# Time Dependence in Pictures of QM

jonbones
I have a question about time dependence in the different pictures of QM. In the Schrodinger Picture, I've read that the time dependence is in the state vector, but one can construct Hamiltonians and other operators that are time dependent in the Schrodinger Picture. For instance, one can construct a time dependent potential in Schrodinger wave mechanics, and one would not be using the Interaction Picture. (But the equation would probably have to be solved numerically.)

Then could states in the Heisenberg Picture be constructed so that they are time dependent as well without invoking the Interaction Picture? I would think it's possible, but I assume it would usually lead to a difficult problem to solve and rarely done. Are there any references that talk about handling such cases?

Gold Member
"Pictures" are a bit like coordinate systems: they have no deep physical meaning and you should usually just work in whatever picture is easiest to handle mathematically.

Now, the Schroeding and Heisenberg pictures are just the two "extremes", you can move (so to speak) time dependences around more or less as you want to simplify your equations and whenever you do that you are in the "interaction picture".
A good example would be if you have a hamiltonian with a time-varying electromagnetic field, sometimes you can then simplify things considerably by moving into a picture where the whole hamiltonian is rotating "with" the field.
(and if you then throw away some higher order terms you get the rotating wave approximation)

Note that you are always free to go back using another transformation.

Hence, this is not really very different from moving from e.g. carteesian coordinates to cylindrical coordinate to simply a PDE solved over a circle.

jonbones
I think I follow you. So, for instance, in the rotating wave approx. where E is the electric field operator in the heisenberg picture,

E = $\sum$kl(akleklfkl(x)exp(-i$\omega$t) + a+kle*klf*kl(x)exp(i$\omega$t))

with <a|E2|a> as the the energy density expectation value, I could choose a state

|a(t)> = cos(t)|0> + sin(t)|2>

and compute <a(t)|E2|a(t)> and still get a "correct" answer. Or do I have to adjust the definition of E as well?

The time-dependent Schrödunger equation

$$i\partial_0\,|\psi,t\rangle = H\,|\psi,t\rangle$$

can be solved using the time evolution operator

$$U(t) = e^{-iHt}$$
$$|\psi,t\rangle = U(t)\,|\psi,0\rangle$$

which you see by differentiating

$$i\partial_0\,U(t) = H\,U(t)$$

Now for any operator A where you are interested in eigenvalues, expectation values or something like that and which you apply to a state |ψ>

$$A\,|\psi,t\rangle$$

you can insert any unitary operator w/o changing physics:

$$A\,|\psi,t\rangle\;\to\;\Omega A \Omega^\dagger \Omega |\psi,t\rangle$$

This is like a basis transformation which - as in linear algebra - acts both on the states (vectors) and on the operators (matrices). So you can introduce transformed states and operators as follows

$$|\psi,t\rangle\;\to\;{}^\Omega|\psi,t\rangle = \Omega |\psi,t\rangle$$
$$A\;\to\;{}^\Omega A = \Omega A \Omega^\dagger$$

In that way you can introduce time-independent coordinate transformations like translations (where Ω is represented as eiap using the momentum operator p) or rotations (where Ω is represented as eiθL using the angular momentum operator L).

But the unitary operator Ω is arbitrary and can be time dependent as well. So we can use something like

$$\Omega(t) = e^{-iGt}$$