Comprehensive Study Notes
In quantum mechanics, the state function of a system changes in two ways. [See E. P. Wigner, Am. J. Phys., 31, 6 (1963).] First, there is the continuous, causal change with time given by the time-dependent Schrödinger equation (7.97). Second, there is the sudden, discontinuous, probabilistic change that occurs when a measurement is made on the system. This kind of change cannot be predicted with certainty, since the result of a measurement cannot be predicted with certainty; only the probabilities (7.73) are predictable. The sudden change in $\Psi$ caused by a measurement is called the reduction (or collapse) of the wave function. A measurement of the property $B$ that yields the result $b{k}$ changes the state function to $g{k}$, the eigenfunction of $\hat{B}$ whose eigenvalue is $b{k}$. (If $b{k}$ is degenerate, $\Psi$ is changed to a linear combination of the eigenfunctions corresponding to $b{k}$.) The probability of finding the nondegenerate eigenvalue $b{k}$ is given by Eq. (7.73) and Theorem 9 as $\left|\left\langle g{k} \mid \Psi\right\rangle\right|^{2}$, so the quantity $\left|\left\langle g{k} \mid \Psi\right\rangle\right|^{2}$ is the probability the system will make a transition from the state $\Psi$ to the state $g_{k}$ when $B$ is measured.
Consider an example. Suppose that at time $t$ we measure a particle's position. Let $\Psi\left(x, t_{-}\right)$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
\(
\begin{equation}
a<x<a+d a \tag{7.104}
\end{equation}
\)
We ask: What is the state function $\Psi\left(x, t{+}\right)$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 $\left|\Psi\left(x, t{+}\right)\right|^{2}$ is the probability density for finding various values of $x$, we conclude that $\Psi\left(x, t{+}\right)$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 $\Psi$ from a function that is spread out over all space to one that is localized in the region (7.104). The change from $\Psi\left(x, t{-}\right)$to $\Psi\left(x, t_{+}\right)$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 $\Psi$ 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. J. 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 system (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 ensemble). 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.
"For the majority of physicists the problem 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).
The probabilistic nature of quantum mechanics has disturbed many physicists, including Einstein, de Broglie, and Schrödinger. These physicists and others have suggested that quantum mechanics may not furnish a complete description of physical reality. Rather, the probabilistic laws of quantum mechanics might be simply a reflection of deterministic laws that operate at a subquantum-mechanical level and that involve "hidden variables." An analogy given by the physicist Bohm is the Brownian motion of a dust particle in air. The particle undergoes random fluctuations of position, and its motion is not completely determined by its position and velocity. Of course, Brownian motion is a result of collisions with the gas molecules and is determined by variables existing on the level of
molecular motion. Analogously, the motions of electrons might be determined by hidden variables existing on a subquantum-mechanical level. The orthodox interpretation (often called the Copenhagen interpretation) of quantum mechanics, which was developed by Heisenberg and Bohr, denies the existence of hidden variables and asserts that the laws of quantum mechanics provide a complete description of physical reality. (Hidden-variables theories are discussed in F. J. Belinfante, A Survey of Hidden-Variables Theories, Pergamon, 1973.)
In 1964, J. S. Bell proved that, in certain experiments involving measurements on two widely separated particles that originally were in the same region of space, any possible local hidden-variable theory must make predictions that differ from those that quantum mechanics makes (see Ballentine, Chapter 20). In a local theory, two systems very far from each other act independently of each other. The results of such experiments agree with quantum-mechanical predictions, thus providing very strong evidence against all deterministic, local hidden-variable theories but do not rule out nonlocal hidden-variable theories. These experiments are described in A. Aspect, in The Wave-Particle Dualism, S. Diner et al. (eds.), Reidel, 1984, pp. 377-390; A. Shimony, Scientific American, Jan. 1988, p. 46; A. Zeilinger, Rev. Mod. Phys, 71, S288 (1999).
Further analysis by Bell and others shows that the results of these experiments and the predictions of quantum mechanics are incompatible with a view of the world in which both realism and locality hold. Realism (also called objectivity) is the doctrine that external reality exists and has definite properties independent of whether or not we observe this reality. Locality excludes instantaneous action-at-a-distance and asserts that any influence from one system to another must travel at a speed that does not exceed the speed of light. Clauser and Shimony stated that quantum mechanics leads to the "philosophically startling" conclusion that we must either "totally abandon the realistic philosophy of most working scientists, or dramatically revise our concept of space-time" to permit "some kind of action-at-a-distance" [J. F. Clauser and A. Shimony, Rep. Prog. Phys., 41, 1881 (1978); see also B. d’Espagnat, Scientific American, Nov. 1979, p. 158; A. Aspect, Nature, 446, 866 (2007); S. Gröblacher et al., Nature, 446, 871 (2007)].
