Bohm’s approach stipulates that a physical particle is the sort of thing that is always located in one particular place or another. In addition, wave functions are not merely mathematical objects but physical ones—physical things. Somewhat like force fields (electric fields or magnetic fields) in classical mechanics, they serve to push particles around or to guide them along their proper courses. The laws that govern the evolutions of wave functions are the standard linear differential equations of motion and are therefore deterministic; the laws that determine how wave functions push their respective particles around, which are unique to Bohm’s theory, are fully deterministic as well.

Thus, the positions of all of the particles in the world at any time, and the world’s complete quantum mechanical wave function at that time, can in principle be calculated with certainty from the positions of all of the particles in the world and the world’s complete quantum mechanical wave function at any earlier time. Any uncertainty in the results of those calculations is necessarily an epistemic uncertainty, a matter of ignorance about the way things happen to be, and not an uncertainty created by an irreducible element of chance in the fundamental laws of the world. Nevertheless, some epistemic uncertainty exists necessarily, or as a matter of principle, since it is entailed by the laws of evolution in Bohm’s theory.

Suppose that a single electron with x-spin = +1 is fed into the apparatus. On Bohm’s theory, the electron will take either the y = +1 path or the y = −1 path—period. Which path it takes will be fully determined by its initial wave function and its initial position (though certain details of those conditions will be impossible in principle to ascertain by measurement). No matter what route the electron takes, however, its wave function, in accordance with the linear differential equations of motion, will split up and take both paths. In the event that the electron takes the y = +1 path, it will be reunited at the black box with that part of its wave function that took the y = −1 path.

One of the consequences of the laws of Bohm’s theory is that, at any given time, only that part of a given particle’s wave function that is occupied by the particle itself at that time can have any effect on the motions of other particles. Thus, any attempt to detect the “empty” part of a wave function that is passing through one of the two paths will fail, since the detecting device itself consists of particles. This accounts for the absence of superposition in actual measurements of electrons emerging from the y-box.

Bohm’s theory accounts for all of the paradoxical behaviours of electrons that are fed into the apparatus without having to appeal to mutually indistinct categories of fundamental laws, as does the standard version of quantum mechanics. Notwithstanding the fact that the linear differential equations of motion are the true equations of the time-evolution of the wave function of the entire universe, there are definite matters of fact about the positions of particles and (consequently) about the indications made by measuring devices.

The theory of Ghirardi, Rimini, and Weber

The second proposed solution to the measurement problem, as noted above, affirms that wave functions are complete representations of physical systems but denies that they are always governed by the linear differential equations of motion. The strategy behind this approach is to alter the equations of motion so as to guarantee that the kind of superposition that figures in the measurement problem does not arise. The most fully developed theory along these lines was put forward in the 1980s by Ghirardi, Rimini, and Weber and is thus sometimes referred to as “GRW”; it was subsequently developed by Philip Pearle and John Stewart Bell (1928–90).

According to GRW, the wave function of any single particle almost always evolves in accordance with the linear deterministic equations of motion, but every now and then—roughly once every 10⁹ years—the particle’s wave function is randomly multiplied by a narrow bell-shaped curve whose width is comparable to the diameter of a single atom of one of the lighter elements. This has the effect of “localizing” the wave function—i.e., of setting its value at zero everywhere in space except within a certain small region. The probability of the bell curve’s being centred at any particular point x depends (in accordance with a precise mathematical rule) on the wave function of the particle at the moment just prior to the multiplication. Then, until the next such jump, everything proceeds as before, in accordance with the deterministic differential equations.

This is the whole theory. No attempt is made to explain the occurrence of these jumps. The fact that such jumps occur, and occur in precisely the way described above, can be thought of as a new fundamental law: a law of the so-called “collapse” of the wave function.

For isolated microscopic systems—those consisting of small numbers of particles—jumps will be so rare as to be completely unobservable. On the other hand, for macroscopic systems—which contain astronomical numbers of particles—the effects of jumps on the evolutions of wave functions can be dramatic. Indeed, a reasonably good argument can be made to the effect that jumps will almost instantaneously convert superpositions of macroscopically different states like particle found in A + particle found in B into either particle found in A or particle found in B.

A third tradition of attempts to solve the measurement problem originated in a proposal by the American physicist Hugh Everett (1930–82) in 1957. According to the so-called “many worlds” hypothesis, the measurement of a particle that is in a superposition of being in region A and being in region B results in the instantaneous “branching” of the universe into two distinct, noninteracting universes, in one of which the particle is observed to be in region A and in the other of which it is observed to be in region B; the universes are otherwise identical to each other. Although these theories have generated a great deal of interest in recent years, it remains unclear whether they are consistent with the probabilistic character of quantum mechanical descriptions of physical systems.

One of the important consequences of attempts at solving the measurement problem for the philosophy of science in general has to do with the general problem of the underdetermination of theory by evidence. Although the various noncollapse proposals, including Bohm’s, differ from each other on questions as profound as whether the fundamental laws of physics are deterministic, it can be shown that they do not differ in ways that could ever be detected experimentally, even in principle. It is thus a real question whether the noncollapse theories differ from each other in any meaningful way.