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Quantum Harmonic Oscillator: Negative Kinetic Energy Question.

  1. Sep 16, 2009 #1
    Hello everybody,

    I noticed these questions are lengthy. If you want to skip my introduction, just scroll down to the questions. I put *** next to each one.

    I just started Quantum Theory I this semester and I have a question (actually two questions) regarding the quantum harmonic oscillator (QHO). (By the way, we are only discussing one dimension here).

    In the QHO (more specifically in the ground state of the QHO), we find that there is a probability that the particle in question may be located beyond the "classical turning points," that is, the particle's potential energy is greater than its total energy, meaning it has a negative kinetic energy. This is weird.

    After coping with this, I figured that the particle's velocity must be an imaginary number (because I'm not willing to cope with a negative mass just yet). Now without proof I was told that "the outcome of a measurement has got to be real" (David J Griffiths - Intro to Quatum Mechanics), and I don't really doubt this (though I've gotten some pretty weird measurements in my labs).

    ***My first question is: Since an imaginary velocity means an imaginary momentum, does this mean that I cannot measure the momentum of a QHO particle if I already measured its position to be beyond the classical turning points? Now I'm aware that after I measure the position, the wave function collapses and I can't measure the momentum, but what I mean is...

    If there is (let's say) a 16% chance that the particle is located beyond the classical turning points (so that its momentum is imaginary), does that mean that there is a 16% chance that when I measure its momentum, I find that it does not have one (or something along these lines).

    ***Now you HAVE to read my second question's intro because I find it necessary that you should know where this question is coming from (sorry for tricking you). I determined that there is about a 15.8% chance that a particle in a QHO has a negative kinetic energy. I determined that this probability does not depend on the particle's mass. In fact, it doesn't depend on anything! (I determined that the probability is that of a normal distribution from z= -1/sqrt(2) to 1/sqrt(2), if that means anything to you.) 15.8% of the time an oscillating particle has an imaginary velocity. This can't be right; when I flick my car's antenna, I don't see any quantum effects.
  2. jcsd
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