Dynamical pictures

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 state vectors are time-independent.

Definition

In the Heisenberg picture of quantum mechanics the state vector, ${\displaystyle |\psi ⟩}$, does not change with time, and an observable A satisfies

${\displaystyle \frac{d}{dt}A(t)=\frac{i}{\hslash }[H,A(t)]+\frac{\mathrm{\partial }A(t)}{\mathrm{\partial }t},}$

where H is the Hamiltonian and [•,•] 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 Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for 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 ${\displaystyle |\psi (t)⟩}$, is given by

$${\displaystyle ⟨A{⟩}_t=⟨\psi (t)|A|\psi (t)⟩.}$$

In the Schrödinger picture, the state ${\displaystyle |\psi ⟩}$ at time t is related to the state ${\displaystyle |\psi ⟩}$ at time 0 by a unitary time-evolution operator, ${\displaystyle U(t)}$: $${\displaystyle |\psi (t)⟩=U(t)|\psi (0)⟩.}$$ If the Hamiltonian does not vary with time, then the time-evolution operator can be written as $${\displaystyle U(t)=e^{-iHt/\hslash },}$$ where H is the Hamiltonian and ħ is the reduced Planck constant. Therefore,

$${\displaystyle ⟨A{⟩}_t=⟨\psi (0)|e^{iHt/\hslash }A{e}^{-iHt/\hslash }|\psi (0)⟩.}$$

Define, then, $${\displaystyle A(t):=e^{iHt/\hslash }A{e}^{-iHt/\hslash }.}$$

It follows that $${\displaystyle \begin{array}{cc}\frac{d}{dt}A(t)& =\frac{i}{\hslash }H{e}^{iHt/\hslash }A{e}^{-iHt/\hslash }+e^{iHt/\hslash }\left(\frac{\mathrm{\partial }A}{\mathrm{\partial }t}\right)e^{-iHt/\hslash }+\frac{i}{\hslash }{e}^{iHt/\hslash }A\cdot (-H)e^{-iHt/\hslash }\\ & =\frac{i}{\hslash }{e}^{iHt/\hslash }\left(HA-AH\right)e^{-iHt/\hslash }+e^{iHt/\hslash }\left(\frac{\mathrm{\partial }A}{\mathrm{\partial }t}\right)e^{-iHt/\hslash }\\ & =\frac{i}{\hslash }\left(HA(t)-A(t)H\right)+e^{iHt/\hslash }\left(\frac{\mathrm{\partial }A}{\mathrm{\partial }t}\right)e^{-iHt/\hslash }.\end{array}}$$

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 $${\displaystyle \frac{d}{dt}A(t)=\frac{i}{\hslash }[H,A(t)]+e^{iHt/\hslash }\left(\frac{\mathrm{\partial }A}{\mathrm{\partial }t}\right)e^{-iHt/\hslash },}$$ 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, 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 standard operator identity, $${\displaystyle e^{B}A{e}^{-B}=A+[B,A]+\frac{1}{2!}[B,[B,A]]+\frac{1}{3!}[B,[B,[B,A]]]+\cdots .}$$ which implies $${\displaystyle A(t)=A+\frac{it}{\hslash }[H,A]-\frac{t^2}{2!{\hslash }^2}[H,[H,A]]-\frac{i{t}^3}{3!{\hslash }^3}[H,[H,[H,A]]]+\dots }$$

This relation also holds for classical mechanics, the classical limit of the above, given the correspondence between Poisson brackets and commutators, $${\displaystyle [A,H]\leftrightarrow i\hslash \{A,H\}}$$ In classical mechanics, for an A with no explicit time dependence, $${\displaystyle \{A,H\}=\frac{d}{dt}A,}$$ 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 x(t₁), x(t₂), p(t₁) and p(t₂). The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator,

${\displaystyle H=\frac{p^2}{2m}+\frac{m{\omega }^2{x}^2}{2}}$ ,

the evolution of the position and momentum operators is given by:

${\displaystyle \frac{d}{dt}x(t)=\frac{i}{\hslash }[H,x(t)]=\frac{p}{m}}$ ,

${\displaystyle \frac{d}{dt}p(t)=\frac{i}{\hslash }[H,p(t)]=-m{\omega }^{2}x}$ .

Differentiating both equations once more and solving for them with proper initial conditions,

${\displaystyle \dot{p}(0)=-m{\omega }^2{x}_0,}$

${\displaystyle \dot{x}(0)=\frac{p_0}{m},}$

leads to

${\displaystyle x(t)=x_0\cos (\omega t)+\frac{p_0}{\omega m}\sin (\omega t)}$ ,

${\displaystyle p(t)=p_0\cos (\omega t)-m\omega x_0\sin (\omega t)}$ .

Direct computation yields the more general commutator relations,

${\displaystyle [x(t_1),x(t_2)]=\frac{i\hslash }{m\omega }\sin (\omega t_2-\omega t_1)}$ ,

${\displaystyle [p(t_1),p(t_2)]=i\hslash m\omega \sin (\omega t_2-\omega t_1)}$ ,

${\displaystyle [x(t_1),p(t_2)]=i\hslash \cos (\omega t_2-\omega t_1)}$ .

