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Prove zero point energy without calculation?

  1. Mar 5, 2009 #1
    What is an elegant way to prove that quantum mechanical system (with bounded particles?) have some non-zero ground state energy?
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  3. Mar 5, 2009 #2

    Vanadium 50

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    How does one prove that the result of a calculation is non-zero (or non-any value) without doing a calculation?
  4. Mar 5, 2009 #3
    Without the full calculation of energy. Basically the easiest way possible, whatever that is.
  5. Mar 5, 2009 #4
    Well, should you take Heisenberg principle as true, there is no much caliculation involved : as an example, take a free particle and localize it ... SNIP ...such uncertanty in momentum, it's easy to derive p2 expectation value thus kinetic energy.

    erorr : I didn't read the question as careful as I thought I did =) Having general QM problem in mind, I am no longer sure ... maybe taking Taylor series around global minimum and recalling oscillator, but that is hardly "without of caliculation".
    Last edited: Mar 5, 2009
  6. Mar 5, 2009 #5


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    A simple hand-waving conceptual answer would be something like: Consider a simple vibrating system, e.g. a harmonic oscillator. The lowest possible energy (classically) would be for the thing to simply not vibrate. Quantum mechanically, it can't do that since it'd mean having a well-defined position and momentum, in violation of the uncertainty principle.

    So the lowest vibrational state must have some energy, on the order of ~[tex]\hbar[/tex].
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