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Name: Unknown
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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.

R.C. Winther

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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