Visualizing wave-functions.

Here is a post to help you picture what is going on with a quantum wave function as a function of time. These are gifs that help you see the oscillations of particular quantum states. The first two are energy eigenstates, so their oscillations are "pure". The 3rd gif shows a mixed (non-stationary) state where the time dependence involves two different frequencies and thus exhibits complex interference patterns that change as a function of time.

Fig 1. The wave-functions shown above are for the ground state and 1st excited state, respectively, of an electron in a harmonic oscillator potential. The wavefunction is complex, hence there are two lines in each  plot. The horizontal axis is x in nm. The vertical axis is wave-function amplitude in \(nm^{-1/2}\). The ground state and 1st excited state oscillation for an electron in an infinite square well would be similar, but with a different spatial shape.

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 Fig 2. Suppose an electron in an HO potential is in a quantum state such that its wave-function is an equal mix of the ground and 1st excited states. Shown here is the complex wave-function for that non-stationary state. The vertical axis is wave-function amplitude in \(nm^{-1/2}\).