This isn't my question but? Frequency quesiton.

Fundamentals of modern physics by R.M.Eisenberg 1967.pg.128-131.


6, The Wilson-Somerfeld Quantization Rules

The success of the Bohr Theory, as measured by it's agreement with experiment, was certainly very striking. But it only accentuated the mysterious nature of the postulates on which it was based. One of the biggest of these mysteries was the question of the relation between Bohr's quantiztion of the angular momentum of an electron moving a circular orbit and Planck's quantization of the total energy of an entity, such as an electron, executing simple harmonic motion. In 1916 some light was shed upon this by Wilson and Sommerfield, who enunciated a set of rules for the quantization of any physical system for which the coordinates are periodic functions of time. These rules included both the Planck and the Bohr quantization of special cases. They were also of considerable use in broadening the range of applicability to quantum theory. These rules can be stated as followed:

For any system in which the coordinates are periodic functions of time, there exists a quantum condition for each coordinate. These quantum conditions are.

(5-20)

Where q is one of the co-ordinates, P_q is the momentum assosciated with the coordinate,n_q is the quantum number which takes the integral values, and $\oint$ means the integration is taken over the co-ordinate q

There are two tables here, showing a wave function and r_oFigure(5.10)

showing the time dependence of coordinates I'll try and reproduce them in mathcad or something at some point, if I can.

The means of these rules can be best illustrated in terms of some specific examples.

Consider a particle of mass m moving with constant angular velocity $\omega$ in a circular orbit radius r_o. The position of the particle can be specified by the polar coordinates and $\theta$. The behaviour of these two coordinates is shown in fig 5-10(apologies not present)as functions of time. They are both periodic functions of time, if we consider r=r_o to be a limiting case of this behaviour. The momentum associated with the angular coordinate $\theta$ is the angular momentum $L=mr^2 \frac{d\theta}{dt}$. The momentum associated with the radial coordinater t is the radial momentum $p_r=m\frac{dr}{dt}$. In the present case r=r_o,$\frac{dr}{dt}=0$ and$\frac{d\theta}{dt}=\omega$ a constant. Thus $L=mr_0^2\omega$ and P_r = 0. We do not need to apply equation (5-20) to the radial coordinate in the limiting case in which the coordinate is a constant. The application of the equation oto the angular coordinate $\theta$ is easy to carry through for the present example. We have $q=\theta$ and $P_q=L$, a constant. Write $n_q=n$; then



So the condition




Here ends part one, more maths to follow when I can be arsed to type the rest out.

[Part 2]Those who dislike bare quantum equations should look away.





Diagram Simple:- sin wave between -x₀ and x₀ with t on the x axis

figure5-11. The time dependence coordinate of a simple harmonic oscillator.

Position of the particle can be specified by the single linear coordinatex. The behavious of this coordinate with time is illustrated in (5.11) and can be expressed.



Where x₀ is the amplitude of oscillation. The momentum of the particle is



to evaluate this integral it is convenient to express in terms of x:



Then



and



The last step depends upon the fact that



Because the integrand is an even function of x, this gives







We would like to write this in terms of the total enrgy E of the particle; it is easy to do if we recall that the total energy of a harmonic oscillator is equal its kinetic energy when x=0. So

$E=\frac{1}{2}m\left(\frac{dx}{dt}\right)_{t=0}^{2}=\frac{1}{2}m \left( x_0\omega cos\omega t \right)_{t=0}^{2}=\frac{1}{2}]mx_0{^2}\omega^2$

Since x = 0 when t = 0. Thus we have.



and

$\frac{E}{v}=n_qh\equiv nh$,

$E=nhv$

Which is identical with Plancks quantization law.

7. Sommerfield's Relativistic theory

One of the important applications of the Wilson-Sommerfield quantization rules was to the case of a hydrogen atom in which it was assumed that the electron could move in elliptical orbits. This was done by Sommerfield in order to explain the fine structure of the hydrogen spectrum. The fine structure is the splitting of the spectral lines, into several distinct components,which is found in all atomic spectra. It can be observed using only by using equipment of very high resolution since separation, in terms of wave number, between adjacent components of single order of the spectral line 10^-4 times the separation between adjacent lines. According to the Bohr theory, this must mean that what we had thought was a single energy state of the hydrogen atom actually consists of several states which are very close together in energy.