation]]. He found the wavefunction for this equation could not be a single complex number, but a [[Dirac spinor|four-component ''spinor'']].<ref name="Quanta 1974"/> [[Spin (physics)|Spin]] automatically entered into the properties of the wavefunction. Later other wave equations were developed: see [[relativistic wave equations]] for further information.
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| == Wave functions and function spaces ==
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| [[Functional analysis]] is commonly used to formulate the wavefunction with a necessary mathematical precision; usually they are [[quadratically integrable function]]s (at least locally) because it is compatible with the Hilbert space formalism mentioned below. The set on which their [[function space]] is defined is the [[configuration space]] of the system. In many situations it is an [[Euclidean space]], that implies that wavefunctions are [[functions of several real variables]]. Superficially, this formalism is simple to understand for the following reasons.
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| *If the wavefunction is to change throughout space and time, one would expect the wave function to be a function of the position and time coordinates. It is solved from the [[Schrödinger equation]] (or other [[relativistic wave equations]]), a [[linear partial differential equation]]:
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| ::<math> i\hbar\frac{\partial }{\partial t}\Psi = \hat{H} \Psi </math>
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| *Functions can easily describe [[wave]]-like motion, using [[periodic function]]s, and [[Fourier analysis]] can be readily done.
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| *Functions are easy to produce, visualize, and interpret, because of the pictorial nature of the [[graph of a function]]. One can plot [[curve]]s, [[surface]]s, [[contour line]]s, more generally any [[level set]]s. If the situation is in a high number of dimensions – one can analyze the function in a lower dimensional slice to see the behavior of the function within that confined region.
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| For concreteness and simplicity, in this article, when [[coordinate system|coordinate]]s are needed we use [[Cartesian coordinates]] so that {{math|'''r'''}} is short for {{math|(''x'', ''y'', ''z'')}}, although [[spherical polar coordinates]] and other [[orthogonal coordinates]] are often useful to solve the Schrödinger equation for potentials with certain geometric symmetries, in which case the position and wavefunction is expressed in these coordinates.
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| One does not have to define wavefunctions necessarily on [[real coordinate space|real spaces]]: appropriate function spaces can be defined wherever a [[measure (mathematics)|measure]] can provide integration. [[Operator theory]] and [[linear algebra]], as shown next, can deal with situations where the [[real analysis]] is not applicable.
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| ===Requirements===
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| [[File:Wavefunction continuity space.svg|180px|thumb|Continuity of the wavefunction and its first spatial derivative (in the ''x'' direction, ''y'' and ''z'' coordinates not shown), at some time ''t''.]]
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| The following constraints on the wavefunction are formulated for the calculations and physical interpretation to make sense:<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><ref name="Rae 2008">{{cite book|author = A.I.M Rae|year = 2008| title = Quantum Mechanics|edition=5th|publisher=Taylor & Francis Group|volume=2| isbn = 1-5848-89705|url=http://books.google.co.uk/books?id=YDhHAQAAIAAJ&q=quantum+mechanics+Alastair+Rae+5th+edition&dq=quantum+mechanics+Alastair+Rae+5th+edition&hl=en&sa=X&ei=S0l8Ueq0L4KH0AXw34CwBw&redir_esc=y}}</ref>
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| * It must everywhere be a [[continuous function]], and [[continuously differentiable]].
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| * It must everywhere satisfy the relevant normalization condition, because the particle or system of particles exists somewhere with 100% certainty. For this to be so, the wavefunction must be [[square integrable]].
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| A requirement less restrictive is that the wavefunction must belong to the [[Sobolev space]] ''W''<sup>1,2</sup>. It means that it is differentiable in the sense of [[Distribution (mathematics)|distributions]], and its [[gradient]] is [[square-integrable]]. This relaxation is necessary for potentials that are not functions but are distributions, such as the [[dirac delta function]].
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| If these requirements are not met, it is not possible to interpret the wavefunction as a probability amplitude.<ref name="Atkins 1974">{{cite book|title=Quanta: A Handbook of Concepts|author=P. W. Atkins|year=1974|page=258|isbn=0-19-855494-X}}</ref>
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| == Definitions (spin-0 particles)==
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| === One spin-0 particle in one spatial dimension ===
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| {{multiple image
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| | align = right
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| | direction = vertical
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| | width = 402
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| | footer = The [[Real and imaginary parts|real part]]s of position and momentum wave functions {{math|Ψ(''x'')}} and {{math|Φ(''p'')}}, and corresponding probability densities {{math|{{!}}Ψ(''x''){{!}}<sup>2</sup>}} and {{math|{{!}}Φ(''p''){{!}}<sup>2</sup>}}, for one spin-0 particle in one {{math|''x''}} or {{math|''p''}} dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (''not'' the wavefunction) of finding the particle at position {{math|''x''}} or momentum {{math|''p''}}.
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| | image1 = Quantum mechanics standing wavefunctions.svg
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| | caption1 = [[Standing wave]]s for a [[particle in a box]], examples of [[stationary state]]s.
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| | image2 = Quantum mechanics travelling wavefunctions.svg
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| | caption2 = Travelling waves of a free particle. }}
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| For now, consider the simple case of a single particle, without [[spin (physics)|spin]], in one spatial dimension. (More general cases are discussed below).
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| ==== Position-space wavefunction ====
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| The state of such a particle is completely described by its wave function:
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| :<math>\Psi(x,t)</math>,
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| where {{math|''x''}} is position and {{math|''t''}} is time. This function is [[complex-valued function|complex-valued]], meaning that {{math|Ψ(''x'', ''t'')}} is a complex number.
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| Interpreted as a probability amplitude, if the particle's position is [[measurement in quantum mechanics|measured]], its location is not deterministic, but is described by a [[probability distribution]]. The probability that its position {{math|''x''}} will be in the interval {{math|[''a'', ''b'']}} (meaning {{math|''a'' ≤ ''x'' ≤ ''b''}}) is:
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| :<math>P_{a\le x\le b} (t) = \int\limits_a^b d x\,|\Psi(x,t)|^2 </math>
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| where {{math|''t''}} is the time at which the particle was measured. In other words, {{math|{{!}}Ψ(''x'', ''t''){{!}}<sup>2</sup>}} is the [[probability density function|''probability density'']] that the particle is at {{math|''x''}}, rather than some other location; see [[probability amplitude]] for details. This leads to the '''normalization condition''':
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| :<math>\int\limits_{-\infty}^\infty d x \, |\Psi(x,t)|^2 = 1\,,</math>
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| because if the particle is measured, there is 100% probability that it will be ''somewhere''.
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| The '''[[inner product]]''' of two wave functions {{math|Ψ<sub>1</sub>(''x'', ''t'')}} and {{math|Ψ<sub>2</sub>(''x'', ''t'')}} is useful and important for a number of reasons, and can be defined as the complex number (at time {{math|''t''}}):
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| :<math>\langle \Psi_1 , \Psi_2 \rangle = \int\limits_{-\infty}^\infty d x \, \Psi_1^*(x, t)\Psi_2(x, t) \,,</math>
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| In the [[Copenhagen interpretation]], the modulus squared of this complex number gives a real number, {{math|{{!}}{{langle}}Ψ<sub>1</sub>, Ψ<sub>2</sub>{{rangle}}{{!}}<sup>2</sup>}}, which is interpreted as the probability of the wavefunction {{math|Ψ<sub>2</sub>}} [[wavefunction collapse|"collapsing"]] to the new wavefunction {{math|Ψ<sub>1</sub>}}.
