Measurement and the Interpretation of Quantum Mechanics

Consider an example. Suppose that at time t we measure a particle s position. Let (x, l) be the state function of the particle the instant before the measurement is made (Fig. 7.6a). We further suppose that the result of the measurement is that the particle is found to be in the small region of space 

We ask What is the state function (x, t+) the instant after the measurement To answer this question, suppose we were to make a second measurement of position at time t+. 

FIGURE 7.6 Reduction of the wave function caused by a measurement of position. 

Since t+ differs from the time t of the first measurement by an infinitesimal amount, we must still find that the particle is confined to the region (7.104). If the particle moved a finite distance in an infinitesimal amount of time, it would have infinite velocity, which is unacceptable. Since (x, t+)p is the probability density for finding various values of X, we conclude that (x, t+) must be zero outside the region (7.104) and must look something like Fig. 7.6b. Thus the position measurement at time t has reduced from a function that is spread out over all space to one that is localized in the region (7.104). The change from (x, L) to (x, t+) is a probabilistic change. 

The measurement process is one of the most controversial areas in quantum mechanics. Just how and at what stage in the measurement process reduction occurs is unclear. Some physicists take the reduction of P as an additional quantum-mechanical postulate, while others claim it is a theorem derivable from the other postulates. Some physicists reject the idea of reduction [see M. Jammer, The Philosophy of Quantum Mechanics, Wiley, 1974, Section 11.4 L. E. Ballentine, Am. /. Phys., 55,785 (1987)]. Ballentine advocates Einstein s statistical-ensemble interpretation of quantum mechanics, in which the wave function does not describe the state of a single systan (as in the orthodox interpretation) but gives a statistical description of a collection of a large number of systems each prepared in the same way (an ensonble). In this interpretation, the need for reduction of the wave function does not occur. [See L. E. Ballentine, Am. J. Phys., 40, 1763 (1972) Rev. Mod. Phys., 42, 358 (1970).] There are many serious problems with the statistical-ensemble interpretation [see Whitaker, pp. 213-217 D. Home and M. A. B. Whitaker, Phys. Rep., 210, 223 (1992) Prob. 10.4], and this interpretation has been largely rejected. 

The probabilistic nature of quantum mechanics has disturbed many eminent physicists, including Einstein, de Broglie, and Schrodinger. These physicists and others 

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For the majority of physicists the problan of finding a consistent and plausible quantum theory of measurement is still unsolved. The immense diversity of opinion. . . concerning quantum measurements. .. [is] a reflection of the fundamental disagreement as to the interpretation of quantum mechanics as a whole (M. Jammer, The Philosophy of Quantum Mechanics, pp. 519,521). 

How to extract from E(qj,t) knowledge about momenta is treated below in Sec. III. A, where the structure of quantum mechanics, the use of operators and wavefunctions to make predictions and interpretations about experimental measurements, and the origin of uncertainty relations such as the well known Heisenberg uncertainty condition dealing with measurements of coordinates and momenta are also treated. 

Parallel Universes—Chapter 3 discussed parallel universes and the many-worlds interpretation of quantum mechanics. Readers interested in a lively and critical discussion of this topic should consult Professor Victor Stenger s The Unconscious Quantum (Prometheus Books, 1995). For example, he doubts very much that the parallel universes (in the many-worlds interpretation) all simultaneously exist. He also does not believe that all branches taken by the universe under the act of measurement are equally real. Stenger discusses other approaches such as the alternate histories theory that suggests every allowed history does not occur. What actually happens is selected by chance from a set of allowed probabilities. 

It is interesting to note that the Gottingen school, who later developed matrix mechanics, followed the mathematical route, while Schrodinger linked his wave mechanics to a physical picture. Despite their mathematical equivalence as Sturm-Liouville problems, the two approaches have never been reconciled. It will be argued that Schrodinger s physical model had no room for classical particles, as later assumed in the Copenhagen interpretation of quantum mechanics. Rather than contemplate the wave alternative the Copenhagen orthodoxy preferred to disperse their point particles in a probability density and to dress up their interpretation with the uncertainty principle and a quantum measurement problem to avoid any wave structure. 

Many applications in chemistry require us to interpret—and even predict—the results of measurements where we have only limited information about the system and the process involved. In such cases the best we can do is identify the possible outcomes of the experiment and assign a probability to each of them. Two examples illustrate the issues we face. In discussions of atomic structure, we would like to know the position of an electron relative to the nucleus. The principles of quantum mechanics tell us we can never know the exact location or trajectory of an electron the most information we can have is the probability of finding an electron at each point in space around the nucleus. In discussing the behavior of a macroscopic amount of helium gas confined at a particular volume, pressure, and temperature we would like to know the speed with which an atom is moving in the container. We do not have experimental means to tag a particular atom. 

The linear superposition principle plays a central role in the theory presented here. It should be noted, however, that the standard Copenhagen interpretation of quantum mechanics is not well adapted to discuss the notion of state amplitudes and measurements in the context required by the GED scheme. A more appropriate theoretical framework for quantum measurement is found in the ideas proposed by Fidder and Tapia [16]. 

III. Experimental observation of Quantum Mechanics. Only this final section should address the rules that govern interpretations of experiments measuring properties of QM systems with macroscopic devices. This includes probability interpretation, uncertainty relations, complementarity and correspondence. Then experiments can be discussed to show how the wave functions manipulated in section I can be used to predict the probabilistic outcome of experiments. 

We wish to account for (i.e., interpret) the Arrhenius parameters A and EA, and the form of the concentration dependence as a product of the factors c (the order of reaction). We would also like to predict values of the various parameters, from as simple and general a basis as possible, without having to measure them for every case. The first of these two tasks is the easier one. The second is still not achieved despite more than a century of study of reaction kinetics the difficulty lies in quantum mechanical... 

In search for an explanation, Aharonov and Bohm worked out quantum mechanics equations based on the measurable physical effect of the vector potential, which is nonnull in a region outside the solenoid. Like many other paradoxes in physics, including the twin paradox, the interpretation of this experiment proposed in 1959 was the subject of an intense controversy among researchers. This controversy is well summarized in a review article [55] and in other references of interest [56-67]. 

If the velocity U of an electron within the beam is constant outside the solenoid, the variation of the vector potential A as a function of time in the medium, and thus also in the solenoid, will induce a modification of the phase, as indicated by the equations written above. This will produce a modification of the boundary conditions on the boundary of the solenoid for the quantities a and b. We must also stress that the modification of the vector potential outside the solenoid is generated by either an external or an internal source feeding the solenoid. This can explain the existence of the Aharonov-Bohm effect for toroidal, permanent magnets. The interpretation of the Aharonov-Bohm effect is therefore classic, but the observation of this effect requires the principle of interference of quantum mechanics, which enables a phase effect to be measured. 

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