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{{Short description|Formulation of quantum mechanics}}
{{Use dmy dates|date=December 2019}}
{{Quantum mechanics|cTopic=Formulations}}

In [[physics]], the '''Heisenberg picture''' or '''Heisenberg representation'''<ref>{{cite web|title=Heisenberg representation|url=http://www.encyclopediaofmath.org/index.php/Heisenberg_representation|publisher=Encyclopedia of Mathematics| access-date=3 September 2013}}</ref> is a [[Dynamical pictures|formulation]] (largely due to [[Werner Heisenberg]] in 1925) of [[quantum mechanics]] in which [[observables]] incorporate a dependency on time, but the [[quantum state|state]]s are time-independent. It stands in contrast to the [[Schrödinger picture]] in which observables are constant and the states evolve in time. 

It further serves to define a third, hybrid, picture, the [[interaction picture]].

==Mathematical details==
In the Heisenberg picture of quantum mechanics the state vectors |''ψ''⟩ do not change with time, while observables {{mvar|A}} satisfy
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|equation =
<math>\frac{d}{dt}A_\text{H}(t)=\frac{i}{\hbar}[H_\text{H}(t),A_\text{H}(t)]+\left( \frac{\partial A_\text{S}}{\partial t} \right)_\text{H} ,</math>
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where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, {{mvar|H}} is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and {{math|[·,·]}} denotes the [[commutator]] of two operators (in this case {{mvar|H}} and {{mvar|A}}). Taking expectation values automatically yields the [[Ehrenfest theorem]], featured in the [[correspondence principle]].

By the [[Stone–von Neumann theorem]], the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a [[transformation theory (quantum mechanics)|basis change]] in [[Hilbert space]]. In some sense, the [[Werner Heisenberg|Heisenberg]] picture is more natural and convenient than the equivalent Schrödinger picture, especially for [[theory of relativity|relativistic]] theories. [[Lorentz invariance]] is manifest in the Heisenberg picture, since the state vectors do not single out the time or space.

This approach also has a more direct similarity to [[classical physics]]: by simply replacing the commutator above by the [[Poisson bracket]], the '''Heisenberg equation''' reduces to an equation in [[Hamiltonian mechanics]].

==Equivalence of Heisenberg's equation to the Schrödinger equation==
For the sake of pedagogy, the Heisenberg picture is introduced here from the subsequent, but more familiar, [[Schrödinger picture]].

According to [[Schrödinger equation|Schrödinger's equation]], the quantum state at time <math>t</math> is <math>|\psi(t)\rangle = U(t) |\psi(0)\rangle</math>, where <math>U(t)=T e^{-\frac{i}{\hbar}\int_0^t ds H_{{\rm S}} (s)}</math> is the [[time-evolution operator]] induced by a Hamiltonian <math>H_{\rm S}(t)</math> that could depend on time, and <math> |\psi(0) \rangle</math> is the initial state. <math>T</math> refers to time-ordering, ''ħ'' is the [[reduced Planck constant]], and ''i'' is the imaginary unit. The [[expectation value]] of an observable <math>A_{\rm S} (t)</math> in the [[Schrödinger picture]], which is a [[Hermitian]] [[linear operator]] that could also be time-dependent, in the state <math>|\psi(t)\rangle</math> is given by
<math display="block"> \lang A \rang _t = \lang \psi (t) | A_{\rm S}(t) | \psi(t) \rang.</math>

