# Unitary evolution while rapid change of potential?

• A
Imagine potential well which in t=0 switches to a different potential well (instantly), like in the picture below.
So in negative times the wavefunction density should tend to be localized in the first well, in positive times to be localized in the second well.

https://dl.dropboxusercontent.com/u/12405967/ehr.jpg [Broken]

The question is what is density evolution of transition between these two wells?
Shouldn't unitary evolution be time-symmetric, so that the density prepares to the switch before it actually happens?

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vanhees71
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2019 Award
Why should it be time symmetric? The formal solution of the time-evolution equation (here for the time evolution operator for the Schrödinger picture)
$$\hat{U}(t)=T_c \exp \left [-\mathrm{i} \int_0^t \mathrm{d} t' \hat{H}(t') \right],$$
which is not time-symmetric at all (##T_c## is the time-ordering operator, which orders operator products at different times such that they are ordered from right to left in terms of time arguments).

So try to perform the propagator you have written from -infinity to 0 (forward) and from +infinity to 0 (backward), you will get different answers.
Think about Feynman path integrals, it is time-symmetric.

The simulation from the picture above comes from (normalized) euclidean path integrals: (diffusion) assuming Boltzmann distribution among paths (fig. 5.1 here), which is time-symmetric diffusion (so called Bernstein process).
What is funny about it is that Ehrenfest equations lead to 2nd Newton law with opposite sign here: to prepare for the switch, the packet needs to first accelerate uphill, then decelerate downhill.

vanhees71
Gold Member
2019 Award
quantum theory is, however, by construction not acausal. So your claim cannot be right.

Hmm, the wavefunction may evolve as per Vanhees71 equation, but what about the conjugate? Don't we require both?

Indeed, at least from the point of view of euclidean path integrals,
rho(t,x) = < phi(t,x) | psi(t,x) >
where phi is a result of propagator from -infinity to t forward, psi from +infinity to t backward.
Usually phi=psi, unless for example a rapid change of potential like above.

Another example of phi != psi is conductance on a torus by assuming a gradient of potential: trajectories prefer circulation in a fixed direction (https://dl.dropboxusercontent.com/u/12405967/conductance.nb [Broken]).

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vanhees71