Unitary evolution while rapid change of potential?

In summary, the conversation discusses the density evolution of a particle transitioning between two potential wells, and whether the time evolution of the wavefunction should be time-symmetric. The use of time-ordering operators and Feynman path integrals is mentioned, and the concept of causality in quantum mechanics is brought up. The discussion also mentions examples where the forward and backward propagators do not match, and how this can affect the behavior of the particle. The concept of CPT symmetry in quantum field theory is also briefly mentioned.
  • #1
jarekduda
82
5
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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  • #2
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).
 
  • #3
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.
 
  • #4
quantum theory is, however, by construction not acausal. So your claim cannot be right.
 
  • #6
Hmm, the wavefunction may evolve as per Vanhees71 equation, but what about the conjugate? Don't we require both?
 
  • #7
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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  • #8
jarekduda said:
Could you elaborate on that?

Causality in quantum mechanics is a very complex topic, for example in Wheeler's delayed choice experiment or delayed choice quantum eraser.
And physicists generally believe that physics (QFT) is CPT symmetric (CPT theorem) - again suggesting some time symmetry of the discussed evolution ...
The delayed-choice experiments in no way invalidate the strict causality of QT, but that's another rather metaphysical topic, which I thus don't like to comment.
 

What is unitary evolution?

Unitary evolution refers to the continuous change of a system over time, without any external influences or disturbances. In other words, it is the natural evolution of a system based on its own inherent properties and dynamics.

What is rapid change of potential?

Rapid change of potential refers to a sudden or significant shift in the potential energy of a system. This can occur due to changes in the system's environment, interactions with other systems, or other external factors.

How are unitary evolution and rapid change of potential related?

Unitary evolution and rapid change of potential are two concepts that are often discussed together in the field of physics. While unitary evolution describes the natural evolution of a system, rapid change of potential refers to external influences that can cause significant changes in the system's behavior.

What are some examples of unitary evolution while rapid change of potential?

One example of unitary evolution while rapid change of potential is the evolution of a quantum system in a changing external field. Another example is the evolution of a biological system in response to environmental changes.

What are the implications of unitary evolution while rapid change of potential for scientific research?

The study of unitary evolution while rapid change of potential is important for understanding how systems evolve and adapt in response to external influences. This knowledge can be applied in various fields, such as physics, biology, and environmental science, to better understand and predict the behavior of complex systems.

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