For ${\displaystyle t_1=t_2}$, 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 into two parts,

${\displaystyle H_S=H_{0,S}+H_{1,S}.}$

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 ${\displaystyle H_{0,S}}$ is well understood and exactly solvable, while ${\displaystyle H_{1,S}}$ 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 ${\displaystyle H_{1,S}}$, leaving ${\displaystyle H_{0,S}}$ time-independent. We proceed assuming that this is the case. If there is a context in which it makes sense to have ${\displaystyle H_{0,S}}$ be time-dependent, then one can proceed by replacing ${\displaystyle e^{\pm i{H}_{0,S}t/\hslash }}$ by the corresponding time-evolution operator in the definitions below.

State vectors

A state vector in the interaction picture is defined as

${\displaystyle |{\psi }_I(t)⟩=e^{i{H}_{0,S}t/\hslash }|{\psi }_S(t)⟩,}$

where ${\displaystyle |{\psi }_S(t)⟩}$ is the same state vector as in the Schrödinger picture.

Operators

An operator in the interaction picture is defined as

${\displaystyle A_I(t)=e^{i{H}_{0,S}t/\hslash }{A}_S(t)e^{-i{H}_{0,S}t/\hslash }.}$

Note that ${\displaystyle A_S(t)}$ will typically not depend on t, and can be rewritten as just ${\displaystyle A_S}$. 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 ${\displaystyle H_0}$ itself, the interaction picture and Schrödinger picture coincide,

${\displaystyle H_{0,I}(t)=e^{i{H}_{0,S}t/\hslash }{H}_{0,S}{e}^{-i{H}_{0,S}t/\hslash }=H_{0,S}.}$

This is easily seen through the fact that operators commute with differentiable functions of themselves. This particular operator then can be called H₀ without ambiguity.

For the perturbation Hamiltonian H_1,I, however,

${\displaystyle H_{1,I}(t)=e^{i{H}_{0,S}t/\hslash }{H}_{1,S}{e}^{-i{H}_{0,S}t/\hslash },}$

where the interaction picture perturbation Hamiltonian becomes a time-dependent Hamiltonian—unless [H_1,s, H_0,s] = 0 .

It is possible to obtain the interaction picture for a time-dependent Hamiltonian H_0,s(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by H_0,s(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 ${\displaystyle {\rho }_I}$ and ${\displaystyle {\rho }_S}$ be the density matrix in the interaction picture and the Schrödinger picture, respectively. If there is probability ${\displaystyle p_n}$ to be in the physical state ${\displaystyle |{\psi }_n⟩}$, then

${\displaystyle {\rho }_I(t)=\sum _n{p}_n(t)|{\psi }_{n,I}(t)⟩⟨{\psi }_{n,I}(t)|=\sum _n{p}_n(t)e^{i{H}_{0,S}t/\hslash }|{\psi }_{n,S}(t)⟩⟨{\psi }_{n,S}(t)|e^{-i{H}_{0,S}t/\hslash }=e^{i{H}_{0,S}t/\hslash }{\rho }_S(t)e^{-i{H}_{0,S}t/\hslash }.}$

Time-evolution equations

States

Transforming the Schrödinger equation into the interaction picture gives:

${\displaystyle i\hslash \frac{d}{dt}|{\psi }_I(t)⟩=H_{1,I}(t)|{\psi }_I(t)⟩.}$

This equation is referred to as the Schwinger–Tomonaga equation.

Operators

If the operator ${\displaystyle A_S}$ is time independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for ${\displaystyle A_I(t)}$ is given by:

${\displaystyle i\hslash \frac{d}{dt}{A}_I(t)=\left[A_I(t),H_0\right].}$

In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian ${\displaystyle H^{\prime }=H_0}$.

Density matrix

Transforming the Schwinger–Tomonaga equation into the language of the density matrix (or equivalently, transforming the von Neumann equation into the interaction picture) gives:

${\displaystyle i\hslash \frac{d}{dt}{\rho }_I(t)=\left[H_{1,I}(t),{\rho }_I(t)\right].}$

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. H = H₀ + V, in the interaction picture it does, at least, if V does not commute with H₀, since

${\displaystyle H_{\mathrm{i}\mathrm{n}\mathrm{t}}(t)\equiv e^{(i/\hslash )t{H}_0}V{e}^{(-i/\hslash )t{H}_0}}$.

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 brackets). 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 physics.

Summary comparison of evolutions

Equivalence

It is evident that the expected values of all observables are the same in the Schrödinger, Heisenberg, and interaction pictures,

${\displaystyle ⟨\psi \mid A(t)\mid \psi ⟩=⟨\psi (t)\mid A\mid \psi (t)⟩=⟨{\psi }_I(t)\mid A_I(t)\mid {\psi }_I(t)⟩,}$

as they must.