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| Although the inner product of two wavefunctions is a complex number, the inner product of a wavefunction {{math|Ψ}} with itself, {{math|{{langle}}Ψ, Ψ{{rangle}}}}, is ''always'' a positive real number. A wavefunction is normalized if that real number is 1: {{math|{{langle}}Ψ, Ψ{{rangle}} {{=}} 1}}.
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| The number <math>\| \Psi \| = \sqrt{\langle \Psi, \Psi \rangle}</math> is called the [[norm (mathematics)|norm]] of the wavefunction. If the wavefunction {{math|Ψ}} is not normalized, then dividing by its norm will normalize it. Note that the complex modulus <math>| \Psi | = \sqrt{ \Psi^{*} \Psi }</math> and the norm <math>\| \Psi \| = \sqrt{\langle \Psi, \Psi \rangle}</math> are not the same.
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| A set of wavefunctions {{math|Ψ<sub>1</sub>, Ψ<sub>2</sub>, ...}} are [[Orthonormality|orthonormal]] if they are each normalized (inner product each wavefunction with itself is 1) and are all [[orthogonal function|orthogonal]] to each other (inner product of any two different wavefunctions is zero):
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| :<math>\langle \Psi_m , \Psi_n \rangle = \int\limits_{-\infty}^\infty d x \, \Psi_m^*(x, t)\Psi_n(x, t) = \delta_{mn} \,,</math>
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| where {{math|''m''}} and {{math|''n''}} each take values 1, 2, ..., and {{math|''δ<sub>mn</sub>''}} is the [[Kronecker delta]].
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| Since the Schrödinger equation is linear, if any number of wavefunctons {{math|Ψ<sub>''n''</sub>}} for {{math|''n'' {{=}} 1, 2, ...}} are solutions of the equation, then so is their sum, and their [[scalar multiplication|scalar multiples]] by complex numbers {{math|''a<sub>n</sub>''}} (taking scalar multiplication and addition together is known as a [[linear combination]]):
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| :<math> \Psi(x,t) = \sum_n a_n \Psi_n(x,t) = a_1 \Psi_1(x,t) + a_2 \Psi_2(x,t) + \cdots </math>
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| This is the ''[[superposition principle]]''. Note that multiplying a wavefunction by any nonzero but constant complex number {{math|''c''}} (also called a phase factor in this context) does not change any information about the quantum system, because {{math|''c''Ψ}} satisfies exactly the same Schrödinger equation {{math|Ψ}} (the constant {{math|''c''}} cancels).
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| Since linear combinations of wavefunctions obtain more wavefunctions, the set of all wavefunctions forms a complex [[vector space]] over the field of complex numbers (more details are given later).
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| ==== Momentum-space wavefunction ====
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| The particle also has a wave function in [[momentum space]]:
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| :<math>\Phi(p,t)</math>
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| where {{math|''p''}} is the [[momentum]] in one dimension, which can be any value from {{math|−∞}} to {{math|+∞}}, and {{math|''t''}} is time. Analogous to the position case, the inner product of two wave functions {{math|Φ<sub>1</sub>(''p'', ''t'')}} and {{math|Φ<sub>2</sub>(''p'', ''t'')}} can be defined as:
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| :<math>\langle \Phi_1 , \Phi_2 \rangle = \int\limits_{-\infty}^\infty d p \, \Phi_1^*(p, t)\Phi_2(p, t) \,,</math>
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| Interpreted as a probability amplitude, if the particle's momentum is [[measurement in quantum mechanics|measured]], the result is not deterministic, but is described by a probability distribution:
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| :<math>P_{a\le p\le b} (t) = \int\limits_a^b d p \, |\Phi(p,t)|^2 </math>,
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| and the normalization condition is similar:
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| :<math> \langle \Phi , \Phi \rangle = \int\limits_{-\infty}^{\infty} d p \, \left | \Phi \left ( p, t \right ) \right |^2 = 1.</math>
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| ==== Relation between wavefunctions ====
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| The position-space and momentum-space wave functions are [[Fourier transform]]s of each other, therefore both contain the same information, and either one alone is sufficient to calculate any property of the particle. For one dimension:<ref>{{cite book|title=Introduction to Quantum Mechanics|author=Griffiths|page=107|edition=1st}}</ref>
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| :<math>\begin{align} \Phi(p,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty d x \, e^{-ipx/\hbar} \Psi(x,t)\\
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| &\upharpoonleft \downharpoonright\\
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| \Psi(x,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty d p \, e^{ipx/\hbar} \Phi(p,t).
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| \end{align}</math>
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| Sometimes the [[wave-vector]] {{math|''k''}} is used in place of [[momentum]] {{math|''p''}}, since they are related by the [[De Broglie relations|de Broglie relation]]
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| :<math>p = \hbar k,</math>
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| and the equivalent space is referred to as [[Momentum space|{{math|''k''}}-space]]. Again it makes no difference which is used since {{math|''p''}} and {{math|''k''}} are equivalent – up to a constant. In practice, the position-space wavefunction is used much more often than the momentum-space wavefunction.
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| ==== Example of normalization ====
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| A particle is restricted to a 1D region between {{math|''x'' {{=}} 0}} and {{math|''x'' {{=}} ''L''}}; its wave function is:
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| :<math>\begin{align}
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| \Psi (x,t) & = Ae^{i(kx-\omega t)}, & 0 \leq x \leq L \\
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| \Psi (x,t) & = 0, & x < 0, x > L \\
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| \end{align} </math>.
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| To normalize the wave function we need to find the value of the arbitrary constant {{math|''A''}}; solved from
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| :<math> \int\limits_{-\infty}^{\infty} dx \, |\Psi|^2 = 1 . </math>
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| From {{math|Ψ}}, we have {{math|{{!}}Ψ{{!}}<sup>2</sup> {{=}} ''A''<sup>2</sup>}}, so the integral becomes;
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| :<math> \int\limits_{-\infty}^0 dx \cdot 0 + \int\limits_0^L dx \, A^2 + \int\limits_L^\infty dx \cdot 0 = 1 , </math>
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| Solving this equation gives {{math|''A'' {{=}} 1/{{sqrt|''L''}}}}, so the normalized wave function in the box is;
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| :<math> \Psi (x,t) = \frac{1}{\sqrt{L}} e^{i(kx-\omega t)}, \quad 0 \leq x \leq L.</math>
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| === One spin-0 particle in three spatial dimensions ===
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| [[Image:Hydrogen Density Plots.png|thumb|400px|The electron probability density for the first few [[hydrogen atom]] electron [[atomic orbital|orbital]]s shown as cross-sections. These orbitals form an [[orthonormal basis]] for the wave function of the electron. Different orbitals are depicted with different scale.]]
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| The position-space wave function of a single particle in three spatial dimensions is similar to the case of one spatial dimension above:
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| :<math>\Psi(\mathbf{r},t)</math>
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| where {{math|'''r'''}} is the [[position vector]] in three-dimensional space, and {{math|''t''}} is time. As always {{math|Ψ('''r''', ''t'')}} is a [[complex number]].
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| The inner product of two wave functions {{math|Ψ<sub>1</sub>('''r''', ''t'')}} and {{math|Ψ<sub>2</sub>('''r''', ''t'')}} can be defined as the complex number:
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| :<math>\langle \Psi_1 , \Psi_2 \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{r} \, \Psi_1^*(\mathbf{r},t)\Psi_2(\mathbf{r},t) \,,</math>
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| Interpreted as a probability amplitude, if the particle's position is measured at time {{math|''t''}}, the probability that it is in a region {{math|''R''}} is:
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| :<math>P_{\mathbf{r}\in R} (t) = \int\limits_R d^3\mathbf{r} \, \left |\Psi(\mathbf{r},t) \right |^2 </math>
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| (a three-dimensional integral over the region {{math|''R''}}, with differential volume element {{math|''d''<sup>3</sup>'''r'''}}, also written "{{math|''dV''}}" or "{{math|''dx dy dz''}}"). The normalization condition is:
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| :<math>\langle \Psi , \Psi \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{r} \, \left | \Psi(\mathbf{r},t)\right |^2 = 1,</math>
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| where the integrals are taken over all of three-dimensional space.
