en>Discospinster: Reverted edits by 223.176.63.35 (talk) to last revision by Magioladitis (HG)

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Reverted edits by 223.176.63.35 (talk) to last revision by Magioladitis (HG)

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Reverted 1 edit by 41.69.236.239 (talk) to last revision by Donner60. (TW)

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{{Merge|Schrödinger picture|Interaction picture|Mathematical formulation of quantum mechanics#Pictures of dynamics|target=Dynamical pictures (quantum mechanics)|discuss=Talk:Dynamical pictures (quantum mechanics)#Merger proposal|date=September 2013}}
{{Quantum mechanics|cTopic=Formulations}}
In [[physics]], the '''Heisenberg picture''' (also called the '''Heisenberg representation'''<ref>{{cite web|title=Heisenberg representation|url=http://www.encyclopediaofmath.org/index.php/Heisenberg_representation|publisher=Encyclopedia of Mathematics|accessdate=3 September 2013}}</ref>) is a formulation (largely due to [[Werner Heisenberg]] in 1925) of [[quantum mechanics]] in which the [[Operator (physics)|operators]] ([[observables]] and others) incorporate a dependency on time, but the [[quantum state|state vector]]s are time-independent, an arbitrary fixed basis rigidly underlying the theory.

It stands in contrast to the [[Schrödinger picture]] in which the operators are constant, instead, and the states evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which corresponds to the difference between [[active and passive transformation]]s. The Heisenberg picture is the formulation of [[matrix mechanics]] in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal.

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

==Mathematical details==
In the Heisenberg picture of quantum mechanics the state vectors, {{ket|''ψ''}}, do not change with time, while observables {{mvar|A}} satisfy
{{Equation box 1
|indent =:
|equation =
<math>\frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\frac{\partial A(t)}{\partial t},</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
where {{mvar|H}} is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and <nowiki>[•,•]</nowiki> 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 [[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]].

==Derivation of Heisenberg's equation==
For pedagogical reasons, the Heisenberg picture is introduced here from the subsequent, but more familiar, [[Schrödinger picture]].
The [[expectation value]] of an observable ''A'', which is a [[Hermitian]] [[linear operator]], for a given Schrödinger state {{math|{{ket|''ψ''(''t'')}}}}, is given by
:<math> \lang A \rang _t = \lang \psi (t) | A | \psi(t) \rang.</math>

In the Schrödinger picture, the state {{math|{{ket|''ψ''(''t'')}}}} at time {{math|''t''}} is related to the state {{math|{{ket|''ψ''(0)}}}} at time 0 by a unitary [[time-evolution operator]], {{math|''U''(''t'')}},
: <math> |\psi(t)\rangle = U(t) |\psi(0)\rangle.</math>
If the [[Hamiltonian (quantum mechanics)|Hamiltonian]] does not vary with time, then the time-evolution operator can be written as
:<math> U(t) = e^{-iHt / \hbar} ,</math>
where {{mvar|H}} is the Hamiltonian and {{mvar|ħ}} is the [[reduced Planck constant]]. Therefore,

:<math> \lang A \rang _t = \lang \psi (0) | e^{iHt / \hbar} A e^{-iHt / \hbar} | \psi(0) \rang .</math>

Peg all state vectors to a rigid basis of {{math|{{ket|''ψ''(0)}}}} then, and define
:<math> A(t) := e^{iHt / \hbar} A e^{-iHt / \hbar} .</math>

It now follows that
:<math> {d \over dt} A(t) = {i \over \hbar} H e^{iHt / \hbar} A e^{-iHt / \hbar} + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right) e^{-iHt / \hbar} + {i \over \hbar} e^{iHt / \hbar} A \cdot (-H) e^{-iHt / \hbar} </math>

:<math> = {i \over \hbar} e^{iHt / \hbar} \left( H A - A H \right) e^{-iHt / \hbar} + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right) e^{-iHt / \hbar} </math>

:<math> = {i \over \hbar } \left( H A(t) - A(t) H \right) + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right)e^{-iHt / \hbar} .</math>

Differentiation was according to the [[product rule]], while ∂''A''/∂''t''
is the time derivative of the initial ''A'', not the ''A''(''t'') operator defined. The last equation holds since {{math|exp(−''iHt''/''ħ'')}} commutes with {{math|''H''}}.

Thus
:<math> {d \over dt} A(t) = {i \over \hbar } [H, A(t)] + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right)e^{-iHt / \hbar} ,</math>
and hence emerges the above Heisenberg equation of motion, since the convective functional dependence on ''x''(0) and ''p''(0) converts to the ''same'' dependence on ''x''(''t''), ''p''(''t''), so that the last term converts to ∂''A(t)''/∂''t'' . [''X'', ''Y''] is the [[commutator]] of two operators and is defined as [''X'', ''Y''] := ''XY'' − ''YX''.

The equation is solved by the ''A(t)'' defined above, as evident by use of the
[[BCH_formula#An_important_lemma|standard operator identity]],
:<math> {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> A(t) = A + \frac{it}{\hbar}[H,A] - \frac{t^{2}}{2!\hbar^{2}}[H,[H,A]] - \frac{it^3}{3!\hbar^3}[H,[H,[H,A]]] + \dots </math>

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

In effect, the arbitrary rigid Hilbert space basis |''ψ(0)''〉has receded from view, and is only considered at the very last step of taking specific expectation values or matrix elements of observables.

