Principles and Concepts

of Quantum Mechanics

of Quantum Mechanics

Implications of

Quantum Mechanics

Quantum Mechanics

12. Localization and the Wave Function.

Summary

The rules of quantum mechanics imply only one localized grain of film is perceived as exposed by a wave function spread out over many grains.

There is one particle-like property that is not related to group representation theory, that of localization. Speaking classically, the carriers of mass and charge seem to be very small, point-like, with highly localized effects. This particle-like localization occurs in an interesting way in quantum mechanical experiments. To illustrate, suppose we have a single light wave function that goes through a single slit and impinges on a screen covered with film grains. The wave function spreads out after going through the slit and hits many grains of film. But surprisingly, a microscopic search will show that only one of the grains will be exposed by the light. It is

*as if*there were a particle of light, a photon, hidden in the wave function, and it is the single grain hit by the particle that is exposed.

As another example, suppose we have a target proton surrounded by a sphere coated with film grains on the inside. An electron (electron-like wave function) is shot at the proton and the wave function of the electron of the scattered electron spreads out in all directions, hitting

*every*grain (see Scattering Experiments). But again, a microscopic examination will show that

*one and only one*grain is (perceived as) exposed. As in the case of light, it is

*as if*a particulate electron embedded in the wave function followed a particular trajectory and hit and exposed only one grain.

Quantum mechanical explanation. Surprisingly, this

*perceived*exposure of only one localized grain by a spread-out wave function can be accounted for strictly from the principles of quantum mechanics, without invoking the existence of particles. The technical explanation is a little complicated and so it is put in a separate section, Details of Localization. But the gist of it can be readily explained.

When the spread out wave function hits all the grains of film, the part that hit a particular grain exposes

*only that grain*. This means that after the electron reaches the screen, the wave function of the electron plus the grains of film will consist of a sum of versions of reality. And we know from principle [P3] (Schrödinger’s cat) that we will perceive one and only one version. But since (1) only one grain is exposed in each version, and (2) we perceive results corresponding to only one version, we will

*perceive*only one grain exposed, even though the wave function has hit all the grains! Thus

[P13] A spread-out wave function produces localized effects.It is remarkable that quantum mechanics alone, with no assumption of the existence of particles, can imitate, in our perceptions, the classical idea of a localized (only one

*localized*grain exposed) particle.

Particle-like trajectories. In cloud and bubble chambers, a spread-out wave function is found to produce more or less continuous, particle-like trajectories. Reasoning similar to the above, applied several times, shows this also follows from quantum mechanics alone. (The ‘continuity’ follows because the wave functions of successive nucleation centers are entangled; in each term, the next nucleation center ‘nucleates’ only if the previous one nucleated.)