Heisenberg Picture with a time-dependent Schrödinger Hamiltonian

Question

So when the Hamiltonian is time-independent, we can define the Heisenberg state vectors by evolving the Schrödinger state vectors back in time:

$$ | \psi \rangle_H = \hat{U}^\dagger (t)|\psi(t) \rangle_S=e^{i\hat{H}t} |\psi(t)\rangle_S $$

and we define operators

$$ \hat{A}_H(t) = \hat{U}^\dagger (t) \hat{A}_S \hat{U}(t)$$

which gives us the Heisenberg equation: $$ \frac{d\hat{A}_H(t)}{dt} = -i[\hat{A}_H(t),\hat{H}]. $$

If, in the Schrödinger picture, we have a time-dependent Hamiltonian, the time evolution operator is given by

$$ \hat{U}(t) = T[e^{-i \int_0^t \hat{H}(t')dt'}] $$

If I define the Heisenberg operators in the same way with the time evolution operators and calculate $ dA_H(t)/dt $ I find

$$ \frac{d}{dt} \hat{A}_H(t)= \frac{d\hat{U}^\dagger(t)}{dt} \hat{A}_S\hat{U}(t) + \hat{U}^\dagger(t) \hat{A}_S \frac{d\hat{U}(t)}{dt} \\ = i \hat{U}^\dagger (t) \hat{H(t)} \hat{A}_S \hat{U}(t) - i\hat{U}^\dagger (t)\hat{A}_S\hat{H}(t)\hat{U}(t). $$

At this point, I am not sure how to proceed. I can't commute $\hat{H}(t)$ through $\hat{U}(t) $ because $[\hat{H}(t),\hat{H}(t')] \neq 0$. How do I show derive Heisenberg's equation for a time-dependent Hamiltonian?

Answer 1

You appear to be replicating all the steps that Dirac utilized with foresight to define his celebrated interaction picture. As indicated in the comment, the proper relationship is already in question 122687. The crucial point is that, as you implicitly noticed, the Heisenberg Hamiltonian is not the Schroedinger Hamiltonian, by contrast to the time-independent case.

That is, motion being a canonical transformation, the two hamiltonians are equivalent but, in general, not equal, $$ H_H=U^\dagger H_S U \neq H_S, $$ evident from the Dyson expansion.

Moreover, thinking of H as an observable, note $$ \frac{d H_H}{dt}= (\partial_t H_S)_H \neq 0, $$ not a constant of the motion.

Conversely, if you take your S-observable A without explicit time dependence, (think of x or p), your final equation is sound, and merely amounts to the customary convective term, $$ \frac{dA_H}{dt}= U^\dagger i[H_S,A_S]U = i[H_H,A_H], $$ as it should. Adding explicit time dependence yields the additional customary explicit term $(\partial_t A_S)_H$.

Answer 2

simply insert $UU^\dagger$ in between $H$ and $A_S$ in both term, here is the first term

$U^\dagger H A_sU=U^\dagger H U U^\dagger A_sU=H_HA_H$

Answer 3

From the Schrodinger Equation and rearranging under the assumption that the kets are time independent, we know $\frac{dU(t)}{dt} = -iH(t)U(t)$, or equivalently $\frac{dU^\dagger(t)}{dt} = iU^\dagger(t) H(t)$ (this is the crux of the problem here, so far as I can tell, and a piece that was incorrect in your original question)

Carrying out $\frac{dA_H(t)}{dt}$ as you did, we now have

$$ \frac{dA_H(t)}{dt} = i U^\dagger(t)H(t)A_SU(t) - i U^\dagger(t) A_S H(t)U(t) + \frac{\partial A_H(t)}{dt} \\ = U^\dagger(t)(i \left[H(t),A_S\right])U(t) + \frac{\partial A_H(t)}{dt}$$

When $U(t)$ does not commute with $H(t)$, I believe that this is as far as you can take things. If $U(t)$ does commute with $H(t)$, then we recover the Heisenberg equation and replace $A_S$ by $A_H(t)$, as we would expect.

Answer 4

From

$$ \frac{dU}{dt} = -iH(t) U(t) $$

I would write

$$ \frac{dU^{\dagger}}{dt} = \biggl(-iH(t) U(t)\biggr)^{\dagger} = iU^{\dagger}(t) H(t). $$

Notice that this is different from what you have.