I was watching this crash course by Geek Lesson on Quantum Mechanics specifically for Quantum Harmonic Oscillator and [at 1:54:54] when video shows the plot of probability density for different states in Quantum Harmonic Oscillator the probability density is shown more at ends. The speaker gives explanation like this

When something is wiggling in a quantum simple harmonic oscillator it is gonna spin more time in the end, because it is going fast in the middle and slow at the ends. Other thing to notice is that probability density falls off asymptomatically but is never zero for any finite value of x.

Unfortunately, I couldn't understand it. Can someone explain this more clearly ?

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Just consider what happens to a classical simple harmonic oscillator. The object moves fast in the middle, goes to the outermost position, stops there, then goes back. Since it stops at the outermost position, it's much more likely to be found near that position. I.e. if we were to take a photo of the oscillator, we'd most likely find the object in the outermost position.

Now this is basically the same in the quantum SHO, just with the specific features added like oscillations of probability density and exponential tails. In particular, in the limit of high excitations we recover the classical probability density.

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