Quantum theory predicts and experiments confirm that when measurements are made on two particles that once interacted but now are separated by an unlimited distance the results obtained in the measurement on one particle depend on the results obtained from the measurement on the second particle and also depend on which property of the second particle is measured. (Such particles are said to be entangled. For more on entanglement, see en.wikipedia.org/wiki/Quantum_entanglement; chaps. 7-10 of J. Baggott, Beyond Measure, Oxford, 2004; L. Gilder, The Age of Entanglement, Vintage, 2009; Part II of A. Whitaker, The New Quantum Age, Oxford, 2012.) Such instantaneous "spooky actions at a distance" (Einstein's phrase) have led one physicist to remark that "quantum mechanics is magic" (D. Greenberger, quoted in N. D. Mermin, Physics Today, April 1985, p. 38).
The relation between quantum mechanics and the mind has been the subject of much speculation. Wigner argued that the reduction of the wave function occurs when the result of a measurement enters the consciousness of an observer and thus "the being with consciousness must have a different role in quantum mechanics than the inanimate measuring device." He believed it likely that conscious beings obey different laws of nature than inanimate objects and proposed that scientists look for unusual effects of consciousness acting on matter. [E. P. Wigner, "Remarks on the Mind-Body Question," in The Scientist Speculates, I. J. Good, ed., Capricorn, 1965, p. 284; Proc. Amer. Phil. Soc., 113, 95 (1969); Found. Phys., 1, 35 (1970).]
In 1952, David Bohm (following a suggestion made by de Broglie in 1927 that the wave function might act as a pilot wave guiding the motion of the particle) devised a nonlocal deterministic hidden-variable theory that predicts the same experimental results as quantum mechanics [D. Bohm, Phys. Rev., 85, 166, 180 (1952)]. In Bohm's theory, a
particle at any instant of time possesses both a definite position and a definite momentum (although these quantities are not observable), and it travels on a definite path. The particle also possesses a wave function $\Psi$ whose time development obeys the time-dependent Schrödinger equation. In Bohm's theory, the wave function is a real physical entity that determines the motion of the particle. If we are given a particle at a particular position with a particular wave function at a particular time $t$, Bohm's theory postulates a certain equation that allows us to calculate the velocity of the particle at that time from its wave function and position; knowing the position and velocity at $t$, we can find the position at time $t+d t$ and can use the time-dependent Schrödinger equation to find the wave function at $t+d t$; then we calculate the velocity at $t+d t$ from the position and wave function at $t+d t$; and so on. Hence the path can be calculated from the initial position and wave function (assuming we know the potential energy). In Bohm's theory, the particle's position turns out to obey an equation like Newton's second law $m d^{2} x / d t^{2}=-\partial V / \partial x$ [Eqs. (1.8) and (1.12)], except that the potential energy $V$ is replaced by $V+Q$, where $Q$ is a quantum potential that is calculated in a certain way from the wave function. In Bohm's theory, collapse of the wave function does not occur. Rather, the interaction of the system with the measuring apparatus follows the equations of Bohm's theory, but this interaction leads to the system evolving after the measurement in the manner that would occur if the wave function had been collapsed.
Bohm's work was largely ignored for many years, but interest in his theory has increased. For more on Bohm's theory, see Whitaker, Chapter 7; D. Bohm and B. J. Hiley, The Undivided Universe, Routledge, 1992; D. Z Albert, Scientific American, May 1994, p. 58; S. Goldstein, "Bohmian Mechanics," plato.stanford.edu/entries/qm-bohm; en.wikipedia .org/wiki/De_Broglie-Bohm_theory. One of the main characters in Rebecca Goldstein's novel Properties of Light (Houghton Mifflin, 2000) is modeled in part on David Bohm.
Some physicists argue that the wave function represents merely our knowledge about the state of the system (this epistemic interpretation is used in the Copenhagen viewpoint), whereas others argue that the wave function corresponds directly to an element of physical reality (the ontic interpretation). A paper published in 2012 used certain mild assumptions to prove a result that the authors argued strongly favored the ontic interpretation; M. F. Pusey, J. Barrett, and T. Rudolph, Nature Phys. 8, 476 (2012) (arxiv.org/abs/1111.3328). For discussion of this result (called the PBR theorem), see www.aps.org/units/gqi/newsletters/upload/vol6num3.pdf; mattleifer.info/2011/11/20/ can-the-quantum-state-be-interpreted-statistically.
Although the experimental predictions of quantum mechanics are not arguable, its conceptual interpretation is still the subject of heated debate. Excellent bibliographies with commentary on this subject are B. S. DeWitt and R. N. Graham, Am. J. Phys., 39, 724 (1971); L. E. Ballentine, Am. J. Phys., 55, 785 (1987). See also B. d'Espagnat, Conceptual Foundations of Quantum Mechanics, 2nd ed., Benjamin, 1976; M. Jammer, The Philosophy of Quantum Mechanics, Wiley, 1974; Whitaker, Chapter 8; P. Yam, Scientific American, June 1997, p. 124. An online bibliography by A. Cabello on the foundations of quantum mechanics lists 12 different interpretations of quantum mechanics (arxiv.org/ abs/quant-ph/0012089); a Wikipedia article lists 14 interpretations of quantum mechanics (en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics).