Harmonic Oscillator Energy Levels
Date: Around 1993
The quantum energy levels for the kinetic energy of a particle in
a box are obtained as the eigenvalues for the wave equation. Is there any
theoretical basis for E=hf in the harmonic oscillator?
If you plug in the potential for the 1-D harmonic oscillator into
the time-independent Schroedinger equation and solve the resulting eigenvalue
problem, you get that the allowed energy values are given by E_n = (n +
1/2)hf for n = 0,1,2,... This is in contrast to Planck's postulated
quantification of E_n = n*hf; the "zero-point" energy is not zero! This is a
consequence of the uncertainty principle.
True, but if you look at the formula for the difference in energy
between state n and state n+1, you get delta E = hf(n+1+1/2) - hf(n+1/2) =
hf! So the harmonic oscillator's energy can change only in units of hf. And,
if the selection rules were right, one would observe that light would only be
absorbed/emitted by such an oscillator with frequency f.
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Update: June 2012