IIIA5. Localization and Particlelike Trajectories

1. Classical Electron Scattering.  

The most direct evidence cited for particles is the phenomenon of  localization, in which a spread-out wave function produces a localized effect. To illustrate this concept, we start with a classical description of electron scattering.  Suppose an electron  scatters off a proton in the middle of a hollow sphere whose inner surface is coated with a film emulsion.  The point electron is deflected by—scatters off—the electric force field of the point proton.  Each electron shot in will scatter in a different direction and be detected by exposing one of the grains of film.

2. Localization in Quantum Electron Scattering. 

The end result is the same in quantum mechanics; one and only one grain of film will be exposed each time an electronlike wave function is shot at the proton (protonlike wave function).  But as was discussed in section IID, the conceptual picture is entirely different.  The electron wave function comes in as a compact cloud of atomic dimensions (but much more spread out than the proton wave function).  When it hits the proton, most of the electron wave function simply continues past it as a still-compact cloud.  But because of the electric force, some of the electronlike wave function is scattered outward in concentric waves centered on the proton, as in figure IID.1.  This means a small part of the scattered electron cloud hits each of the grains of film.  But, in spite of this, the effect is localized; only one localized grain is perceived as exposed. 

3. Conventional Underlying-Particle Explanation. 

The conventional interpretation of this result is that, concealed within the electron wave function is an actual electron particle, and it is the film grain hit by the actual particle that is exposed.  This particle-based explanation, however, is not required, because quantum mechanics—with one logical snafu—can explain the results by itself, without assuming there are particles.

4. Quantum Explanation. 

To see this, visualize the electronlike wave function just a split second before it hits the N grains.  Because the theory is linear, the wave function can be broken up into a sum of segments—as many segments as there are grains—with segment i about to hit grain i.  The total wave function for the electron wave function plus grains is thus

(IIIA5-1)                     

where the ’s are the wave functions of the unexposed grains.  Now according to the mathematics of quantum mechanics, segment i of the electron wave function hits and exposes only grain i.  So after the wave function has hit all N grains, the total wave function, electron plus grains, is a sum of N terms, with term i having only grain i activated.

(IIIA5-2)                                       

where the asterisk indicates an exposed grain.  Each of these N terms is a branch of the wave function.  And one of the interpretive rules of quantum mechanics, discussed in section IIIA3, is that we perceive one and only one branch.   Therefore, because only one grain is activated on each branch, and because we perceive only one branch, we will perceive only one grain activated even though the wave function hits all of them! 

Thus quantum mechanics alone (plus the “only-one” interpretive rule of section IIIA3), without the need for the introduction of particles, explains why we perceive only one localized grain as activated!  There is no need to postulate a localized, objectively existing electron.  (Everett showed this in his "Many-Worlds" intepretation.

5. Photons and Localization. 

The same localization phenomenon occurs for light.  A lightlike wave function can be spread out over many grains of film, but only one will be found to be exposed.  Exactly the same reasoning shows that the localized light wave effect does not require the objective existence of a photon to explain it.

6. The Logical Snafu. 

There is a logic problem in the above explanation, although it is relatively minor and doesn’t affect the conclusion.  We assumed that, for whatever reason, one and only one of the many branches of the (total) wave function would be perceived.  In section IIIA3, we argued that, because each branch of the wave function constituted a separate universe, totally isolated from each other branch, one could be physically (communicably) aware of only one branch.  So we are on firm ground in asserting that quantum mechanics alone allows the perception of only one branch.

However, quantum mechanics does not explain why we perceive the particular branch we perceive, be it in the Schrödinger’s cat example or localization examples.  It is conceivable that the explanation is that there are objectively existing particles and they pick out the particular choice.  But localization gives no evidence that the existence of particles is the explanation for the particular choice.  Thus localization provides no evidence for particles.

7. Quantum Versus Classical Waves. 

In a classical wave, each small part of the wave carries a correspondingly small part of the total energy of the wave.  So if the electronlike wave function behaved like a classical wave, the small segment, , that hits grain i, would have only a very small amount of the total energy, E, of  “ the electron,” and this would not be enough to shift the energy to the "exposed" state of the grain.  The wave function, however, does not behave like a classical wave, at least in this sense; each segment of the wave carries the full complement of energy (and momentum, spin and charge). 

To see this, suppose for simplicity that the scattering is elastic so that the energy of the (total) wave function, both before and after scattering, is E. Then, with H being the Hamiltonian, using ket notation, and noting that the Hamiltonian applies separately to each (free-particle) segment, we have

(IIIA5-3)                   

If we take the scalar product of the fourth and the last term with  and note that the different segments of the wave function are orthogonal, we find that  for each j.  This shows that each small segment of the quantum wave function, , has the full complement, E, of energy.  Thus the wave function is very different from a classical wave in this respect. 

Note that these results do not violate conservation of energy because energies only add for product wave functions; they do no add across sums within a particular wave function.  Note also that the same conclusion holds for mass, momentum, spin and charge; a small piece of the wave function "carries" the full complement of each of these. 

Finally, note that a small piece of a classical light wave carries a correspondingly small part of the energy because the classical light wave consists of many localized light wave functions, each carrying the small energy hf.

8. Particlelike Trajectories. 

The particlelike trajectories one sees in a cloud chamber or bubble chamber appear to provide convincing visual evidence that electrons must be particles.  However, quantum mechanics, by itself, without the presumed existence of particles, also gives particlelike trajectories.  So, convincing as they seem, these trajectories do not constitute evidence for particles.  (Mott point out that quantum mechanics yielded particlelike trajectories in the 1930's.)

To see this, we will modify our electron scattering experiment.  We suppose the detector consists of not just one layer of film grains, but, say, three layers, and that the grains in successive layers are arranged exactly behind the grains in front of it.  We further suppose the electronlike wave function passes through each grain, exposing it, and then goes on through the grain in the next layer without being scattered from its path.  Then the total wave function after the electron has gone through 0, 1, 2 and 3 layers, is

(IIIA5-4)               

(IIIA5-5)           

(IIIA5-6)   

(IIIA5-7)   

where again the asterisks indicate an exposed grain.   is the sum of N terms, each of which is a branch of the wave function.  One and only one of these branches will be perceived.  And we see that on each branch, the electron wave function exposes all the grains in its straight-line flight path, but it exposes no other grains.  Thus the trail of exposed grains on each branch, and therefore the trail that is perceived, is the same as would be the trail left by an actual particle. Quantum mechanics yields particlelike trajectories on each branch of the wave function.

 Allowing each grain to scatter the wave function flight path slightly does not affect the conclusion; one would still see a particlelike path—that is, a reasonably smooth, sharply defined path, rather than a disjoint or diffuse path—but it would not be a straight line.

9. Short “Derivation.” 

There is a shorter (but much less acceptable) way of deriving the same result:  If quantum mechanics predicted anything other than the perceived particlelike trajectories, then that would be a proof that the theory was incorrect.  And we do not expect such a successful theory to be so blatantly incorrect.  The same (mostly unacceptable) reasoning applies to the localization argument.