Heisenberg picture

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In physics, the Heisenberg picture or Heisenberg representation<ref>Template:Cite web</ref> is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the states 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 Template:Math do not change with time, while observables Template:Mvar satisfy Template:Equation box 1{\partial t} \right)_\text{H} ,</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F9FFF7}} where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, Template:Mvar is the Hamiltonian and Template:Math denotes the commutator of two operators (in this case Template:Mvar and Template:Mvar). 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 basis change in Hilbert space. In some sense, the Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for 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 over the reduced Planck constant 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'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_Template:\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, Template:Math is the reduced Planck constant, and Template:Math is the imaginary unit. Given 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>, its expectation value 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> 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_Template:\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 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 <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_Template:\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_Template:\rm S \equiv A</math> is also time-independent, then the last term vanishes and <math display="block">\frac {d}{dt} A_{\rm H}(t)=\frac {i}{\hbar} [H, A_{\rm H} (t)] ,</math> where <math>A_Template:\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 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 correspondence between Poisson brackets 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 Template:Math is the Taylor expansion around Template:Math.

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 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 Template:Math, Template:Math, Template:Math and Template:Math. 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 Template:Math and Template: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 Template:Math, where Template:Math is the free Hamiltonian, Template:Pictures in quantum mechanics

See also

References

Template:Reflist

  • 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 [1]
  • 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 [2]
  • The computations for the hydrogen atom in the Heisenberg representation originally from a paper of Pauli [3]

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