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{{Short description|Formulations of quantum mechanics}}
{{distinguish|text=[[Dynamic Pictures]], a defunct manufacturer of 3D graphics cards}}
{{Quantum mechanics|cTopic=Formulations}}
{{Use dmy dates|date=December 2019}}
In [[quantum mechanics]], '''dynamical pictures''' (or ''representations'') are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system.

The two most important ones are the '''[[Heisenberg picture]]''' and the '''[[Schrödinger picture]]'''.  These differ only by a basis change with respect to time-dependency, analogous to the [[Lagrangian and Eulerian specification of the flow field]]:  in short, time dependence is attached to [[quantum states]] in the Schrödinger picture and to [[Operator (physics)|operators]] in the Heisenberg picture.

There is also an intermediate formulation known as the '''[[interaction picture]]''' (or '''Dirac picture''') which is useful for doing computations when a complicated [[Hamiltonian (quantum mechanics)|Hamiltonian]] has a natural decomposition into a simple "free" Hamiltonian and a [[Perturbation theory (quantum mechanics)|perturbation]].

Equations that apply in one picture do not necessarily hold in the others, because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others. Not all textbooks and articles make explicit which picture each operator comes from, which can lead to confusion.

==Schrödinger picture==

===Background===
In elementary quantum mechanics, the [[quantum state|state]] of a quantum-mechanical system is represented by a complex-valued [[wavefunction]] {{math|''ψ''(''x'', ''t'')}}. More abstractly, the state may be represented as a state vector, or [[Bra-ket notation|''ket'']], |''ψ''⟩. This ket is an element of a ''[[Hilbert space]]'', a vector space containing all possible states of the system. A quantum-mechanical [[Operator (physics)#Operators in quantum mechanics|operator]] is a function which takes a ket |''ψ''⟩ and returns some other ket |''ψ&prime;''⟩.

The differences between the Schrödinger and Heiseinberg pictures of quantum mechanics revolve around how to deal with systems that evolve in time: the time-dependent nature of the system ''must'' be carried by some combination of the state vectors and the operators. For example, a [[quantum harmonic oscillator]] may be in a state |''ψ''⟩ for which the [[Expectation value (quantum mechanics)|expectation value]] of the momentum, <math>\langle \psi | \hat{p} | \psi \rangle</math>, oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector |''ψ''⟩, the momentum operator <math>\hat{p}</math>, or both. All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture.

The Schrödinger picture is useful when dealing with a time-independent Hamiltonian {{mvar|H}}, that is, <math>\partial_tH=0 </math>.

===The time evolution operator===

====Definition====

The time-evolution operator ''U''(''t'', ''t''<sub>0</sub>) is defined as the operator which acts on the ket at time ''t''<sub>0</sub> to produce the ket at some other time ''t'':
:<math> | \psi(t) \rangle = U(t,t_0) | \psi(t_0) \rangle.</math>

For [[Bra-ket notation|bras]], we instead have
:<math> \langle \psi(t) | = \langle \psi(t_0) |U^{\dagger}(t,t_0).</math>

====Properties====

=====Unitarity=====
The time evolution operator must be [[Unitary operator|unitary]]. This is because we demand that the [[Norm (mathematics)|norm]] of the state ket must not change with time. That is,
:<math> \langle \psi(t)| \psi(t) \rangle = \langle \psi(t_0)|U^{\dagger}(t,t_0)U(t,t_0)| \psi(t_0) \rangle  = \langle \psi(t_0) | \psi(t_0) \rangle.</math>

Therefore,
:<math> U^{\dagger}(t,t_0)U(t,t_0)=I.</math>

=====Identity=====
When ''t''&nbsp;= ''t''<sub>0</sub>, ''U'' is the [[identity operator]], since
:<math> | \psi(t_0) \rangle = U(t_0,t_0) | \psi(t_0) \rangle.</math>

=====Closure=====
Time evolution from ''t''<sub>0</sub> to ''t'' may be viewed as a two-step time evolution, first from ''t''<sub>0</sub> to an intermediate time ''t''<sub>1</sub>, and then from ''t''<sub>1</sub> to the final time ''t''. Therefore,
:<math>U(t,t_0) = U(t,t_1)U(t_1,t_0).</math>

