Principles and Concepts
of Quantum Mechanics
Implications of
Quantum Mechanics

13. Small Parts of the Wave Function.

Summary
In contrast to the classical case, a small part of the wave function of quantum mechanics carries the full mass, energy, momentum, spin and charge.

Einstein received the Nobel Prize, in part, for his particulate explanation of the photoelectric effect (light knocking electrons out of a metal). Perhaps the main reason Einstein proposed his particulate photon model of light was that a classical light wave could not transfer a sufficient amount of energy to electrons to quickly knock them out of a metal. The reasoning was that a small portion of the (classical) light wave carried a correspondingly small part of the energy, and since an electron was small, it could not quickly accumulate enough energy from the light wave. But with a proper understanding of the properties of the wave function, we can see that Einstein’s reasoning is not applicable.

The basic idea is that when a wave function splits into several parts, each part carries the full mass, energy, momentum, spin, and charge. This holds because the operators associated with these quantities are linear and local. The details of the derivation are given in Details of Small Parts.
[P14] In contrast to the classical case, small parts of the wave function carry the full mass, energy, momentum, spin and charge.
This implies the small part of the light wave function that hits one electron in the photoelectric (or Compton) effect carries the full energy and momentum of the wave function. Hence it was not necessary for Einstein to assume those quantities were carried in a concentrated form by a point particle. (Note: Einstein proposed his reasoning in 1905, before quantum mechanics had been discovered and of course before all the properties of the wave function were appreciated.)

Conservation Laws. The full light (or electron) wave function is the sum of the small parts. One might imagine that if each small part carries the full energy and momentum, we have given too much energy and momentum to the full wave function. But energy and momentum (and charge, etc.) do not add across sums; they only add across products of wave functions in quantum mechanics (because the corresponding operators are essentially first-order derivatives).

Classical Electromagnetic Waves. Why does a classical electromagnetic wave behave differently in this respect from the light wave functions making up the wave? Because the classical wave is made up of many localized light-like wave functions, and each small portion of the classical wave contains only the energy and momentum of those photon-like wave functions localized in that region of space.  