## Re: Wilson-Somerfield quantization rules.

I don't know if this is going to be of interest to anyone or even useful or even if its been touched upon before, but since I spent so long typing it out, I figured it might peak someone's interest in a historical curiosity sort of sense.

Anyway. There is a footnote for the laymen, this was originally aimed at people of various levels.

In the interest of well just about anyone who cares why things are the way they are, here is an excerpt copied as best I can from the original work.

Why waves are quantized as e=nhv

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 quantization 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.
$$\displaystyle\oint P_q\,dq=n_qh\,\,\,\,(5-20)$$
Where q is one of the co-ordinates, Pq is the momentum associated with the coordinate,nq 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 rho, rho table missing (apologies) (5.10)

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 ro. 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=ro 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 coordinate t is the radial momentum $p_r=m\frac{dr}{dt}$. In the present case $r=ro,\frac{dr}{dt}=0$ and $\frac{d\theta}{dt}=\omega$ a constant. Thus $L=mr_0^2\omega$ and Pr = 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 onto 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
$$\dsiplaystyle\oint P_q\,dq=\oint L\,d\theta=L\oint\,d\theta =L\int_0^{2\pi}=2\pi=2\pi L$$
So the condition
$$\displaystyle\oint P_q\,dq=n_qh$$
$$\displaystyle 2\pi L=nh$$
$$\displaystyle L=nh/2\pi\equiv n\hbar$$

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

Position of the particle can be specified by the single linear co-ordinate x. The behaviours of this coordinate with time is illustrated in (5.11) and can be expressed.
$$\displaystyle x=x_0 \sin 2\pi vt=x_0 \sin \omega t, \omega\equiv 2\piv$$
$$\displaystyle p=m\frac{dx}{dt}=mx_0\omega \cos \omega t$$
to evaluate this integral it is convenient to express $cos \omegat$ in terms of x:
$$\displaystyle x^2=x_0^2 \sin^2\omega t =x_0^2(1-\cos^2\omega t)$$
Then
$$\displaystyle \cos^2\omega t=\frac{x_0^2-x^2}{x_0^2}, \cos\omega t=\frac{\sqrt {x_0^2-x^2}}{x_0}$$
and
$$\displaystyle \oint p\,dx=mx_0\omega\oint \frac{\sqrt {x_0^2-x^2}}{x_0} dx=m\omega... 4\int_0^{x_0} \sqrt{x_0^2-x^2}dx$$
The last step depends upon the fact that
$$\displaystyle \oint = \int_0^{x_0} + \int_{x_0}^0 + \int_0^{-x_0} + \int_{-x_0}^{0} = 4\int_{0}^{x_0}$$
Because the integrand is an even function of x, this gives
$$\displaystyle \oint p\,dx=m\omega 4\left[ \frac{x\sqrt x_0^2-x^2}{2} +\frac{x_0^2}{2} \sin^{-1}\cdot\frac{x}{x_0}\right]_{0}^{x_0}$$
$$\displaystyle \oint p\,dx=m\omega 2\,(x_0^2\sin^{-1}\cdot 1-x_0^2\sin^{-1} 0)$$
$$\displaystyle p\,dx=mx_0^2\omega\pi$$
We would like to write this in terms of the total energy 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
$$\displaystyle E=\frac{1}{2}m\left[\frac{dx}{dt}\right]_{t=0}^{2} =\frac{1}{2}m \left[ x_0\omega\cdot \cos\omega t \right]_{t=0}^{2}=\left(\frac{1}{2}\right)mx_0{^2}\omega^2$$
Since x = 0 when t = 0. Thus we have.
$$\displaystyle \oint p\,dx=\frac{1}{2}mx_0{^2}}\cdot\omega^2 \frac{2\pi}{\omega}=\frac{E}{v}$$
and
$$\displaystyle \frac{E}{v}=n_qh\equiv nh,$$
$$\displaystyle E=nhv$$
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.

This pre-dates Schrödingers operator approach.

Quantising the angular momentum and relating this to the linear momentum under a simple harmonic oscillator will indeed reveal energies quantised in units of $\hbar\nu.$ A more sophisticated approach solves the Schrödinger equation for a simple harmonic oscillator potential using operator methods, and the energy eigenvalues reveal the same relationship. They also show a positive energy ground state, which the Sommerfeld approach does not, because a zero-energy quantum state has no wavefunction, and thus does not exist.

Footnotes:

This might need a bit of explanation to the laymen so.

Integration is the area under the curve on a graph so:-

Simple sinosoidal wave. expressed by $\sin\theta$

In this case the area under the peaks and troughs of the wave is the Integral. Essentially what the above equation does is place the particles momentum and thus it's energy in terms of $\hbar$ or planks constant by using our initial equation for the momentum of the particle between -xo and xo or the amplitude of wave at maximum value and minimum.

And then solve the initial equation in terms of the simple harmonic oscillator for an electron, the answer given correlates with the energy of the wave, which in term agrees with the equation e=hnv and explains why unlike other waves an electron or a photon or whatever has the equation e=hnv
where h is planks constant: instead of $e=Av^2$ Where A is a constant as in a water wave or guitar string or whatever.

Provided By DR E.P.A.Bailey, latex by me. To answer his own question. A bit of physics history.
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