February 2022 – Quantum Mechanical Animations

Double-Well Potential

Michael DoeringFebruary 22, 2022February 10, 2023Leave a comment

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?

Time-dependent potentials 1

Michael DoeringFebruary 21, 2022February 10, 20231 Comment

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).

Wave Packet on a Potential Step 8

Michael DoeringFebruary 19, 2022February 10, 20238 Comments

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.