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This animation shows the reflection/transmission on a potential step:

Wave packet colliding with a potential step.

Items to be discussed:

  • Real and imaginary part of the Gaussian wave packets move with a different speed than |Ψ(x,t)|². This Illustrates phase velocity vs. group velocity
  • Once the packet hits the step (also shown in animation), parts of it are reflected, and parts are transmitted, which helps understand the respective undergraduate standard formulae. One could even position an observer at some positive x-value, that integrates |Ψ(x,t)|² over time to measure how much of the wave packet is transmitted.
  • The last thought can then also be set into relation with conservation of probability and the continuity equation.

In quantum mechanical scattering, there can be bound states and resonances. Both phenomena are connected through analytic continuation of the scattering energy or momentum. This animation shows the connection by tracing poles as the depth of a spherical well gets deeper and deeper.

Poles of the T-matrix in the complex momentum plane with varying potential depth.

The above movie exhibits many phenomena of scattering. In the following, a screen shot illustrates some key moments in the animation:

Key moments for the pole trajectories and their transition from resonances to bound states

The left column shows the S-wave T-matrix, |t0|, in the complex-momentum plane (arb. units), the right column shows the phase shift. (a) For a shallow potential, there is no bound state, but only virtual state 1 and resonances 2 and 3. In (b), infinite scattering length is reached which motivates a discussion of universality. In (c), pole 1 became a deeply bound state. Pole 2 and its mirror pole 2’ have met on the imaginary k-axis and then separated again as virtual states 2 and 2', with 2 on its way to become a bound state and 2' a deeper-bound virtual state. Such intriguing S-wave pole trajectories
have only been discovered ten years ago. Levinson’s theorem is also nicely illustrated within this animation.

This is a time-independent potential with a Gaussian wave packet initially in the left pit. Over time, parts of the wave packet populate the other well to the right. Parts of the wave packet start an oscillation across the entire double-well.

Double-well potential with a wave packet initially confined to the left pit. The fact that there is a non-zero chance to find the particle to the right is due to tunneling through the central barrier. It is also due to the fact that the Gaussian wave packet does possesses high-momentum components with an energy E>0 that can just cross the central barrier "classically".

Further question: Can we understand the formation of almost standing wave in the right pit?

The time-dependent solution of the Schrödinger equation, for a time-dependent potential, can be obtained with the Crank-Nicholson method. Some examples are shown below.


A Gaussian wave packet in a time-dependent potential (based on an original ideal of GW graduate A. Lange).

The Gaussian wave packet in this example maintains its shape, even if the harmonic oscillator potential has a nontrivial time dependence; is there a proof for that?

Another Gaussian wave packet in another potential (based on an original idea by Andy Sargent).