====Differential equation for time evolution operator====
We drop the ''t''<sub>0</sub> index in the time evolution operator with the convention that {{nowrap|''t''<sub>0</sub> {{=}} 0}} and write it as ''U''(''t''). The [[Schrödinger equation]] is
:<math> i \hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle,</math>
where ''H'' is the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. Now using the time-evolution operator ''U'' to write <math>|\psi(t)\rangle = U(t) |\psi(0)\rangle</math>, we have
:<math> i \hbar {\partial \over \partial t} U(t) | \psi (0) \rangle = H U(t)| \psi (0)\rangle.</math>

Since <math>|\psi(0)\rangle</math> is a constant ket (the state ket at {{nowrap|''t'' {{=}} 0}}), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation
:<math> i \hbar {\partial \over \partial t} U(t) = H U(t).</math>

If the Hamiltonian is independent of time, the solution to the above equation is<ref>Here we use the fact that at {{nowrap|''t'' {{=}} 0}},  ''U''(''t'') must reduce to the identity operator.</ref>
:<math> U(t) = e^{-iHt / \hbar}.</math>

Since ''H'' is an operator, this exponential expression is to be evaluated via its [[Taylor series]]:
:<math> e^{-iHt / \hbar} = 1 - \frac{iHt}{\hbar} - \frac{1}{2}\left(\frac{Ht}{\hbar}\right)^2 + \cdots .</math>

Therefore,
:<math>| \psi(t) \rangle = e^{-iHt / \hbar} | \psi(0) \rangle.</math>

Note that <math>|\psi(0)\rangle</math> is an arbitrary ket. However, if the initial ket is an [[eigenstate]] of the Hamiltonian, with eigenvalue ''E'', we get:
:<math>| \psi(t) \rangle = e^{-iEt / \hbar} | \psi(0) \rangle.</math>

Thus we see that the eigenstates of the Hamiltonian are ''stationary states'': they only pick up an overall phase factor as they evolve with time.

If the Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as
:<math> U(t) = \exp\left({-\frac{i}{\hbar} \int_0^t H(t')\, dt'}\right),</math>

If the Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written as
:<math> U(t) = \mathrm{T}\exp\left({-\frac{i}{\hbar} \int_0^t H(t')\, dt'}\right),</math>
where T is [[time-ordering]] operator, which is sometimes known as the Dyson series, after [[Freeman Dyson]].

The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator.  Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static.  This is the Heisenberg picture (below).

==Heisenberg picture==

The Heisenberg picture is a formulation (made by [[Werner Heisenberg]] while on [[Heligoland]] in the 1920s) of [[quantum mechanics]] in which the operators ([[observables]] and others) incorporate a dependency on time, but the [[quantum state|state vector]]s are time-independent.

===Definition===
In the Heisenberg picture of quantum mechanics the state vector, <math> |\psi \rang </math>, does not change with time, and an observable ''A'' satisfies
{{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 ''H'' is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and <nowiki>[•,•]</nowiki> denotes the [[commutator]] of two operators (in this case ''H'' and ''A''). Taking expectation values 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.  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. This approach also has a more direct similarity to [[classical physics]]: by replacing the commutator above by the [[Poisson bracket]], the '''Heisenberg equation''' becomes an equation in [[Hamiltonian mechanics]].

===Derivation of Heisenberg's equation===
The [[expectation value]] of an observable ''A'', which is a [[Hermitian]] [[linear operator]] for a given state <math>|\psi(t)\rang </math>, is given by

<math display="block"> \lang A \rang _t = \lang \psi (t) | A | \psi(t) \rang.</math>

In the [[Schrödinger picture]], the state <math>|\psi\rang </math> at time ''t'' is related to the state <math>|\psi\rang </math> at time 0 by a unitary [[time-evolution operator]], <math>U(t)</math>:
<math display="block"> |\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 display="block"> U(t) = e^{-iHt / \hbar} ,</math>
where ''H'' is the Hamiltonian and ħ is the [[reduced Planck constant]]. Therefore,