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| The 3d Fourier transform of the position space wavefunction gives the corresponding momentum space wavefunction<ref>Quantum Mechanics (3rd Edition), Eugen Merzbacher, 1998, John Wiley & Sons, ISBN 0-471-88702-1</ref>
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| :<math>\Phi(\mathbf{p},t) = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r} \, e^{-i \mathbf{r}\cdot \mathbf{p} /\hbar} \Psi(\mathbf{r},t) </math>
| |
| | |
| where {{math|'''p'''}} is the momentum in 3-dimensional space, and {{math|''t''}} is time.
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| | |
| The inner product of two wave functions {{math|Φ<sub>1</sub>('''p''',''t'')}} and {{math|Φ<sub>2</sub>('''p''', ''t'')}} can be defined as the complex number:
| |
| | |
| :<math>\langle \Phi_1 , \Phi_2 \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{p} \, \Phi_1^*(\mathbf{p},t)\Phi_2(\mathbf{p},t) \,,</math>
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| | |
| Interpreted as a probability amplitude, the probability of measuring the momentum vector in a region of momentum space {{math|''M''}} is given by:
| |
| | |
| :<math>P_{\mathbf{p} \in M} (t) = \int_M d^3 \mathbf{p} \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 ,</math>
| |
| | |
| where, analogous to position space, {{math|''d''<sup>3</sup>'''p''' {{=}} ''dp<sub>x</sub> dp<sub>y</sub> dp<sub>z</sub>''}} is a differential 3-momentum volume element in momentum space. The normalization condition is:
| |
| | |
| :<math> \langle \Phi , \Phi \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{p} \, \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 = 1.</math>
| |
| | |
| To get back to the position space wavefunction, we apply the inverse Fourier transform on the momentum space wavefunction:
| |
| | |
| :<math>\Psi(\mathbf{r},t) = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{\mathrm{ all \, space}} d^3\mathbf{p} \, e^{i \mathbf{r}\cdot \mathbf{p} /\hbar} \Phi(\mathbf{p},t). </math>
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| | |
| ===Many spin-0 particles in three spatial dimensions===
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| | |
| [[File:Two particle wavefunction.svg|right|402px|thumb|Travelling waves of two free particles, with two of three dimensions suppressed. Top is position space wavefunction, bottom is momentum space wavefunction, with corresponding probability densities.]]
| |
| | |
| If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that ''one'' wave function describes ''many'' particles is what makes [[quantum entanglement]] and the [[EPR paradox]] possible. The position-space wave function for {{math|''N''}} particles is written:<ref name="Quanta 1974"/>
| |
| :<math>\Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t)</math>
| |
| where {{math|'''r'''<sub>i</sub>}} is the position of the {{math|''i''}}th particle in three-dimensional space, and {{math|''t''}} is time.
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| | |
| The inner product of two wave functions {{math|Ψ<sub>1</sub>('''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub>, ''t'')}} and {{math|Ψ<sub>2</sub>('''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub>, ''t'')}} can be defined as the complex number:
| |
| | |
| :<math>\langle \Psi_1 , \Psi_2 \rangle = \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_N \, \Psi_1^*(\mathbf{r}_1 \cdots \mathbf{r}_N,t)\Psi_2(\mathbf{r}_1 \cdots \mathbf{r}_N,t) \,,</math>
| |
| | |
| (altogether, this is 3''N'' one-dimensional integrals).
| |
| | |
| Interpreted as a probability amplitude, if the particles' positions are all measured simultaneously at time {{math|''t''}}, the probability that particle 1 is in region {{math|''R''<sub>1</sub>}} ''and'' particle 2 is in region {{math|''R''<sub>2</sub>}} and so on is:
| |
| :<math>P_{\mathbf{r}_1\in R_1,\mathbf{r}_2\in R_2 \cdots \mathbf{r}_N\in R_N}(t) = \int\limits_{R_1} d ^3\mathbf{r}_1 \int\limits_{R_2} d ^3\mathbf{r}_2\cdots \int\limits_{R_N} d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2</math>
| |
| The normalization condition is:
| |
| :<math>\langle \Psi , \Psi \rangle = \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2 = 1</math>
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| | |
| == Definitions (particles with spin) ==
| |
| | |
| === One particle with spin in three dimensions ===
| |
| | |
| For a particle with [[spin (physics)|spin]], the wave function can be written in "position–spin space" as:
| |
| :<math>\Psi(\mathbf{r},s_z,t)</math>
| |
| where {{math|'''r'''}} is a position in three-dimensional space, {{math|''t''}} is time, and {{math|''s''<sub>z</sub>}} is the [[spin (physics)|spin projection quantum number]] along the {{math|''z''}} axis. (The {{math|''z''}} axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The {{math|''s<sub>z</sub>''}} parameter, unlike {{math|'''r'''}} and {{math|''t''}}, is a ''discrete variable''. For example, for a [[spin-1/2]] particle, {{math|''s''<sub>z</sub>}} can only be {{math|+1/2}} or {{math|−1/2}}, and not any other value. (In general, for spin {{math|''s''}}, {{math|''s<sub>z</sub>''}} can be {{math|''s'', ''s'' − 1, ... , −''s''}}.)
| |
| | |
| The inner product of two wave functions {{math|Ψ<sub>1</sub>('''r'''<sub>1</sub>, ''s''<sub>''z''</sub>, ''t'')}} and {{math|Ψ<sub>2</sub>('''r'''<sub>1</sub>, ''s''<sub>''z''</sub>, ''t'')}} can be defined as the complex number:
| |
| | |
| :<math> \langle \Psi_1 , \Psi_2 \rangle = \sum_{\mathrm{all\, }s_z} \int\limits_{\mathrm{ all \, space}} \, d^3\mathbf{r} \Psi^{*}_1(\mathbf{r},t,s_z)\Psi_2(\mathbf{r},t,s_z) </math>
| |
| | |
| Interpreted as a probability amplitude, if the particle's position and spin is measured simultaneously at time {{math|''t''}}, the probability that its position is in {{math|''R''<sub>1</sub>}} ''and'' its spin projection quantum number is a certain value {{math|''s<sub>z</sub>'' {{=}} ''m''}} is:
| |
| :<math>P_{\mathbf{r}\in R,s_z=m} (t) = \int\limits_{R} \, d ^3\mathbf{r} |\Psi(\mathbf{r},t,m)|^2</math>
| |
| The normalization condition is:
| |
| :<math>\langle \Psi , \Psi \rangle = \sum_{\mathrm{all\, }s_z} \int\limits_{\mathrm{ all \, space}} \, d^3\mathbf{r} |\Psi(\mathbf{r},t,s_z)|^2 = 1</math>.
| |
| Since the spin quantum number has discrete values, it must be written as a sum rather than an integral, taken over all possible values.
