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{{Main|operator (physics)}}
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In [[quantum mechanics]], [[energy]] is defined in terms of the '''energy operator''', acting on the [[wavefunction]] of the system.
 
==Definition==
 
It is given by: <ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>
 
:<math>\hat{E} = i\hbar\frac{\partial}{\partial t} \,\!</math>
 
It acts on the wavefunction (the [[probability amplitude]] for different [[configuration space|configurations]] of the system)
 
:<math>\Psi\left(\mathbf{r},t\right) \,\!</math>
 
==Application==
 
The energy operator [[correspondence principle|corresponds]] to the full energy of a system. The [[Schrödinger equation]] describes the space- and time-dependence of slow changing (non-[[theory of relativity|relativistic]]) wavefunction of quantum systems. The solution of this equation for bound system is discrete (a set of permitted states, each characterized by an [[energy level]]) which results in the concept of [[quantum|quanta]].
 
===Schrödinger equation===
 
Using the classical equation for [[conservation of energy]] of a [[particle]]:
 
:<math>E = H = T + V \,\!</math>
 
where ''E'' = total [[energy]], ''H'' = [[Hamiltonian mechanics#Basic physical interpretation|hamiltonian]], ''T'' = [[kinetic energy]] and ''V'' = [[potential energy]] of the particle, substituting the energy and Hamiltonian operators and multiplying by the wavefunction obtains the Schrödinger equation
 
:<math>\begin{align}
& \hat{E} = \hat{H} \\
& \hat{E}\Psi = \hat{H} \Psi \\
\end{align}\,\!</math>
 
that is
 
:<math>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = \hat H \Psi(\mathbf{r},t) \,\!</math>
 
where ''i'' is the [[imaginary unit]], ''ħ'' is the [[reduced Planck constant]], and <math>\hat H</math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] [[operator (physics)|operator]].
 
===Klein-Gordon equation===
 
The [[Mass in special relativity#The relativistic energy-momentum equation|relativistic mass-energy relation]]:
 
:<math>E^2 = (pc)^2 + (mc^2)^2 \,\!</math>
 
where again ''E'' = total energy, ''p'' = total 3-[[momentum]] of the particle, ''m'' = [[invariant mass]], and ''c'' = [[speed of light]], can similarly yield the [[Klein-Gordon equation]]:
 
:<math>\begin{align}
& \hat{E}^2 = c^2\hat{p}^2 + (mc^2)^2 \\
& \hat{E}^2\Psi = c^2\hat{p}^2\Psi + (mc^2)^2\Psi \\
\end{align}\,\!</math>
 
that is:
 
:<math>\frac{\partial^2 \Psi}{\partial t^2} =  c^2\nabla^2\Psi - \left(\frac{mc^2}{\hbar}\right)^2\Psi \,\!</math>
 
==Derivation==
 
The energy operator is easily derived from using the [[free particle]] wavefunction ([[plane wave]] solution to Schrödinger's equation). <ref>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> Starting in one dimension the wavefunction is
 
:<math> \Psi = e^{i(kx-\omega t)} \,\!</math>
 
The time derivative of ''Ψ'' is
 
:<math> \frac{\partial \Psi}{\partial t} = -i \omega e^{i(kx-\omega t)} = - i \omega \Psi \,\!</math>.
 
By the [[De Broglie relation]]:
 
:<math> E=\hbar \omega \,\!</math>,
 
we have
 
:<math> \frac{\partial \Psi}{\partial t} = - i \frac{E}{\hbar} \Psi \,\!</math>.
 
Re-arranging the equation leads to  
 
:<math> E\Psi = i\hbar\frac{\partial \Psi}{\partial t} \,\!</math>,
 
where the energy factor ''E'' is a [[Scalar (mathematics)|scalar]] value, the energy the particle has and the value that is measured. Cancellation of ''Ψ'' leads to
 
:<math> E = i\hbar\frac{\partial }{\partial t} \,\!</math>
 
The [[partial derivative]] is a [[linear operator]] so this expression ''is'' the operator for energy:
 
:<math> \hat{E} = i\hbar\frac{\partial }{\partial t} \,\!</math>.
 
It can be concluded that the scalar ''E'' is the [[eigenvalue]] of the operator, while <math> \hat{E} \,\!</math> is the operator. Summarizing these results:
 
:<math> \hat{E}\Psi = i\hbar\frac{\partial }{\partial t}\Psi=E\Psi \,\!</math>
 
For a 3-d plane wave
 
:<math> \Psi = e^{i(\bold{k}\cdot\bold{r}-\omega t)} \,\!</math>
 
the derivation is exactly identical, as no change is made to the term including time and therefore the time derivative. Since the [[linear operator|operator is linear]], they are valid for any [[linear combination]] of plane waves, and so they can act on any wavefunction without affecting the properties of the wavefunction or operators. Hence this must be true for any wavefunction. It turns out to work even in [[relativistic quantum mechanics]], such as the [[Klein-Gordon equation]] above.
 
==See also==
 
*[[Planck constant]]
*[[Schrödinger equation]]
*[[Momentum operator]]
*[[Hamiltonian (quantum mechanics)]]
*[[Conservation of energy]]
*[[Complex number]]
*[[Stationary state]]
 
==References==
 
{{reflist}}
 
{{Physics operator}}
 
[[Category:Energy]]
[[Category:Partial differential equations]]
[[Category:Quantum mechanics]]

Latest revision as of 23:23, 8 January 2015

The UK best essay writing services do have a point. • Are you intending to tell the narrative objectively. Whenever students are assigned to write high school vs. This is important as if the essays assigned are not submitted on time, students will fail the class. If the essay contains any facts, check their accuracy with reliable sources.

Try to get hold of the latest information which will enhance the quality of your essay by incorporating the modern development(s) on the subject. He deals with thousand of application a day, he doesn't want to know how many words you have stored in your vocabulary , instead he want to know why should they consider you for the admission so it is better that you try to convince him in a more subtle manner. Being familiar with the discursive essay let's discuss first what a discursive essay is. Now, those thoughts are very important, but also a bit generic and can be applied to any admissions essay. overloaded with so much work that he cannot get time to write any academic writing.

At the same time, you will be assigned one writer or a team of writers, according to the difficulty of your project and the time frame you want it done in. Buy APA legitimate custom economic papers, MLA economic essays and other internationally recognized formats. Essay writing in the university will enable students to develop research techniques. Think about how this reflects ideas in the poem, for example a very uneven structure could represent ideas of chaos or lack of control. Obviously it's best to write your own essays; don't let laziness goad you into taking a shortcut.

Our well qualified and experienced staff will assist you in writing a high quality research paper. A narrative paper can also be a challenge to students because it requires a lot of time in drafting, pondering on the details significance and presenting an outline of the experience. Equally, most of the author's students, members in the initially indigenous online populace, favor reading virtual texts to looking at genuine books: For them, it implies that just about every college course should acknowledge the increasing online assets (from educational e-books to social book-renting web sites) offered to college students who commit extra time engaged together with the World wide web than glued into the Tv set. Consequently, we suggest you to apply Exclusive-Essays. Quality is not debatable if you are to get a high score in your term paper.

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