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

Define, then,
<math display="block"> A(t) := e^{iHt / \hbar} A e^{-iHt / \hbar} .</math>

It follows that
<math display="block">\begin{align}
 \frac{d}{dt} A(t) &= \frac{i}{\hbar} H e^{iHt / \hbar} A e^{-iHt / \hbar} + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right) e^{-iHt / \hbar} + \frac{i}{\hbar} e^{iHt / \hbar} A \cdot (-H) e^{-iHt / \hbar} \\
 &= \frac{i}{\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} \\
 &= \frac{i}{\hbar} \left( H A(t) - A(t) H \right) + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right)e^{-iHt / \hbar} .
\end{align}</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 exp(−''iHt''/''ħ'') commutes with ''H''.

Thus
<math display="block"> \frac{d}{dt} A(t) = \frac{i}{\hbar} [H, A(t)] + e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right)e^{-iHt / \hbar} ,</math>
whence the above Heisenberg equation of motion emerges, 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'',&nbsp;''Y''] is the [[commutator]] of two operators and is defined as {{math|1=[''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 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{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 display="block"> [A,H] \leftrightarrow  i\hbar\{A,H\} </math>
In classical mechanics, for an ''A'' with no explicit time dependence,
<math display="block"> \{A,H\} = \frac{d}{dt}A \,, </math>
so, again, the expression for ''A''(''t'') is the Taylor expansion around ''t'' = 0.

===Commutator relations===
Commutator relations may look different from 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>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.

==Interaction picture==

The interaction Picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian".

===Definition===

Operators and state vectors in the interaction picture are related by a change of basis ([[unitary transformation]]) to those same operators and state vectors in the Schrödinger picture.

To switch into the interaction picture, we divide the Schrödinger picture [[Hamiltonian (quantum mechanics)|Hamiltonian]] into two parts,
{{Equation box 1
|indent =:
|equation = <math>H_S = H_{0,S} + H_{1, S} ~.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that <math>H_{0,S}</math> is well understood and exactly solvable, while <math>H_{1,S}</math> contains some harder-to-analyze perturbation to this system.

If the Hamiltonian has ''explicit time-dependence'' (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with <math>H_{1,S}</math>, leaving <math>H_{0,S}</math> time-independent. We proceed assuming that this is the case. If there ''is'' a context in which it makes sense to have <math>H_{0,S}</math> be time-dependent, then one can proceed by replacing <math>e^{\pm i H_{0,S} t/\hbar}</math> by the corresponding [[Schrödinger picture|time-evolution operator]] in the definitions below.

====State vectors====

A state vector in the interaction picture is defined as<ref>[http://www.nyu.edu/classes/tuckerman/stat.mechII/lectures/lecture_21/node2.html The Interaction Picture], online lecture notes from New York University (Mark Tuckerman)</ref>
{{Equation box 1
|indent =:
|equation =  <math> | \psi_{I}(t) \rangle = e^{i H_{0, S} t / \hbar} | \psi_{S}(t) \rangle ~,</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where <math>| \psi_{S}(t) \rangle </math> is the same state vector as in the Schrödinger picture.

====Operators====

An operator in the interaction picture is defined as
{{Equation box 1
|indent =:
|equation =  <math>A_{I}(t) = e^{i H_{0,S} t / \hbar} A_{S}(t) e^{-i H_{0,S} t / \hbar}.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
Note that <math>A_S(t)</math> will typically not depend on ''t'', and can be rewritten as just <math>A_S</math>. It only depends on ''t'' if the operator has "explicit time dependence", for example due to its dependence on an applied, external, time-varying electric field.

=====Hamiltonian operator=====

For the operator <math>H_0</math> itself, the interaction picture and Schrödinger picture coincide,
:<math>H_{0,I}(t) = e^{i H_{0,S} t / \hbar} H_{0,S} e^{-i H_{0,S} t / \hbar} = H_{0,S} .</math>
This is easily seen through the fact that operators [[commutativity|commute]] with differentiable functions of themselves. This particular operator then can be called ''H''<sub>0</sub> without ambiguity.

