Suppose I know(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

H \psi(x) = \left( -\frac{1}{2m} \Delta_x + V(x) \right) \psi(x) = E\psi(x).

[/tex]

Then

[tex]

\psi(x,t) = e^{-iEt}\psi(x)

[/tex]

solves the time-dependent Schrodinger equation

[tex]

\left( i \frac{\partial}{\partial t} + \frac{1}{2m} \Delta_x - V(x) \right)\psi(x,t) = 0.

[/tex]

I've done some computations, and it looks like

[tex]

\Psi(x,t) = e^{-imvx}e^{-imv^2t/2}\psi(x+vt)

[/tex]

is a solution to the time-dependent Schrodinger equation

[tex]

\left( i \frac{\partial}{\partial t} + \frac{1}{2m} \Delta_x - V(x+vt) \right)\Psi = 0.

[/tex]

I have a couple of questions about this:

- What is going on here physically? That is, what are those two phase factors telling me?
- What does this mean the propagator is? When [itex]H[/itex] is time-independent, [itex]U(t) = e^{-iEt}[/itex]...but what is it in the time-dependent case? Is there a neat little formulation of it?

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# Propagators for time-dependent Hamiltonians

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