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Quantum Harmonic Oscillator Question

  1. Feb 9, 2012 #1
    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).
    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.

    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.

    Attached Files:

  2. jcsd
  3. Feb 9, 2012 #2
    That is definitely an interesting pattern. It might have to do with the probability of actually finding a particle at a certain distance away from the nucleus as you increase in orbitals.
    If you can imagine the pattern in that graph continuing, the distance away from the nucleus begins to sort of "level-off" in that inverse sort of way where it keeps increasing by a tinier and tineir amount but never actually reaches a stop.
    So basically, as you increase the orbital number, the distance that the most probable location grows by becomes less and less. I might be getting it the other way around, but there is some correlation like that.

    I was thinking like this, though I guess it may be different like, "spin states" of the same orbital and as you icnrease that, the same orbital levels off.


    I guess there isn't a leveling off from one orbital to another, but perhaps in the different states of one orbital or subshells.
    Someone with more QM background can probably help out with this.
    Last edited: Feb 9, 2012
  4. Feb 9, 2012 #3


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    Schroedinger's (time-independent) equation is: [tex]\frac{-\hbar^2}{2m}\frac{d^2 \psi}{dx^2}+V(x)\psi = E\psi[/tex]

    One can see that from the equation, it's quite obvious that the second derivative will change signs where E=V, which is exactly the classical turning points.

    I'm not sure what more you want?
  5. Feb 10, 2012 #4
    Both geometrically and algebracaly the case is pretty obvious. What I am asking again is if this fact has some sort of physical significance. Does it represent something or is it some sort of restriction upon something? That king of thing, if this is any more helpful...
  6. Feb 10, 2012 #5


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    These points represent the points beyond which the kinetic energy operator changes sign and has negative expectation value. Is that "something physical" that you're looking for?
  7. Feb 10, 2012 #6
    It's a good starting point. I'll have to do some further reading into that and see if I can get anything out of that. Thanks for your timely reply.
  8. Feb 10, 2012 #7
    Wait what? Within the same line system it represents that, but what does that physically mean or what does it mean from intersection to intersection? If the sine changes that's where it oscillates in a way where based on evolution through time you would have a high or low probability of measuring it in a specific location? Or, why does the x-value distance between those intersections from system to the next system seem to decrease?
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