For the perturbation Hamiltonian ''H''<sub>1,''I''</sub>, however,
:<math>H_{1,I}(t) = e^{i H_{0,S} t / \hbar} H_{1,S} e^{-i H_{0,S} t / \hbar} ,</math>
where the interaction picture perturbation Hamiltonian becomes a time-dependent Hamiltonian—unless [''H''<sub>1,s</sub>, ''H''<sub>0,s</sub>] = 0 .

It is possible to obtain the interaction picture for a time-dependent Hamiltonian ''H''<sub>0,s</sub>(''t'') as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by    ''H''<sub>0,s</sub>(''t''), or more explicitly with a time-ordered exponential integral.

=====Density matrix=====

The [[density matrix]] can be shown to transform to the interaction picture in the same way as any other operator. In particular, let <math>\rho_I</math> and <math>\rho_S</math> be the density matrix in the interaction picture and the Schrödinger picture, respectively. If there is probability <math>p_n</math> to be in the physical state <math>|\psi_n\rang</math>, then

:<math>\rho_I(t) = \sum_n p_n(t) |\psi_{n,I}(t)\rang \lang \psi_{n,I}(t)| = \sum_n p_n(t) e^{i H_{0, S} t / \hbar}|\psi_{n,S}(t)\rang \lang \psi_{n,S}(t)|e^{-i H_{0, S} t / \hbar}  = e^{i H_{0, S} t / \hbar} \rho_S(t) e^{-i H_{0, S} t / \hbar}.</math>

===Time-evolution equations===

====States====

Transforming the [[Schrödinger equation]] into the interaction picture gives:

:<math> i \hbar \frac{d}{dt} | \psi_{I} (t) \rang = H_{1, I}(t) | \psi_{I} (t) \rang. </math>

This equation is referred to as the '''[[Julian Schwinger|Schwinger]]–[[Sin-Itiro Tomonaga|Tomonaga]] equation'''.

====Operators====

If the operator <math>A_{S}</math> is time independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for <math>A_I(t)</math> is given by:

:<math> i\hbar\frac{d}{dt}A_I(t)=\left[A_I(t),H_0\right].\;</math>

In the interaction picture the operators evolve in time like the operators in the [[Heisenberg picture]] with
the Hamiltonian <math>H'=H_0</math>.

====Density matrix====

Transforming the Schwinger–Tomonaga equation into the language of the [[density matrix]] (or equivalently, transforming the [[Density Matrix#The Von Neumann equation for time evolution|von Neumann equation]] into the interaction picture) gives:

:<math> i\hbar \frac{d}{dt} \rho_I(t) = \left[ H_{1,I}(t), \rho_I(t)\right].</math>

===Existence===

The interaction picture does not always exist. In interacting quantum field theories, [[Haag's theorem]] states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. {{math|''H'' {{=}} ''H''<sub>0</sub> + ''V''}}, in the interaction picture it does, at least, if {{math|''V''}} does not commute with {{math|''H''<sub>0</sub>}}, since
:<math>H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_0}\,V\,e^{{(-i/\hbar})tH_0}</math>.

==Comparison of pictures==

The Heisenberg picture is closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly correspond to classical [[Poisson bracket]]s).
The Schrödinger picture, the preferred formulation in introductory texts, is easy to visualize in terms of [[Hilbert space]] rotations of state vectors, although it lacks natural generalization to Lorentz invariant systems. The Dirac picture is most useful in nonstationary and covariant perturbation theory, so it is suited to [[quantum field theory]] and [[many-body theory|many-body physics]].

===Summary comparison of evolutions===
{{Pictures in quantum mechanics}}

===Equivalence===
It is evident that the expected values of all observables are the same in the Schrödinger, Heisenberg, and Interaction pictures,
:<math>\langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle   =\langle\psi_I(t)\mid A_I(t)\mid\psi_I(t)\rangle ~,         </math>
as they must.

==See also==
*[[Hamilton–Jacobi equation]]
*[[Bra-ket notation]]

==Notes==
{{reflist|group="note"}}
{{reflist}}

==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) p.&nbsp;430-1 {{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.

{{Quantum mechanics topics}}

{{DEFAULTSORT:DYNAMICAL PICTURE}}
[[Category:Quantum mechanics]]