Harmonic Oscillator Energy Levels

Question: 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?
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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.

R.C. Winther
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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.

Robert Topper
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