| |
| | |
| It is convenient to write the wavefunction as a column vector, in which there are as many entries in the column vector as there are allowed values of {{math|''s<sub>z</sub>''}}, and the entries are indexed by the spin quantum number:<ref name="Abers p 115">{{cite book|title=Quantum Mechanics|author=E. Abers|publisher=Pearson Ed., Addison Wesley, Prentice Hall Inc|year=2004|page=115|isbn=978-0-13-146100-0}}</ref>
| |
| | |
| :<math>\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi_s(\mathbf{r},t) \\ \Psi_{s-1}(\mathbf{r},t) \\ \vdots \\ \Psi_{-(s-1)}(\mathbf{r},t) \\ \Psi_{-s}(\mathbf{r},t) \\ \end{bmatrix}</math>
| |
| | |
| and the normalization condition is equivalent to:
| |
| | |
| :<math> \int\limits_{\mathrm{ all \, space}} \, d ^3\mathbf{r} \Psi^\dagger (\mathbf{r},t)\Psi(\mathbf{r},t) = 1</math>.
| |
| | |
| where the dagger denotes the [[Hermitian adjoint|Hermitian conjugate]] ([[complex conjugate]] [[transpose]] of the [[column vector]] into a [[row vector]]).
| |
| | |
| === Many particles with spin in three dimensions ===
| |
| | |
| Likewise, the wavefunction for ''N'' particles each with spin is:
| |
| | |
| :<math>\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t)</math>
| |
| | |
| The inner product of two wave functions {{math|Ψ<sub>1</sub>('''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub>, ''s''<sub>''z'' 1</sub>, ''s''<sub>''z'' 2</sub>, ..., ''s''<sub>''z N''</sub>, ''t'')}} and {{math|Ψ<sub>2</sub>('''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub>, ''s''<sub>''z'' 1</sub>, ''s''<sub>''z'' 2</sub>, ..., ''s''<sub>''z N''</sub>, ''t'')}} can be defined as the complex number:
| |
| | |
| :<math> \langle \Psi_1 , \Psi_2 \rangle = \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3 \mathbf{r}_N \Psi^{*}_1 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right )\Psi_2 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) </math>
| |
| | |
| Now there are {{math|3''N''}} one-dimensional integrals followed by {{math|''N''}} sums.
| |
| | |
| The probability that particle 1 is in region {{math|''R''<sub>1</sub>}} with spin {{math|''s''<sub>''z''1</sub> {{=}} ''m''<sub>1</sub>}} ''and'' particle 2 is in region {{math|''R''<sub>2</sub>}} with spin {{math|''s''<sub>''z''2</sub> {{=}} ''m''<sub>2</sub>}} etc. reads:
| |
| | |
| :<math>P_{\mathbf{r}_1\in R_1,s_{z\,1} = m_1, \ldots, \mathbf{r}_N\in R_N,s_{z\,N} = m_N} (t) = \int\limits_{R_1} d ^3\mathbf{r}_1 \int\limits_{R_2} d ^3\mathbf{r}_2\cdots \int\limits_{R_N} d ^3\mathbf{r}_N \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,m_1\cdots m_N,t \right ) \right |^2</math>
| |
| | |
| The normalization condition is:
| |
| | |
| :<math> \langle \Psi , \Psi \rangle = \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3 \mathbf{r}_N \left | \Psi \left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 = 1</math>
| |
| | |
| == Distinguishable and identical particles ==
| |
| | |
| {{further|Identical particles#Wavefunction representation}}
| |
| | |
| In quantum mechanics there is a fundamental distinction between [[identical particles]] and distinguishable particles. For example, any two electrons are fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it.<ref>Griffiths, p. 179 of the first edition</ref> This translates to a requirement on the wavefunction: For example, if particles 1 and 2 are indistinguishable, then:
| |
| | |
| :<math>\Psi \left ( \mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \left ( -1 \right )^{2s} \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )</math>
| |
| | |
| where {{math|''s''}} is the [[spin quantum number]] of the particle: [[integer]] for [[bosons]] ({{math|''s'' {{=}} 1, 2, 3, ...}}) and half-integer for [[fermions]] ({{math|''s'' {{=}} 1/2, 3/2, ...}}).
| |
| | |
| The wavefunction is said to be ''symmetric'' (no sign change) under boson interchange and ''antisymmetric'' (sign changes) under fermion interchange. The latter feature of the wavefunction leads to the [[Pauli exclusion principle|Pauli principle]]. Generally, bosonic and fermionic symmetry requirements are the manifestation of [[particle statistics]] and are present in other quantum state formalisms.
| |
| | |
| For {{math|''N''}} interacting ''distinguishable'' particles (all the particles are different, no two are [[identical particles|identical]]), which do not interact mutually and move independently in a ''time-independent'' potential, the (spatial part of the) wavefunction can be separated into a product of separate wavefunctions for each particle:<ref>{{cite book|author=N. Zettili|title=Quantum Mechanics: Concepts and Applications|edition=2nd|page=459|isbn=978-0-470-02679-3}}</ref>
| |
| | |
| :<math>\Psi (\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N) = \prod_{i=1}^N\psi(\mathbf{r}_i) = \psi(\mathbf{r}_1)\psi(\mathbf{r}_2)\cdots\psi(\mathbf{r}_N).</math>
| |
| | |
| This [[separation of variables]] is a simple method for solving partial differential equations like the Schrödinger equation. If the potential is ''time-dependent'', then the wavefunction cannot be separated into the separate wavefunctions of the particles.
| |
| | |
| For non-interacting distinguishable particles in a time-independent potential, the spatial part of the wavefunction is the product of separate wavefunctions for each particle:
| |
| | |
| :<math>\Psi (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}, s_{z\,2} \cdots s_{z\,N} ) = \prod_{i=1}^N\psi(\mathbf{r}_i,s_{z\,i}) = \psi(\mathbf{r}_1,s_{z\,1})\psi(\mathbf{r}_2,s_{z\,2})\cdots\psi(\mathbf{r}_N,s_{z\,N}).</math>
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| | |
| == Units of the wavefunction ==
| |
| | |
| Even though wavefunctions are complex numbers, both the real and imaginary parts each have the same units (the [[imaginary unit]] {{math|''i''}} is a number without unit). The units of {{math|''ψ''}} depend on the number of particles the wavefunction describes, and the number of spatial or momentum dimensions of the system. In general, for ''N'' particles with positions {{math|'''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub>}} in {{math|''n''}} spatial dimensions, the normalization conditions require {{math|''ψ''}} to have units of [length]<sup>−''Nn''/2</sup>. The square root of the length unit is removed when one finds {{math|{{abs|''ψ''}}<sup>2</sup>}}, which has units of [length]<sup>−''Nn''</sup>.
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| | |
| In momentum space, length is replaced by momentum, and the units are [momentum]<sup>−''Nn''/2</sup>.
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| | |
| These results are true for particles with or without spin, since for particles with spin, the summations are over dimensionless spin quantum numbers.
| |
| | |
| == Wave functions as elements of an abstract vector space ==
| |
| | |
| {{main|Quantum state}}
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| | |
| The set of all possible wave functions (at any given time) forms an abstract mathematical [[vector space]]. Specifically, the ''entire'' wave function is treated as a ''single'' abstract vector:
| |
| :<math>\Psi(\mathbf{r}) \leftrightarrow |\Psi\rangle </math>
| |
| where {{math|{{ket|Ψ}}}} is a "ket" (a vector) written in [[bra–ket notation]]. As always, the state vector for the system is solved from the [[Schrödinger equation]] (or other [[Dynamical pictures (quantum mechanics)|dynamical pictures of quantum mechanics]]):
| |
| | |
| :<math> i\hbar\frac{d}{dt}|\Psi\rangle = \hat{H} |\Psi\rangle</math>
| |
| | |
| This vector space is infinite-[[Dimension (vector space)|dimensional]], because there is no finite set of functions which can be added together in various combinations to create every possible function. It is a [[Hilbert space]], for the following reasons.
