Re: [Phys-l] eigenenergies

[John Mallinckrodt <ajm@csupomona.edu>]

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 ...)

or

  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:
>
> http://usna.edu/Users/physics/mungan/Scholarship/Eigenenergies.pdf
>
> 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
> mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
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A. JOHN MALLINCKRODT
Professor of Physics (Ret'd), Cal Poly Pomona
     http://www.csupomona.edu/~ajm
Consulting Editor, AMERICAN JOURNAL of PHYSICS
     http://www.kzoo.edu/ajp

email: ajm@csupomona.edu
phone: 909-869-4054

Physics Department
Building 8, Room 223
Cal Poly Pomona
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Pomona, CA 91768-4031