In the Heisenberg picture, the quantum state is assumed to remain constant at its initial value <math> |\psi(0) \rangle</math>, whereas operators evolve with time according to the definition 
<math display="block"> A_{\rm H} (t) := U^{\dagger}(t) A_{\rm S} (t) U(t) \, .</math>
This readily implies <math>\langle A \rangle_t =\langle \psi(0) | A_{\rm H}(t) | \psi(0) \rangle</math>, so the same expectation value can be obtained by working in either picture. The Schrödinger equation for the time-evolution operator is
<math display="block"> \frac{d}{dt} U(t) = -\frac{i}{\hbar} H_{\rm S}(t) U(t) . </math><nowiki> </nowiki>It follows that
<math display="block">\begin{align}
 \frac{d}{dt} A_{\rm H} (t) &=  \left( \frac{d}{dt} U^\dagger (t) \right) A_{\rm S}(t) U(t) + U^\dagger(t) A_{\rm S} (t)  \left( \frac{d}{dt} U (t) \right) + U^\dagger (t) \left(\frac{\partial A_{\rm S}} {\partial t}\right) U(t) \\
 & = \frac{i}{\hbar} U^{\dagger}(t) H_{\rm S}(t) A_{{\rm S}} (t) U(t)  - \frac{i}{\hbar} U^{\dagger}(t) A_{\rm S}(t) H_{\rm S} (t) U(t) + U^{\dagger}(t) \left(\frac{\partial A_{\rm S}}{\partial t}\right) U(t)\\
 & = \frac{i}{\hbar} U^{\dagger}(t) H_{\rm S} (t) U(t) U^\dagger(t) A_{\rm S}(t) U(t) - \frac{i}{\hbar} U^{\dagger}(t) A_{\rm S} (t) U(t) U^\dagger(t) H_{\rm S}(t) U(t) + \left(\frac{\partial A_{\rm S} }{\partial t}\right)_{\rm H}\\
 & = \frac{i}{\hbar} [H_{\rm H}(t), A_{\rm H} (t)] + \left(\frac{\partial A_{\rm S}}{\partial t} \right) _{\rm H} ,
\end{align}</math>
where differentiation was carried out according to the [[product rule]]. This is Heisenberg's equation of motion. Note that the [[Hamiltonian (quantum mechanics)|Hamiltonian]] that appears in the final line above is the Heisenberg Hamiltonian <math>H_{\rm H}(t)</math>, which may differ from the Schrödinger Hamiltonian <math>H_{\rm S} (t)</math>.

An important special case of the equation above is obtained if the [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>H_{\rm S}</math> does not vary with time.  Then the time-evolution operator can be written as
<math display="block"> U(t) = e^{-\frac{i} {\hbar} t H_{\rm S}} ,</math>
and hence <math> H_{\rm H} \equiv H_{\rm S}\equiv H</math> since <math>U(t)</math> now commutes with <math>H</math>. Therefore,
<math display="block"> \lang A \rang _t = \lang \psi (0) | e^{\frac i \hbar t H} A_{{\rm S}} (t) e^{-\frac i \hbar t H} | \psi(0) \rang </math>
and following the previous analyses, 
<math display="block">\begin{align}
 \frac{d}{dt} A_{\rm H}(t) 
 & = \frac{i}{\hbar} [H,A_{\rm H}(t)]+ e^{\frac i \hbar t H}\left(\frac{\partial A_{\rm S}}{ \partial t} \right) e^{-\frac i \hbar t H}.
\end{align}</math>

Furthermore, if <math>A_{{\rm S}} \equiv A</math> is also time-independent, then the last term vanishes and

<math>\frac {d}{dt} A_{\rm H}(t)=\frac {i}{\hbar} [H, A_{\rm H} (t)] ,</math>

where <math>A_{{\rm H}}(t) \equiv A(t)= e^{\frac {i} {\hbar} t H} A  e^{- \frac {i} {\hbar} t H} </math> in this particular case. The equation is solved by use of the [[BCH formula#An important lemma|standard operator identity]],<math display="block"> {e^B A e^{-B}} = A + [B,A] + \frac{1}{2!} [B,[B,A]] + \frac{1}{3!}[B,[B,[B,A]]] + \cdots\, ,</math>
which implies
<math display="block"> A(t) = A + \frac{i t}{\hbar}[H,A] + \frac{1}{2!}\left(\frac{i t}{\hbar}\right)^2 [H,[H,A]] + \frac{1}{3!} \left(\frac{i t}{\hbar}\right)^3 [H,[H,[H,A]]] + \cdots </math>

A similar relation also holds for [[classical mechanics]], the [[classical limit]] of the above, given by the [[Moyal bracket|correspondence]] between [[Poisson bracket]]s and [[commutators]]:
<math display="block"> [A,H] \quad \longleftrightarrow \quad i\hbar\{A,H\}. </math>
In classical mechanics, for an ''A'' with no explicit time dependence,
<math display="block"> \{A,H\} = \frac{dA}{dt}~,</math> 
so again the expression for ''A''(''t'') is the Taylor expansion around ''t'' = 0.

In effect, the initial state of the quantum system has receded from view, and is only considered at the final step of taking specific expectation values or matrix elements of observables that evolved in time according to the Heisenberg equation of motion. A similar analysis applies if the initial state is [[Quantum state|mixed]].

The time evolved state <math> |\psi(t)\rangle </math> in the Schrödinger picture is sometimes written as <math> |\psi_{\rm S}(t)\rangle </math> to differentiate it from the evolved state <math> |\psi_{\rm I}(t)\rangle </math> that appears in the different [[interaction picture]].