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| | |
| *The statement that "wave functions form an abstract vector space" means that it is possible [[scalar multiplication|multiply]] wave functions by complex numbers and add together different wave functions in a [[quantum superposition|coherent superposition]]. If {{math|{{ket|''ψ''}}}} and {{math|{{ket|''ϕ''}}}} are two states in the Hilbert space, and {{math|''a''}} and {{math|''b''}} are two complex numbers, then the [[linear combination]]
| |
| | |
| ::<math>|\Psi \rangle = a|\psi\rangle + b|\phi\rangle</math>
| |
| | |
| :(subject to normalization, see below) is also in the Hilbert space. The [[dual vector]]s are denoted as "bras", {{math|{{bra|Ψ}}}}, which do not live in the same space as {{math|{{ket|Ψ}}}}, but instead the [[dual space]]:
| |
| | |
| ::<math>\langle \Psi | = a^{*} \langle \psi | + b^{*} \langle \phi | </math>
| |
| | |
| :where * denotes [[complex conjugate]].
| |
| | |
| *The inner product of wave functions can be defined.
| |
| | |
| See the [[quantum state]] article for more explanation of the Hilbert space formalism and its consequences to quantum physics.
| |
| | |
| There are several advantages to understanding wave functions as elements of an abstract vector space:
| |
| *All the powerful tools of [[linear algebra]] can be used to manipulate and understand wave functions. For example:
| |
| **Linear algebra explains how a vector space can be given a [[Basis (linear algebra)|basis]], and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
| |
| **[[Bra–ket notation]] can be used to manipulate wave functions.
| |
| *The idea that [[quantum state]]s are vectors in an abstract vector space (technically, a complex [[projective space|projective]] [[Hilbert space]]) is completely general in all aspects of quantum mechanics and [[quantum field theory]], whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.
| |
| | |
| Following is a breakdown of the [[bra–ket notation|bra–ket]] formalism. Kets are analogous to the more elementary [[Euclidean vector]]s, although the components are complex-valued.
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| | |
| === Discrete and continuous bases ===
| |
| | |
| {{multiple image
| |
| | left
| |
| | footer = Components of complex vectors plotted against index number; discrete {{math|''k''}} and continuous {{math|''x''}}. Two probability amplitudes out of infinitely many are highlighted.
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| | width1 = 225
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| | image1 = Discrete complex vector components.svg
| |
| | caption1 = Discrete components {{math|''A''<sub>''k''</sub>}} of a complex vector {{math|{{ket|''A''}} {{=}} ∑<sub>''k''</sub> ''A''<sub>''k''</sub>{{ket|''e<sub>k</sub>''}}}}, which belongs to a ''countably infinite''-dimensional Hilbert space; there are countably infinitely many {{math|''k''}} values and basis vectors {{math|{{ket|''e<sub>k</sub>''}}}}.
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| | width2 = 230
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| | image2 = Continuous complex vector components.svg
| |
| | caption2 = Continuous components {{math|''ψ''(''x'')}} of a complex vector {{math|{{ket|''ψ''}} {{=}} ∫''dx'' ''ψ''(''x''){{ket|''ψ''}}}}, which belongs to an ''uncountably infinite''-dimensional [[Hilbert space]]; there are uncountably infinitely many ''x'' values and basis vectors {{math|{{ket|''x''}}}}.
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| }}
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| | |
| A Hilbert space with a discrete basis {{math|{{ket|''ε<sub>i</sub>''}}}} for {{math|''i'' {{=}} 1, 2...''n''}} is [[orthonormal]] if the inner product of all pairs of basis kets are given by the [[Kronecker delta]]:
| |
| | |
| :<math> \langle \varepsilon_i | \varepsilon_j \rangle = \delta_{ij}\,.</math>
| |
| | |
| Orthonormal bases are convenient to work with because the inner product of two vectors have simple expressions. A wave function {{math|{{ket|''ψ''}}}} expressed in this discrete basis of the Hilbert space, and the corresponding bra in the dual space, are respectively given by:
| |
| | |
| :<math> | \psi \rangle
| |
| = \sum_{i = 1}^n c_i | \varepsilon_i \rangle
| |
| = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} \,,\quad \langle \psi | = | \psi \rangle^{*} = \sum_{i = 1}^n c^{*}_i \langle \varepsilon_i | = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} \end{bmatrix} \,,</math>
| |
| | |
| where the complex numbers {{math|''c<sub>i</sub>'' {{=}} {{bra-ket|''ε<sub>i</sub>''|''ψ''}}}} are the components of the vector. The [[column vector]] is a useful representation in terms of matrices. The entire vector {{math|{{ket|''ψ''}}}} is independent of the basis, but the components depend on the basis. If a [[change of basis]] is made, the components of the vector must also change to compensate.
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| | |
| A Hilbert space with a continuous basis {{math|{ {{ket|''ε''}} }}} is orthonormal if the inner product of all pairs of basis kets are given by the [[Dirac delta function]]:
| |
| | |
| :<math> \langle \varepsilon | \varepsilon' \rangle = \delta (\varepsilon-\varepsilon')\,.</math>
| |
| | |
| As with the discrete bases, a symbol {{math|''ε''}} is used in the basis states, two common notations are {{math|{{ket|''ε''}}}} and sometimes {{math|{{ket|''ψ<sub>ε</sub>''}}}}. A particular basis ket may be subscripted {{math|{{ket|''ε''<sub>0</sub>}} ≡ {{ket|''ψ''<sub>''ε''<sub>0</sub></sub>}}}} or primed {{math|{{ket|''ε''′}} ≡ {{ket|''ψ''<sub>''ε''′</sub>}}}}.
| |
| | |
| While discrete basis vectors are summed over a discrete index, continuous basis vectors are integrated over a continuous index (a variable of a function). In what follows, all integrals are with respect to the [[real number|real]]-valued basis variable {{math|''ε''}} (not complex-valued), over the required range. Usually this is just the [[real line]] or [[subset]]s of it. The state {{math|{{ket|''ψ''}}}} in the continuous basis of the Hilbert space, with the corresponding bra in the dual space, are respectively given by:<ref>{{cite book|title=Quantum Mechanics|author=D. McMahon|edition=1st|series=Demystified|isbn=0-07-145546-9|page=196}}</ref><ref name="Schaum QM">{{cite book|title=Quantum Mechanics|author=Y. Peleg, R. Pnini, E. Zaarur, E. Hecht|series=Schaum's Outlines|isbn=978-0-07-162358-2|edition=2nd|page=64-65}}</ref>
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| | |
| :<math> | \psi \rangle = \int d \varepsilon | \varepsilon \rangle \psi(\varepsilon) \,,\quad \langle \psi | = \int d \varepsilon \langle \varepsilon | {\psi(\varepsilon)}^{*} \,,</math>
| |
| where the components are the complex-valued functions {{math|''ψ''(''ε'') {{=}} {{bra-ket|''ε''|''ψ''}}}} of a real variable {{math|''ε''}}.
