Hello everybody, recently in my quantum mechanical course we were introduced to the concept of the quantum harmonic oscillator. My question is: is there a physical significance attached to the fact that the classical turning points overlap with the sign change of the second derivative of the quantum mechanical wavefunction (which in turn occurs where the wavefunction intercepts the potential curve)? That is both geometrically and algebraically it is easy to see that these points will be the same for all states (ground state and all excited states) but can this be interpreted physically somehow? As you can see in the picture below, the quadratic/parabolic/harmonic function intersects each wavefunction at an inflection point where the concavity of the curve changes (which indicates a change in the second derivative of the wavefunction at that particular point).(adsbygoogle = window.adsbygoogle || []).push({});

http://postimage.org/image/a4q43w97v/

It might be a trivial question or this might not have a physical significance at all, but anyway it troubled me and my lecturer as well, who said that he had never thought about that. I haven't given much thought about it either, nor have I settled down to play around with the maths of it, but if any of you has a clue as to which direction I should be heading towards that would save me much valuable time.

Thanks in advance for your time and effort, if you bother replying to this post.

Achilles

P.S. For some odd reason I can't manage to incorporate the image in the text using an image hosting website. Find it attached or simply google it, sketch it, look it up in a book or your notes. It's just a casual random quantum harmonic oscillator potential and wavefunction plot.

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# Quantum Harmonic Oscillator Question

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