For operator $A$, the famous equation from Heisenberg picture (see: Time Variation of Expectation Value ) stated that, since $$\frac{d A}{dt}=\frac{i}{\hbar}\left[H,A\right]+\left(\frac{\partial A}{\partial t}\right) ,$$ if $[H,A]=0$ and $A$ does not explicitly dependent on time, then the expectation value of $\langle A\rangle$ was a constant.

However, I'm wondering if that's true only for system possible to be described by the Heisenberg picture. (Possible mistake I.) Is it possible for it to be different in other pictures?

In that case, do we need to prove that all system can be written in either Heisenberg picture / Schrödinger picture? Or the "minimal condition for the system" be able to expressed in either Heisenberg picture / Schrödinger picture?)


Example 1: In Rabi cycle, the energy states and expectation $\langle H\rangle$ definitely changes.

Example 2: In interaction picture, like Jaynes Cumming model, with the presents of $\hat{H}_{int}$, things also become complicated. (Is it guaranteed that $[\hat{H}_{int},\hat{H}_{atom}]\neq 0$ because of the physical interpretation?)

What's the necessary and sufficient condition for the expectation value of $A$ to be constant? Does the statement in Heisenberg true for all the system? Or if there's any more generalized statement?


1 Answer 1

  1. The dynamics of every quantum system can be expressed in both the Schrödinger and Heisenberg pictures, and the two descriptions are formally equivalent. In other words, for whatever calculation you would like to perform, the outcome of the calculation does not depend on the chosen picture. (This situation is quite analogous to mechanics, where you are allowed to choose any inertial coordinate system for your calculations.)

  2. The expectation value of an operator depends on the state of your system. We must thus distinguish two different questions. (a) Under which condition is the expectation value of $A$ constant, no matter which state the system is in? (b) Under which condition is the expectation value of $A$ constant, given that the initial state of the system is $|\psi\rangle$?

  3. Both questions can be answered in the Heisenberg picture using the time evolution equation you provided.
    (a) The expectation value of $A$ is constant in all states if and only if $\frac{\mathrm dA}{\mathrm dt} = 0$, i.e., $$ \frac{\mathrm i}{\hbar} [H,A] + \frac{\partial A}{\partial t} = 0 . $$ (b) The expectation value $\langle A \rangle_\psi \equiv \langle \psi | A | \psi \rangle$ is constant if and only if $\langle \frac{\mathrm dA}{\mathrm dt} \rangle_\psi = 0$, i.e., $$ \frac{\mathrm i}{\hbar} \langle \psi | [H,A] | \psi \rangle + \langle \psi | \frac{\partial A}{\partial t} | \psi \rangle = 0 . $$

  4. In Rabi oscillations, the Hamiltonian is explicitly time-dependent, therefore $\frac{\mathrm dH}{\mathrm dt} = \frac{\partial H}{\partial t} \neq 0$.


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