| |
| | |
| === Completeness conditions ===
| |
| | |
| The '''[[completeness condition]]s''' (also called '''closure relations''') are
| |
| | |
| :<math> \sum_{i=1}^n | \varepsilon_i \rangle \langle \varepsilon_i | = 1 \,,\quad \int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | = 1 \,. </math>
| |
| | |
| for the discrete and continuous orthonormal bases, respectively. An orthonormal set of kets form bases if and only if they satisfy these relations.<ref name="Schaum QM"/> In each case, the equality to unity means this is an [[identity operator]]; its action on any state leaves it unchanged. Multiplying any state on the right of these gives the representation of the state {{math|{{ket|''ψ''}}}} in the basis. The inner product of a first state {{math|{{ket|''χ''}}}} with a second {{math|{{ket|''ψ''}}}} can also be obtained by multiplying {{math|{{bra|''χ''}}}} on the left and {{math|{{ket|''ψ''}}}} on the right of the relevant completeness condition.
| |
| | |
| === Inner product ===
| |
| | |
| Physically, the nature of the inner product is dependent on the basis in use, because the basis is chosen to reflect the quantum state of the system.
| |
| | |
| If {{math|{{ket|''ψ''}}}} is a state in the above basis with components {{math|''c''<sub>1</sub>, ''c''<sub>2</sub>, ..., ''c''<sub>''n''</sub>}} and {{math|{{ket|''χ''}}}} is another state in the same basis with components {{math|''z''<sub>1</sub>, ''z''<sub>2</sub>, ..., ''z''<sub>''n''</sub>}}, the inner product is the complex number:
| |
| | |
| :<math> \langle \chi | \psi \rangle = \left(\sum_i z^*_i \langle\varepsilon_i | \right)\left(\sum_j c_j |\varepsilon_j\rangle \right) =\sum_{ij} z^*_i c_j \delta_{ij} =\sum_i z^*_i c_i \,.</math>
| |
| | |
| If {{math|{{ket|''ψ''}}}} is a state in the above continuous basis with components {{math|''ψ''(''ε'')}}, and {{math|{{ket|''χ''}}}} is another state in the same basis with components {{math|''χ''(''ε''′)}}, the inner product is the complex number:
| |
| | |
| :<math> \langle \chi | \psi \rangle = \left( \int d\varepsilon' {\chi(\varepsilon')}^{*} \langle \varepsilon' | \right)\left(\int d\varepsilon \psi(\varepsilon) |\varepsilon \rangle \right) = \int d\varepsilon' \int d\varepsilon {\chi(\varepsilon')}^{*} \psi(\varepsilon) \delta (\varepsilon' - \varepsilon ) = \int d\varepsilon {\chi(\varepsilon)}^{*} \psi(\varepsilon) \,.</math>
| |
| | |
| where the integrals are taken over all {{math|''ε''}} and {{math|''ε''′}}.
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| === Normalization ===
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| | |
| The square of the '''[[Norm (mathematics)|norm]] (magnitude)''' of the state vector {{math|{{ket|''ψ''}}}} is given by the inner product of {{math|{{ket|''ψ''}}}} with itself, a real number:
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| :<math> \|\psi\|^2 = \langle \psi | \psi \rangle = \sum_{j=1}^n | c_j |^2 \,,\quad \|\psi\|^2 = \langle \psi | \psi \rangle = \int d \varepsilon \, | \psi(\varepsilon) |^2 </math>
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| for the discrete and continuous bases, respectively. Each say the projection of a complex probability amplitude onto itself is real. If {{math|{{ket|''ψ''}}}} is normalized, these expressions would be unity. If the state is not normalized, then dividing by its magnitude normalizes the state to:
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| :<math> | \psi_N \rangle = \frac{1}{\| \psi \|} | \psi \rangle </math>
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| | |
| === Normalized components and probabilities ===
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| For the discrete basis, projecting the normalized state {{math|{{ket|''ψ<sub>N</sub>''}}}} onto a particular state the system may collapse to, {{math|{{ket|''ε<sub>q</sub>''}}}}, gives the complex number;
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| :<math> \langle \varepsilon_q | \psi_N \rangle = \langle \varepsilon_q | \frac{1}{\|\psi\|} \left ( \sum_{i = 1}^n c_i | \varepsilon_i \rangle \right ) = \frac{c_q}{\|\psi\|} \,,</math>
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| | |
| so the modulus squared of this gives a real number;
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| :<math> P(\varepsilon_q) = \left| \langle \varepsilon_q | \psi_N \rangle \right|^2 = \frac{\left|c_q\right|^2}{\|\psi\|^2} \, , </math>
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| In the [[Copenhagen interpretation]], this is the probability of state {{math|{{ket|''ε<sub>q</sub>''}}}} occurring.
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| In the continuous basis, the projection of the normalized state onto some particular basis {{math|{{ket|''ε''′}}}} is a complex-valued function;
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| :<math> \langle \varepsilon' | \psi_N \rangle = \langle \varepsilon' | \left( \frac{1}{\|\psi\|}\int d \varepsilon | \varepsilon \rangle \psi(\varepsilon) \right) = \frac{1}{\|\psi\|}\int d \varepsilon \langle \varepsilon' | \varepsilon \rangle \psi(\varepsilon) = \frac{1}{\|\psi\|}\int d \varepsilon \delta( \varepsilon' - \varepsilon ) \psi(\varepsilon) = \frac{\psi(\varepsilon')}{\|\psi\|} \,,</math>
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| | |
| so the squared modulus is a real-valued function
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| :<math> \rho(\varepsilon') = \left| \langle \varepsilon' | \psi_N \rangle \right|^2 = \frac{\left|\psi(\varepsilon')\right|^2}{\|\psi\|^2} </math>
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| In the [[Copenhagen interpretation]], this function is the ''[[probability density function]]'' of measuring the observable {{math|''ε''′}}, so integrating this with respect to {{math|''ε''′}} between {{math|''a'' ≤ ''ε''′ ≤ ''b''}} gives:
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| :<math> P_{a \leq \varepsilon \leq b} = \frac{1}{\|\psi\|^2}\int_a^b d\varepsilon' | \psi(\varepsilon') |^2 = \frac{1}{\|\psi\|^2}\int_a^b d\varepsilon' | \langle \varepsilon' | \psi \rangle |^2 \,, </math>
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| the probability of finding the system with {{math|''ε''′}} between {{math|''ε''′ {{=}} ''a''}} and {{math|''ε''′ {{=}} ''b''}}.
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| === Wave function collapse ===
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| | |
| The physical meaning of the components of {{math|{{ket|''ψ''}}}} is given by the ''wave function [[collapse postulate]] also known as [[Wave function collapse]]''. If the observable(s) {{math|''ε''}} (momentum and/or spin, position and/or spin, etc.) corresponding to states {{math|{{ket|''ε<sub>i</sub>''}}}} has distinct and definite values, {{math|''λ<sub>i</sub>''}}, and a measurement of that variable is performed on a system in the state {{math|{{ket|''ψ''}}}} then the probability of measuring {{math|''λ<sub>i</sub>''}} is {{math|{{abs|{{bra-ket|''ε<sub>i</sub>''|''ψ''}}}}<sup>2</sup>}}. If the measurement yields {{math|''λ<sub>i</sub>''}}, the system "collapses" to the state {{math|{{ket|''ε<sub>i</sub>''}}}}, irreversibly and instantaneously.
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| === Time dependence ===
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| In the [[Schrödinger picture]], the states evolve in time, so the time dependence is placed in {{math|{{ket|''ψ''}}}} according to:<ref>{{cite book|title=Quantum mechanics|edition=2nd|year=2010|publisher=McGraw Hill|author=Y. Peleg, R. Pnini, E. Zaarur, E. Hecht|series=Schaum's outlines|pages=68–69|isbn=978-0-07-162358-2}}</ref>
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| :<math>|\psi(t)\rangle = \sum_i \, | \varepsilon_i \rangle \langle \varepsilon_i | \psi(t)\rangle = \sum_i c_i(t) | \varepsilon \rangle </math>
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| for discrete bases, or
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| :<math>|\psi(t)\rangle = \int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | \psi(t)\rangle = \int d\varepsilon \, \psi(\varepsilon,t) | \varepsilon \rangle </math>
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| | |
| for continuous bases. However, in the [[Heisenberg picture]] the states {{math|{{ket|''ψ''}}}} are constant in time and time dependence is placed in the Heisenberg operators, so {{math|{{ket|''ψ''}}}} is not written as {{math|{{ket|''ψ''(''t'')}}}}.
