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Re: [Phys-l] eigenenergies

The derivation of the energy levels for the SHO is cute, but is clearly an example of getting the right answer *only* by knowing it in advance. Borrowing the hydrogen atom quantization rule by claiming that the SHO is a projection of circular motion might seem at least somewhat reasonable, but throwing out the even values of n on the basis of an analogy with closed pipes seems to me to be pure bogosity (look it up!)

It's not hard to get the right answer using quantum if one knows in advance what the answer is. I present below my own semi-classical "derivation" for the energy levels of the SHO that is no less bogus but, I would submit, offers more physical insight.


For a given energy, E:

1. The classical turning point is x_max = sqrt(2E/k) where k is the force constant.

2. The space averaged kinetic energy is K = (2/3) E.

3. The corresponding momentum is p = sqrt(2mK).

Using the quantization condition that an integer number of half wavelengths must fit between the turning points gives

lambda_n = (4/n) x_max (with n = 1, 2, 3 ...)

and using the DeBroglie relation between p and lambda gives

h/lambda_n = sqrt(2mK)

Substituting and rearranging yields the quantized energies

E_n = (pi/4) sqrt(3/2) n h_bar omega = 0.962 n h_bar omega

with omega = sqrt(k/m) and h_bar = h/(2 pi) as usual.

Now, 0.962 is too close to 1 to be an accident. Moreover, the analysis here should be expected to work far better at high energies, where the turning points are relatively "hard" than at low energies where they are relatively "soft." Thus, we should expect

1. Energy levels that closely approach n h_bar omega (from *below* due to the "slop over" of the wavefunction past the turning points) at high n and

2. A significantly lower energy than h_bar omega (say by half?) for the lowest energy level because of the large relative "slop over."

We can accomplish both objectives by letting

E_n = (n - 1/2) h_bar omega (with n = 1, 2, 3 ...)


E_n = (n + 1/2) h_bar omega (with n = 0, 1, 2 ...)

QE almost D


John Mallinckrodt
Cal Poly Pomona

On Jul 10, 2008, at 7:35 AM, Carl Mungan wrote:

I have briefly summarized a paper from the current issue of Physics
Education on the semiclassical derivation of energy levels:

I include an attempt at an explanation of why one only gets odd
harmonics for the SHO case. If you have the interest to take a look,
I'd appreciate comments on:

(a) whether you think there's any merit to my explanation; and/or
(b) any alternative explanation you come up with.

Thanks, Carl
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002
Forum for Physics Educators

Professor of Physics (Ret'd), Cal Poly Pomona

phone: 909-869-4054

Physics Department
Building 8, Room 223
Cal Poly Pomona
3801 W. Temple Ave.
Pomona, CA 91768-4031