Harmonic oscillator momentum distribution

[andresordonez]

Hi, I don't understand why the momentum probability distribution of the quantum mechanical oscillator has the same shape as the position probability distribution (with peaks at the extremes), I mean, I understand the mathematics but I don't understand the concept.

This is my reasoning (which I'm sure is wrong, but I don't see why)

The position probability distribution has peaks (for high energies) approximately at the positions corresponding to the amplitude of the classical oscillator. The physical explanation of this, is that the oscillator spends more time in the extreme positions because there its velocity is close to zero.

But then what this means is that if I measured (classically) many times the position of the oscillator (at a fixed energy) I would get that most of the measurements would have the oscillator in the extreme positions (where the momentum would be close to zero) right?

How come then, that if I measure many times the momentum I get most of the measurements with a high momentum?? (in other words, that the momentum probability distribution has peaks at the extremes)

Thanks

 

[andresordonez]

Well, this is what happens:

When the oscillator is near the extreme positions, the momentum changes rapidly (the acceleration is maximum at the extreme positions) and the momentum measurements are distributed over a wide range, that is, while the position changes slowly, the momentum changes rapidly, so if you have, say, 3 consecutive measurements of the position near the extremes, you would get the first measurement of the momentum (corresponding to the first measurement of the position) far from zero, the 2nd measurement close to zero, and the third far from zero again. The opposite would be true when the position is close to zero. It's kinda hard to visualize (at least it was for me), so I'll attach a plot where it can be seen clearly.

The vertical bars correspond to a measurement, and I considered $$\pm$$[1,.9] to be near the extremes and [-.1,.1] to be near zero.

The purple line corresponds to the position and the blue line to the momentum.