Editing Quantum mechanics (section)

=== Time evolution of a quantum state ===

The time evolution of a quantum state is described by the Schrödinger equation:
<math display=block>i\hbar {\frac {\partial}{\partial t}} \psi (t) =H \psi (t). </math>
Here <math>H</math> denotes the [[Hamiltonian (quantum mechanics)|Hamiltonian]], the observable corresponding to the [[total energy]] of the system, and <math>\hbar</math> is the reduced [[Planck constant]]. The constant <math>i\hbar</math> is introduced so that the Hamiltonian is reduced to the [[Hamiltonian mechanics|classical Hamiltonian]] in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the [[correspondence principle]].

The solution of this differential equation is given by
<math display=block> \psi(t) = e^{-iHt/\hbar }\psi(0). </math>
The operator <math>U(t) = e^{-iHt/\hbar } </math> is known as the time-evolution operator, and has the crucial property that it is [[Unitarity (physics)|unitary]]. This time evolution is [[deterministic]] in the sense that&nbsp;– given an initial quantum state <math>\psi(0)</math> – it makes a definite prediction of what the quantum state <math>\psi(t)</math> will be at any later time.<ref>{{cite book |title=Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature |first1=Steven |last1=Weinberg |publisher=Random House |year=2010 |isbn=978-1-4070-6396-6 |page=[https://books.google.com/books?id=OLrZkgPsZR0C&pg=PT82 82] |url=https://books.google.com/books?id=OLrZkgPsZR0C}}</ref>
{{anchor|fig1}}
[[File:Atomic-orbital-clouds spd m0.png|thumb|upright=1.25|Fig. 1: [[Probability density function|Probability densities]] corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: ''n'' = 1, 2, 3, ...) and angular momenta (increasing across from left to right: ''s'', ''p'', ''d'', ...). Denser areas correspond to higher probability density in a position measurement.{{pb}}Such wave functions are directly comparable to [[Chladni's figures]] of [[acoustics|acoustic]] modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The [[angular momentum]] and energy are [[quantization (physics)|quantized]] and take <em>only</em> discrete values like those shown – as is the case for [[resonant frequencies]] in acoustics.]]

Some wave functions produce probability distributions that are independent of time, such as [[eigenstate]]s of the Hamiltonian.<ref name="Zwiebach2022">{{cite book |first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8}}</ref>{{rp|133–137}} Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the [[atomic nucleus]], whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an [[atomic orbital|''s'' orbital]] ([[#fig1|Fig. 1]]).

Analytic solutions of the Schrödinger equation are known for [[List of quantum-mechanical systems with analytical solutions|very few relatively simple model Hamiltonians]] including the [[quantum harmonic oscillator]], the [[particle in a box]], the [[dihydrogen cation]], and the [[hydrogen atom]]. Even the [[helium]] atom&nbsp;– which contains just two electrons&nbsp;– has defied all attempts at a fully analytic treatment, admitting no solution in [[Closed-form expression|closed form]].<ref>{{Cite journal |last1=Zhang |first1=Ruiqin |last2=Deng |first2=Conghao |date=1993 |title=Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems  |journal=Physical Review A |volume=47 |issue=1 |pages=71–77 |doi=10.1103/PhysRevA.47.71 |pmid=9908895 |bibcode=1993PhRvA..47...71Z |issn=1050-2947}}</ref><ref>{{Cite journal |last1=Li |first1=Jing |last2=Drummond |first2=N. D. |last3=Schuck |first3=Peter |last4=Olevano |first4=Valerio |date=2019-04-01 |title=Comparing many-body approaches against the helium atom exact solution |journal=SciPost Physics |volume=6 |issue=4 |page=40 |doi=10.21468/SciPostPhys.6.4.040 |doi-access=free |arxiv=1801.09977 |bibcode=2019ScPP....6...40L |issn=2542-4653}}</ref><ref>{{cite book |last=Drake |first=Gordon W. F. |chapter=High Precision Calculations for Helium |date=2023 |title=Springer Handbook of Atomic, Molecular, and Optical Physics |series=Springer Handbooks |pages=199–216 |editor-last=Drake |editor-first=Gordon W. F. |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-030-73893-8_12 |isbn=978-3-030-73892-1}}</ref>

However, there are techniques for finding approximate solutions. One method, called [[perturbation theory (quantum mechanics)|perturbation theory]], uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak [[potential energy]].<ref name="Zwiebach2022" />{{rp|793}} Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<ref name="Zwiebach2022" />{{rp|849}}