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First energy level

  1. Jul 4, 2009 #1
    Hallo.

    my teacher wrote for the first energy level of a particle in a certain potential


    [tex]\Delta X \Delta P \approx \frac{\hbar}{2}[/tex]

    exist.

    is it a general result for all energy level or there is specific meaning for the first energy level?
     
  2. jcsd
  3. Jul 7, 2009 #2

    alxm

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    Science Advisor

    Well [tex]\Delta X\Delta P \ge \hbar/2[/tex] always, per the uncertainty principle. For the lowest energy state of a harmonic oscillator, a gaussian wave-packet has the lowest ground-state energy, and for the ground state [tex]\Delta X\Delta P = \hbar/2[/tex].

    The ground-state energy is not always related to the uncertainty principle like this; it depends on whether or how the terms are related to the energy levels described. With a harmonic oscillator potential, the energy levels are vibrational modes - so momentum and position are related in a fairly straightforward way.

    It demonstrates the uncertainty principle and shows that bound particles cannot be perfectly stationary - and hence, that they must have a certain amount of kinetic energy even in their ground state, known as zero-point energy.
     
  4. Jul 8, 2009 #3
    For the n-th energy level you have

    [tex]\Delta X \Delta P = n \frac{\hbar}{2}[/tex]

    this is called Bohr-Sommerfeld quantization condition and say that the allowed orbits are those with an integer number of cycle. It is a periodicity condition.
     
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