@JamesHMcLaren What you are seeing is the probability amplitude of where the particle might be; The wave does interact with itself! The initial localization is a hand-crafted solution in order for it to have some initial shape. As it progresses and encounters the potential barrier, and will be reflected which causes the complicated interaction.

@brandonb_phy Yes I assumed that’s what you were doing. Part way through it looks like a classic beat frequency pattern for a while, which of course implies a change in frequency of the probability wave if that’s what’s happening. But why would the frequency change? Losing energy to the walls? Did you model that? Really neat to think about anyway.

@JamesHMcLaren All I did was implement the solution to the time-independent Schrödinger equation! The rest of the behavior evolves out of the solution for the time-dependent equation. (See photo)

In general, the wave will 'want' to dissipate but because there are potential barriers (that in this simulation can not penetrate), the whole wave will spread out in the box and self-interact. So you are seeing constructive and destructive interference. The wave equation itself is a combination of stationary waves that will have different frequencies. Because these are all re-calculated for each time frame, it is the case that the frequency of the probability density will change with time as well.