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| == Position representations (spinless particles) ==
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| === State space for one spin-0 particle in 1d ===
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| For a spinless particle in one spatial dimension (the ''x''-axis or real line), the state {{math|{{ket|''ψ''}}}} can be expanded in terms of a continuum of states; {{math|{{ket|''x''}}}}, also written {{math|{{ket|''ψ<sub>x</sub>''}}}}, corresponding to the set of all position coordinates {{math|''x''}}.
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| | |
| If the particle is confined to a region {{math|''R''}} (a subset of the ''x''-axis), the state is:
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| :<math> | \psi \rangle = \int\limits_R d x \, | x \rangle \langle x | \psi \rangle = \int\limits_R d x \, \psi(x) | x \rangle </math>
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| | |
| leading to the closure relation
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| | |
| :<math> 1 = \int\limits_R d x \, | x \rangle \langle x | </math>
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| | |
| and the inner product as stated at the beginning of this article (in that case {{math|''R'' {{=}} (−∞, ∞)}}):
| |
| | |
| :<math> \langle \chi | \psi \rangle = \int\limits_R d x \, \langle \chi | x \rangle \langle x | \psi \rangle = \int\limits_R d x \, \chi(x)^{*} \psi(x) \,.</math>.
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| | |
| The "wavefunction" described previously is simply a component of the complex state vector. Projecting {{math|{{ket|''ψ''}}}} onto a particular position state {{math|{{ket|''x''<sub>0</sub>}}}}, where {{math|''x''<sub>0</sub>}} is in {{math|''R''}}:
| |
| | |
| :<math> \langle x_0 | \psi \rangle = \int\limits_R d x \, \langle x_0 | x \rangle \psi(x) = \int\limits_R d x \, \delta( x_0 - x ) \psi(x) = \psi(x_0) \,.</math>
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| | |
| === State space for one spin-0 particle in 3d ===
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| | |
| In three dimensions, {{math|{{ket|''ψ''}}}} can be expanded in terms of a continuum of states with definite position, {{math|{{ket|'''r'''}}}}, also written {{math|{{ket|''x'', ''y'', ''z''}}}} or {{math|{{ket|''ψ''<sub>'''r'''</sub>}}}}, corresponding to each {{math|'''r''' {{=}} (''x'', ''y'', ''z'')}}.
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| | |
| If the particle is confined to a region {{math|''R''}} (a subset of 3d space), the state is;
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| | |
| :<math> | \psi \rangle = \int\limits_R d^3\mathbf{r} \, | \mathbf{r} \rangle \langle \mathbf{r} | \psi\rangle = \int\limits_R d^3\mathbf{r} \, \psi(\mathbf{r}) | \mathbf{r} \rangle </math>
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| | |
| The closure relation is
| |
| | |
| :<math> 1 = \int\limits_R d^3\mathbf{r} \, | \mathbf{r} \rangle \langle \mathbf{r} | </math>
| |
| | |
| leading to the inner product of {{math|{{ket|''ψ''}}}} with itself leads to the normalization conditions in the three-dimensional definitions above:
| |
| | |
| :<math> \langle \chi | \psi \rangle = \int\limits_R d^3\mathbf{r} \, \langle \chi | \mathbf{r} \rangle \langle \mathbf{r} | \psi \rangle = \int\limits_R d^3\mathbf{r} \, \chi(\mathbf{r})^{*} \psi(\mathbf{r}) </math>.
| |
| | |
| Projecting {{math|{{ket|''ψ''}}}} onto a particular position state {{math|{{ket|'''r'''<sub>0</sub>}}}}, where {{math|'''r'''<sub>0</sub>}} is in {{math|''R''}}:
| |
| | |
| :<math> \langle \mathbf{r}_0 | \psi \rangle = \int\limits_R d^3 \mathbf{r} \, \langle \mathbf{r}_0 | \mathbf{r} \rangle \psi(\mathbf{r}) = \int\limits_R d^3 \mathbf{r} \, \delta( \mathbf{r}_0 - \mathbf{r} ) \psi(\mathbf{r}) = \psi(\mathbf{r}_0) </math>
| |
| | |
| === State space for many spin-0 particles in 3d ===
| |
| | |
| In three dimensions, {{math|{{ket|''ψ''}}}} can be expanded in terms of a continuum of states for all the particles each with definite position, {{math|{{ket|'''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub>}}}}, corresponding to each {{math|'''r'''<sub>''j''</sub> {{=}} (''x''<sub>''j''</sub>, ''y''<sub>''j''</sub>, ''z''<sub>''j''</sub>)}}.
| |
| | |
| If particle 1 is in region {{math|''R''<sub>1</sub>}}, particle 2 is in region {{math|''R''<sub>2</sub>}}, and so on, the state in this position representation is:
| |
| | |
| :<math> \begin{align} | \psi \rangle & = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \, | \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N \rangle \langle \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N | \psi\rangle \\
| |
| & = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \, \psi ( \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N ) | \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N \rangle
| |
| \end{align} </math>
| |
| | |
| The closure relation is
| |
| | |
| :<math> 1 = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \, | \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N \rangle \langle \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N | </math>
| |
| | |
| leading to the inner product of {{math|{{ket|''ψ''}}}} with itself leads to the normalization conditions in the three-dimensional definitions above:
| |
| | |
| :<math> \langle \chi | \psi \rangle = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \chi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)^{*} \psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N) </math>.
| |
| | |
| Projecting {{math|{{ket|''ψ''}}}} onto a particular position state {{math|{{ket|'''r'''<sub>1 0</sub>, '''r'''<sub>2 0</sub>, ..., '''r'''<sub>''N'' 0</sub>}}}}, where {{math|'''r'''<sub>1 0</sub>}} is in {{math|''R''<sub>1</sub>}}, {{math|'''r'''<sub>2 0</sub>}} is in {{math|''R''<sub>2</sub>}}, etc., gives {{math|''ψ''('''r'''<sub>1 0</sub>, '''r'''<sub>2 0</sub>, ..., '''r'''<sub>''N'' 0</sub>)}}.
