Very good book! I strongly recommend reading it! But not right now! We first have to talk…. …about troubles with some other things.

Trouble with the de Broglie waves In many situations particles behave like “proper particles” – i.e., like an object of very small size, with all its mass enclosed within that size. Examples of effects, in which particles reveal their “particle nature”: ● Electrons, for instance, can be detected by a photographic film – a single electron pro- duces a tiny dark spot on the film. Or, one can use a fluorescent screen: then a single electron produces a microscopic “flash” on it. We can determine “where it was*” as well as “when it was there”. * Within certain accuracy limits, we will get to this shortly.

● Moving charged particles leave visible “tracks” in cloud chambers and in bubble chambers. The list can be much longer… Cloud chamber: first obs- ervation of the positron (electron’s antiparticle) Particle tracks in a liquid hydrogen bubble chamber at CERN, Switzerland

However, we have learned that in other types of experiments the same particles behave like typical waves! (Bragg diffraction, double-slit interference). SO, WHAT’S GOING ON?! ARE PARTICLES REALLY PARTICLES? PERHAPS THEY ARE WAVES? NO – definitely one cannot say that “particles are waves”. Why? Because particles, as we have said, are “localized objects”. We will see shortly that a particle position cannot be determined with an infinitely high “precision” – but certainly one can determine with micrometer accuracy “where the particle is”.

And how about waves? Recall – the simplest wave (a plane wave) propagating along certain direction (call it x ) can be mathematically described as follows:

Now, the equation has a simpler form: Now, please tell me: where is this wave? Answer: EVERYWHERE! This function spans from x = -  to x = + . Over the entire Universe! A wave is not localized, so a particle cannot be a single wave!

PAY ATTENTION, PLEASE ! I use Microsoft Equation Editor for preparing my slide presentations (any other choice? :o) ) The appearance of some Roman and Greek characters in MSEqEd is very similar: Velocity sym- bol in the Text Frequency symbol in the Textbook Therefore, when velocity and frequency appear together, we will use capital V for velocity

Is this the only reason why not? No! Another reason is the velocity. The wave propagation velocity, as you certainly remember from Ph212, is:

Now, consider a non-relativistic particle: From the de Broglie Equations we get: Comparing the two results for K/p, we obtain for the particle:

The particle velocity is twice as large as the de Broglie wave’s velocity! CONCLUSION Because of its “delocalized” character, and its velocity which is inconsistent with the velocity of the particle it represents, a wave – at least a simple plane wave – cannot be used as a mathematical description of a particle.

Epilogue: Our goal We have to construct a mathematical description of a particle that provides a proper localization and velocity, but still accounts for the wave-like properties revealed by experiments.