Simple harmonic oscillators-Quantum mechanics

  1. 1. The problem statement, all variables and given/known data
    An ion in a harmonic ion trap sees a potential which is effectively that of a simple harmonic
    oscillator. It has a natural oscillation frequency given by v = 1 MHz. Ignoring any internal
    excitations, it is known to be in a superposition of the n = 0, 1 and 2 SHO energy states.
    A measurement is then made and it is found to be in the n = 2 level.

    a)What is the energy of the ion after the measurement has been made?

    3. The attempt at a solution
    Why is the answer E_n = (2n+1)/2 [tex]\hbar\omega[/tex]

    I do not understand the (2n+1) / 2

    Thanks!
     
  2. jcsd
  3. Chegg
    Hi,
    The average energy in the nth state (or in the phonon picture: number of phonons in a mode associated with frequency [tex]\omega[/tex]) for a single harmonic oscillator is given by:
    [tex]E_n=\frac{2n+1}{2}\hbar\omega=(n+\frac{1}{2})\hbar\omega.[/tex]
    where
    [tex]h\nu=\hbar\omega.[/tex]
     
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