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| | |
| == Position and spin representations ==
| |
| | |
| === State space for one particle with spin in 3d ===
| |
| | |
| In Dirac notation, for a particle with spin {{math|''s''}}, in all three spatial dimensions, the basis states {{math|{{ket|'''r''', ''s<sub>z</sub>''}}}} are a combination of the discrete variable {{math|''s<sub>z</sub>''}} and the continuous variable {{math|'''r'''}},<ref name="Abers p 115"/> more specifically a [[Bra–ket notation#Composite bras and kets|tensor product]] of the spin basis {{math|{{ket|''s<sub>z</sub>''}}}} and position basis {{math|{{ket|'''r'''}}}}, which exists in a new space from the spin space and position space alone.<ref>{{cite book|author=N. Zettili|title=Quantum Mechanics: Concepts and Applications|edition=2nd|page=300|isbn=978-0-470-02679-3}}</ref> Applying the above formalism, the state can be written:
| |
| | |
| :<math> | \Psi \rangle = \sum_{s_z} \int\limits_R d^3 \, \mathbf{r} \Psi(\mathbf{r},s_z) | \mathbf{r}, s_z \rangle </math>
| |
| | |
| and therefore the closure relation is:
| |
| | |
| :<math> 1 = \sum_{s_z} \int\limits_R d^3 \, \mathbf{r} | \mathbf{r},s_z\rangle \langle \mathbf{r} , s_z | </math>
| |
| <!--- The following for completeness seems correct, though for now unable to find a source for the orthogonality relation for the position-spin state space...--->
| |
| Projecting {{math|{{math|Ψ}}}} onto a particular position-spin state {{math|{{ket|'''r'''<sub>0</sub>, ''m''}}}}, where {{math|'''r'''<sub>0</sub>}} is in {{math|''R''}}:
| |
| | |
| :<math> \langle \mathbf{r}_0, m | \Psi \rangle = \sum_{s_z}\int\limits_R d^3 \mathbf{r} \, \langle \mathbf{r}_0, m | \mathbf{r}, s_z \rangle \Psi(\mathbf{r}, s_z) = \sum_{s_z}\int\limits_R d^3 \mathbf{r} \, \delta_{m \, s_z}\delta( \mathbf{r}_0 - \mathbf{r} ) \Psi(\mathbf{r}, s_z) = \Psi(\mathbf{r}_0, m) \,.</math>
| |
| | |
| where the joint orthogonality relation
| |
| | |
| :<math>\langle \mathbf{r}_0, m | \mathbf{r}, s_z \rangle = \delta_{m\,s_z}\delta( \mathbf{r}_0 - \mathbf{r} )</math>
| |
| | |
| has been employed. For more particles, there are sums over the allowed spins for each particle, and integrals over the position vector for each particle.
| |
| | |
| == Ontology ==
| |
| | |
| {{main|Interpretations of quantum mechanics}}
| |
| | |
| Whether the wave function really exists, and what it represents, are major questions in the [[interpretation of quantum mechanics]]. Many famous physicists of a previous generation puzzled over this problem, such as [[Erwin Schrödinger|Schrödinger]], [[Albert Einstein|Einstein]] and [[Niels Bohr|Bohr]]. Some advocate formulations or variants of the [[Copenhagen interpretation]] (e.g. Bohr, [[Eugene Wigner|Wigner]] and [[John von Neumann|von Neumann]]) while others, such as [[John Archibald Wheeler|Wheeler]] or [[Edwin Thompson Jaynes|Jaynes]], take the more classical approach<ref>E. T. Jaynes. ''Probability Theory: The Logic of Science'', [[Cambridge University Press]] (2003),</ref> and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, Einstein, [[David Bohm|Bohm]] and [[Hugh Everett III|Everett]] and others, argued that the wave function must have an objective, physical existence. The latter argument is consistent with the fact that whenever two observers both think that a system is in a [[quantum state|pure quantum state]], they will always agree on exactly what state it is in (but this may not be true if one or both of them thinks the system is in a [[density matrix|mixed state]]).<ref name=pusey>{{cite journal|last=Pusey|first=Matthew F.|coauthors=Jonathan Barrett, Terry Rudolph|title=The quantum state cannot be interpreted statistically|journal=arXiv.org|date=14 November 2011|pages=arxiv:1111.3328v1|url=http://arxiv.org/abs/1111.3328}}</ref> For more on this topic, see [[Interpretations of quantum mechanics]].
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| | |
| ==Examples==
| |
| | |
| ===One-dimensional quantum tunnelling===
| |
| {{main|Finite potential barrier|Quantum tunnelling}}
| |
| [[Image:Finitepot.png|thumb|right|Scattering at a finite potential barrier of height <math>V_0</math>. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. <math>E > V_0</math> for this illustration.]]
| |
| | |
| One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics) [[potential energy|force potential]]. In the one-dimensional case of particles with energy less than <math>V_0</math> in the square potential
| |
| :<math>V(x)=\begin{cases}V_0 & |x|<a \\ 0 & \text{otherwise,}\end{cases}</math>
| |
| the steady-state solutions to the wave equation have the form (for some constants <math>k, \kappa</math>)
| |
| :<math>\psi (x) = \begin{cases}
| |
| A_{\mathrm{r}}\exp(ikx)+A_{\mathrm{l}}\exp(-ikx) & x<-a, \\
| |
| B_{\mathrm{r}}\exp(\kappa x)+B_{\mathrm{l}}\exp(-\kappa x) & |x|\le a, \\
| |
| C_{\mathrm{r}}\exp(ikx)+C_{\mathrm{l}}\exp(-ikx) & x>a.
| |
| \end{cases}
| |
| </math>
| |
| Note that these wave functions are not normalized; see [[scattering theory]] for discussion.
| |
| | |
| The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative {{math|''x''}}): setting {{math|''A''<sub>r</sub> {{=}} 1}} corresponds to firing particles singly; the terms containing {{math|''A''<sub>r</sub>}} and {{math|''C''<sub>r</sub>}} signify motion to the right, while {{math|''A''<sub>l</sub>}} and {{math|''C''<sub>l</sub>}} – to the left. Under this beam interpretation, put {{math|''C''<sub>l</sub> {{=}} 0}} since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.
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| | |
| ===Other===
| |
| Here are examples of wavefunctions for specific applications:
| |
| | |
| * [[Free particle]]
| |
| * [[Particle in a box]]
| |
| * [[Finite square well]]
| |
| * [[Delta potential]]
| |
| * [[Quantum harmonic oscillator]]
| |
| * [[Hydrogen atom#Mathematical summary of eigenstates of hydrogen atom|Hydrogen atom]] and [[Hydrogen-like atom]]
| |
| | |
| == See also ==
| |
| | |
| *[[Boson]]
| |
| *[[Double-slit experiment]]
| |
| *[[Faraday wave]]
| |
| *[[Fermion]]
| |
| *[[Schrödinger equation]]
| |
| *[[Wave function collapse]]
| |
| *[[Wave packet]]
| |
| | |
| ==References==
| |
| | |
| {{Reflist}}
| |
| 2.Quantum Mechanics (Non-Relativistic Theory), L.D. Landau and E.M. Lifshitz, ISBN 0-08-020940-8
| |
| | |
| == Further reading ==
| |
| *{{Cite book | author=Griffiths, David J.|authorlink=David Griffiths (physicist)|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn=0-13-111892-7}}
| |
| *{{Cite journal| last =Yong-Ki Kim| title =Practical Atomic Physics| journal =National Institute of Standards and Technology
| |
| | pages =1 (55 pages)
| |
| | publisher =
| |
| | location =Maryland
| |
| | date =September 2, 2000
| |
| | url =http://amods.kaeri.re.kr/mcdf/lectnote.pdf
| |
| | accessdate =2010-08-17}}
| |
| *{{Cite book | author=Polkinghorne, John |authorlink=John Polkinghorne | title=Quantum Theory, A Very Short Introduction | publisher=Oxford University Press | year=2002 | isbn=0-19-280252-6}}
| |
| | |
| ==External links==
| |
| * [http://www.eng.fsu.edu/~dommelen/quantum/style_a/complexs.html], [http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html], [http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm], [http://vergil.chemistry.gatech.edu/notes/quantrev/node34.html]
| |
| * [http://cat.middlebury.edu/~chem/chemistry/class/physical/quantum/help/normalize/normalize.html] Normalization.
| |
| * [https://www.edx.org/courses/BerkeleyX/CS191x/2013_Spring/about] Quantum Mechanics and Quantum Computation at BerkeleyX
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| {{DEFAULTSORT:Wave Function}}
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