Wilson-Sommerfeld Quantization Condition
To make a good first stab at the wave nature of matter, we will derive what is known as the Wilson-Sommerfeld quantization condition. Both Wilson and Sommerfeld derived the equation in terms of an action integral from classical mechanics, and as such did not seem to recognize the relationship to the phase of a wave.
In dealing only with phase arguments, we will derive the quantization condition in a way that follows directly from the way we've handled waves until now, and as far as I know, it has never been done this way outside of this body of work. Taking this view has one clear advantage: It becomes a fundamental consequence of the behavior of waves and phase addition. I cannot find evidence of it being viewed that way in the early days of quantum theory. If you (the student) end up finding evidence otherwise, please pass it on to me.
So here's how we start: When we wrote in past chapters, we assumed k was constant. We will soon consider cases where k might change with location as it would have in the context of Snell's law. We will also write it in terms of momentum, using de Broglie's hypothesis This means we can write our wave number as , and our phase condition as . Momentum p must be allowed to change with location along the path L, and possibly in a continuous fashion, so we need to change this into an integral and write . Exactly what might cause momentum to change along a path will be discussed shortly. As a last step of the derivation we wish for constructive interference of these matter waves, and set , and clean up the algebra. This gives us the final quantization condition:
This condition holds for any system for which the coordinates are periodic. The integral must therefore be taken over a closed path, indicated by the little circle through the integral sign. This is just like when we calculated path length difference. We always meant that it was the phase difference along a path that returned back to the starting place - like when we used 2L (twice the length of a string) for the path length difference when solving for standing waves on a string. In that case, the path of the integration would be, for instance, from one end of the string to the other end, and back - while including the two reflections at the end points as well.
In reference to the mathematics, in many cases this quantization condition will reduce to a product rather than an integral (when p does not vary over the path), but this is the general version of the expression that is true regardless.
Limitations and Applications
While the Wilson-Sommerfeld (WS) quantization condition is certainly not the final word on quantum theory, and is generally in the category of "old quantum theory", it is a good starting place. It is nice because it emphasizes that we are really dealing with waves and not discrete entities, or particles. It is nice because it emphasizes phase and constructive interference as the primary notions that allow waves of any sort to persist in nature. In other words, phase and interference likely underlie all of existence. (Recall that everything in nature is a wave!)
We will soon look at some of the same problems that we will do in a full quantum mechanical formalism in later sections. But that will be only after discussing wave equations and some later work. What's nice to know about the WS condition, is that it is often much easier than a full quantum mechanical solution, and still gives either a completely correct solution, or a nearly correct one.
I should also note that the WS quantization condition is still used today to facilitate many calculations in quantum mechanics with applications like quantum tunneling and the WKB (Wenzel, Kramers & Brillouin) approximation that is used for many quantum calculations when they can't be solved exactly. We will look at tunneling later, and the WKB approximation is beyond the scope of our present course, but feel free to search and read about it elsewhere. I mention it just to point out the usefulness of the present work.
One obvious limitation of the WS quantization is that it seems to require bounded systems such that closed trajectories of waves are present. In other words, the system must be periodic. In spite of this limitation, combined with other concepts it can be a very useful tool even in unbounded systems. As mentioned above, it sometimes leads to incorrect answers. As we'll see, the version of the W-S quantization condition misses the zero-point energy in quantum oscillators unless modifications are made that Wilson and Sommerfeld did not make, for instance, but which seems to follow directly from my wave-based derivation. We will discuss this in a later section.