==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''1), ''x''(''t''2), ''p''(''t''1)}} and {{math| ''p''(''t''2)}}. The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator,
:<math>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>{d \over dt} x(t) = {i \over \hbar } [ H , x(t) ]=\frac {p}{m}</math> ,
:<math>{d \over dt} p(t) = {i \over \hbar } [ H , p(t) ]= -m \omega^{2} x</math> .

Differentiating both equations once more and solving for them with proper initial conditions,
:<math>\dot{p}(0)=-m\omega^{2} x_0 ,</math>
:<math>\dot{x}(0)=\frac{p_0}{m} ,</math>
leads to
:<math>x(t)=x_{0}\cos(\omega t)+\frac{p_{0}}{\omega m}\sin(\omega t) </math> ,
:<math>p(t)=p_{0}\cos(\omega t)-m\omega\!x_{0}\sin(\omega t) </math> .

Direct computation yields the more general commutator relations,

:<math>[x(t_{1}), x(t_{2})]=\frac{i\hbar}{m\omega}\sin(\omega t_{2}-\omega t_{1}) </math> ,
:<math>[p(t_{1}), p(t_{2})]=i\hbar m\omega\sin(\omega t_{2}-\omega t_{1}) </math> ,
:<math>[x(t_{1}), p(t_{2})]=i\hbar \cos(\omega t_{2}-\omega t_{1}) </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==

<center>
{| tableborder="1" cellspacing="0" cellpadding="8" style="padding: 0.3em; clear: right;margin: 0px 0px 5px 1em; border:1px solid #999; border-bottom:2px solid; border-right-width: 2px; text-align:center;line-height: 1.2em; font-size: 90%"
| bgcolor="#E0FFEE" style="border-left:1px solid; border-top:1px solid;" | Evolution
| colspan="3" bgcolor="#E6F6FF" style="border-left:1px solid; border-right:1px solid; border-top:1px solid;" | '''Picture'''
|-----
| bgcolor="#E0FFEE" style="border-left:1px solid; border-top:1px solid;" | of:
| bgcolor="#E0F0FF" style="border-left:1px solid; border-top:1px solid;" | Heisenberg
| bgcolor="#E0F0FF" style="border-left:1px solid; border-top:1px solid;" | [[Interaction picture|Interaction]]
| bgcolor="#E0F0FF" style="border-left:1px solid; border-right:1px solid; border-top:1px solid;" | [[Schrödinger picture|Schrödinger]]
|-----
| style="border-left:1px solid; border-top:1px solid; background:#D0FFDD;" | [[Bra-ket notation|Ket state]]
| style="border-left:1px solid; border-top:1px solid;" | constant
| style="border-left:1px solid; border-top:1px solid;" |<math> | \psi_{I}(t) \rang = e^{i H_{0, S} ~t / \hbar} | \psi_{S}(t) \rang </math>
| style="border-left:1px solid; border-top:1px solid; border-right:1px solid;" | <math> | \psi_{S}(t) \rang = e^{-i H_{ S} ~t / \hbar} | \psi_{S}(0) \rang </math>
|-----
| style="border-left:1px solid; border-top:1px solid; background:#D0FFDD;" | [[Observable]]
| style="border-left:1px solid; border-top:1px solid;" | <math>A_H (t)=e^{i H_{ S}~ t / \hbar} A_S e^{-i H_{ S}~ t / \hbar}</math>
| style="border-left:1px solid; border-top:1px solid;" | <math>A_I (t)=e^{i H_{0, S} ~t / \hbar} A_S e^{-i H_{0, S}~ t / \hbar} </math>
| style="border-left:1px solid; border-top:1px solid; border-right:1px solid;" | constant
|-----
| style="border-left:1px solid; border-top:1px solid; border-bottom:1px solid; background:#D0FFDD;" | [[Density matrix]]
| style="border-left:1px solid; border-top:1px solid; border-bottom:1px solid;" | constant
| style="border-left:1px solid; border-top:1px solid; border-bottom:1px solid;" | <math>\rho_I (t)=e^{i H_{0, S} ~t / \hbar} \rho_S (t) e^{-i H_{0, S}~ t / \hbar}</math>
| style="border-left:1px solid; border-top:1px solid; border-right:1px solid; border-bottom:1px solid;" | <math>\rho_S (t)= e^{-i H_{ S} ~t / \hbar} \rho_S(0) e^{i H_{ S}~ t / \hbar} </math>
|-----
|}
</center>

==See also==
* [[Interaction picture]]
* [[Bra-ket notation]]
* [[Schrödinger picture]]

== References ==
<references />
* {{cite book
| last = Cohen-Tannoudji
| first = Claude
| authorlink = Claude Cohen-Tannoudji
| coauthors = Bernard Diu, 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.

==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.
{{Use dmy dates|date=December 2010}}

{{DEFAULTSORT:Heisenberg Picture}}
[[Category:Quantum mechanics]]

[[es:Imagen de evolución temporal]]
[[ja:ハイゼンベルグ描像]]
[[ru:Представление Гейзенберга]]

Structural analysis - Revision history

en>Discospinster: Reverted edits by 223.176.63.35 (talk) to last revision by Magioladitis (HG)

en>Donner60: Reverted 1 edit by 41.69.236.239 (talk) to last revision by Donner60. (TW)