==Commutator relations==
Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators. For example, consider the operators {{math|''x''(''t''<sub>1</sub>), ''x''(''t''<sub>2</sub>), ''p''(''t''<sub>1</sub>)}} and {{math|''p''(''t''<sub>2</sub>)}}. The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator,
<math display="block">H = \frac{p^2}{2m} + \frac{m\omega^2 x^2}{2} ,</math>
the evolution of the position and momentum operators is given by:
<math display="block">\frac{d}{dt} x(t) = \frac{i}{\hbar} [ H , x(t) ] = \frac {p}{m},</math>
<math display="block">\frac{d}{dt} p(t) = \frac{i}{\hbar} [ H , p(t) ] = -m \omega^2 x .</math>

Note that the Hamiltonian is time independent and hence <math> x(t),p(t) </math> are the position and momentum operators in the Heisenberg picture. Differentiating both equations once more and solving for them with proper initial conditions,
<math display="block">\dot{p}(0) = -m \omega^2 x_0 ,</math>
<math display="block">\dot{x}(0) = \frac{p_0}{m} ,</math> 
leads to
<math display="block">x(t) = x_0 \cos(\omega t) + \frac{p_0}{\omega m}\sin(\omega t) ,</math>
<math display="block">p(t) = p_0 \cos(\omega t) - m \omega x_0 \sin(\omega t) .</math>

Direct computation yields the more general commutator relations,
<math display="block">[x(t_1), x(t_2)] = \frac{i\hbar}{m\omega} \sin\left(\omega t_2 - \omega t_1\right) ,</math>
<math display="block">[p(t_1), p(t_2)] = i\hbar m\omega \sin\left(\omega t_2 - \omega t_1\right) ,</math>
<math display="block">[x(t_1), p(t_2)] = i\hbar \cos\left(\omega t_2 - \omega t_1\right) .</math>

For <math>t_1 = t_2</math>, one simply recovers the standard canonical commutation relations valid in all pictures.

==Summary comparison of evolution in all pictures==

For a time-independent Hamiltonian ''H''<sub>S</sub>, where ''H''<sub>0,S</sub> is the free Hamiltonian,
{{Pictures in quantum mechanics}}

==See also==
* [[Bra–ket notation]]
* [[Interaction picture]]
* [[Schrödinger picture]]
* [[Heisenberg–Langevin equations]]
* [[Phase space formulation]]

== References ==
<references />
* {{cite book
 | last = Cohen-Tannoudji
 | first = Claude
 | author-link = Claude Cohen-Tannoudji
 |author2=Bernard Diu |author3=Frank Laloe
 | title = Quantum Mechanics (Volume One)
 | publisher = Wiley
 | year = 1977
 | location = Paris
 | pages = 312–314
 | isbn = 0-471-16433-X }}
* [[Albert Messiah]], 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
*[[Eugen Merzbacher|Merzbacher E.]], ''Quantum Mechanics'' (3rd ed., John Wiley 1998) pp.&nbsp;430–431 {{ISBN|0-471-88702-1}}
*{{cite book
 |author=[[Lev Landau|L.D. Landau]], [[Evgeny Lifshitz|E.M. Lifshitz]]
 |year=1977
 |title=Quantum Mechanics: Non-Relativistic Theory
 |edition=3rd |volume=3
 |publisher=[[Pergamon Press]]
 |isbn=978-0-08-020940-1}} [https://archive.org/details/QuantumMechanics_104 Online copy] 
*[[Ramamurti Shankar|R. Shankar]] (1994); ''Principles of Quantum Mechanics'', Plenum Press, {{ISBN|978-0306447907}}.
*[[J. J. Sakurai]] (1993); ''[[Modern Quantum Mechanics]]'' (Revised Edition), {{ISBN|978-0201539295}}.

==External links==
*[http://www.quantumfieldtheory.info Pedagogic Aides to Quantum Field Theory] Click on the link for Chap. 2 to find an extensive, simplified introduction to the Heisenberg picture.
* Some expanded derivations and an example of the harmonic oscillator in the Heisenberg picture [https://courses.physics.illinois.edu/phys580/fa2013/heisenberg.pdf]
* The original Heisenberg paper translated (although difficult to read, it contains an example for the anharmonic oscillator): Sources of Quantum mechanics B.L. Van Der Waerden [https://physicstoday.scitation.org/doi/10.1063/1.3035086]
* The computations for the hydrogen atom in the Heisenberg representation originally from a paper of Pauli [https://arxiv.org/ftp/arxiv/papers/1501/1501.05894.pdf]
{{Quantum mechanics topics}}

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[[Category:Quantum mechanics]]
[[Category:Werner Heisenberg]]