chem-istry, like the properties of the periodic table, molecular geometry, etc. Among the books providing accessible introductions to QM we mention: the very nice elementary text by Enge (1972), the concise introduction by Landshoff (1998), McGervey (1995) which focus on wave mechanics, and Heitler (1956) which focus on quantum chemistry.
Quantum Mechanics was also the basis for the development of completely new tech-nologies. Among the most distinguished examples are solid-state or condensed matter electronic devices such as transistors, integrated circuits, lasers, liquid crystals, etc.. These devices constitute, in turn, the basic components of modern digital computers. Finally, one can argue that computer based information processing tools are among the most revolutionary technologies introduced in human society, having had an impact in its or-ganization comparable only to a handful of other technologies (perhaps the steam and internal combustion engines, or electric power), see XX (20xx).
Nevertheless, all this success was not for free. Quantum Mechanics required the re-thinking and re-interpretation of some of the most fundamental concepts of science. In this and the next sections we analyze the impact of Quantum Mechanics on the most important concept of statistical science, namely, probability.
Although Scr¨odinger arrived at the appropriate functional form of a wave equation for Quantum Mechanics, the adequate interpretation for the wave function,ψ, was given only a few months later by Max Born. According to Born’s interpretation: The probability density of “finding” the particle at position x, is proportional to the square of the wave function absolute amplitude, |ψ(x)|2. Since, in the general case,ψ is a complex function, the last quantity can also be written as the product of the wave function by its complex conjugate, that is, |ψ(x)|2 =ψ∗ψ.
From this interpretation of the wave function, we can understand Max Born’s formu-lation of ‘the core metaphor of wave mechanics’, as quoted in Pais (1988, ch.12, sec.d, p.258),
“The essence of wave mechanics: ‘The motion of particles follows probabil-ity laws but the probabilprobabil-ity itself propagates according to the law of causalprobabil-ity.”
This is a revolutionary interpretation, that attributes to the concept of probability a new and distinct ‘objective’ character. Hence, it is interesting to have some insight on the genesis of Born’s interpretation. Born’s own recollections are presented at Pais (1988, ch.12, sec.d, p.258-259):
“What made Born take his step?
In 1954 Born was awarded the Nobel Prize ‘for his fundamental research, specially for his statistical interpretation of the wave function’. In his ac-ceptance speech Born, then in his seventies, ascribed his inspiration for the
statistical interpretation to ‘an idea of Einstein’s [who] had tried to make the duality of particles - light-quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the ψ-function: |ψ|2 ought to represent the probability density of electrons.’ ”
5.6.3 Classic and Quantum Probability
One of the favorite metaphors used by the orthodox Bayesian school describes the sci-entist’s work as a game against nature, with the objective of scoring a good guess on
“nature’s true state”. Implicit in this metaphor is the assumption that such a “true state of nature” exists and is, at least in principle, accessible. In this paradigm, omniscience is usually a matter of money, that is, with enough economic resources all pertinent in-formation can, at least in principle, be acquired, see Blackwell and Girshick (1954), for example.
“Statistics can be viewed as a game against nature.” (p.75).
“...games where one of the players is not faced with an intelligent opponent but rather with an unknown state of nature.” (p.121).
“The same theory that served to delineate optimal strategies in games played against an intelligent opponent will serve to delineate classes of op-timal strategies in games played against nature.” (p.123).
“What prevents the statistician from getting full knowledge of ω [the state of nature] by unlimited experimentation is the cost of experiments.” (p.78).
This paradigm seems incompatible with, or at least very unfriendly to, Born’s proba-bilistic interpretation of Quantum Mechanics and Heisenberg’s uncertainty principle. We believe that, in the context of quantum mechanics, the strictly subjective interpretation of probability is, please forgive the pun, a very risky metaphor, and that pushing this metaphor where it does not belong will lead to endless paradoxes. In Chapter 7 of his book, The Physics of Chance, for example, Charles Ruhla presents the adventures of the simple-minded hero Monsieur de La Palice, struggling to understand some basic quantum experiments.
For a strict subjectivist the situation is even worse, and the use of Quantum Mechanics is at risk of being considered illegal. A statement giving the current best estimate of h (Planck’s constant) toghether with its standard deviation was presented in section 5.6.1. Since h appears at the right hand side of Heisenberg’s uncertainty principle, an uncertainty about the value of h implies a second order uncertainty. The propagation of the uncertainty about the value of fundamental physical constants generates similar second order probabilistic statements about the detection, mesurement or observation
5.6 VARIETIES OF PROBABILITIES 157 of quantum phenomena. For example, section 5.7.2 arrives at statements giving the (probabilistic) uncertainty of (probabilistic) transition rates. All these are prototypical examples of statements that are categorically forbidden in orthodox Bayesian statistics, as bombastically proclaimed in the following quotations from Finetti (1977, p.1,5 and 1972, p.190), see also Mosleh and Bier (1996) and Wechsler et al. (2005).
“Does it make sense to ask what is the probability that the probability of a given event has a given value, pi ? ... It makes no sense to state that the probability of an event E is to be regarded as unknown in that its true value is one of the pi’s, but we do not know which one.”
“Speaking of unknown probabilities [or of probability of a probability] must be forbidden as meaningless.”
A similar statement of de Finetti was analyzed in section 4.7. Such an awkward position, at least for a modern physicist, was seen by the founding fathers of orthodox Bayesian statistics as an unavoidable consequence of the subjectivist doctrine, according to which,
“Probabilities are states of mind, not of nature.” Savage (1981, p.674).
From a constructivist perspective, fundamental physical constants, including of course Planck’s constant, correspond to very objective (very sharp, stable, separable and com-posable) eigenvalues of Physics’ research program, and it is perfectly admissible to speak about the uncertainty of their estimated values. Of course that is what physicists need to do, and have done for almost a century, regardless of being disapproved by the Bayesian orthodoxy (theoretically coherent, but understandably very shy and timid). There have also been some attempts to reconcile a strict subjectivist position with modern physics, through long and sophisticated translations of simple “crude” statements like the ones quoted above. Some of these translations are as bizarre and / or intricately involved as similar attempts to translate epistemic probabilistic statements that are categorically for-bidden in frequentist statistics into “acceptable” frequentist probabilistic statements, see section 2.5 and Rouanet et al. (1998, Preamble). Richard Feynman (2002, p.14), makes the following comments on some ideas behind some of such interpretations:
“Now, the philosophical question before us is, when we make an observation of our track in the past, does the result of our observation become real in the same sense that the final state would be defined if an outside observer were to make the observation? This is al very confusing, especially when we consider that even though we may consistently consider ourselves always to be the outside observer when we look at the rest of the world, the rest of the world is at the same time observing us, and that often we agree on what we
see in each other. Does this mean that my observations become real only when I observe an observer observing something as it happens? This is an horrible viewpoint. Do you seriously entertain the thought that without observer there is no reality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Was there no reality before 109 B.C. before life began?
Or are you the observer? Then there is no reality to the world after you are dead? I know a number of otherwise respectable physicists who have bought life insurance. By what philosophy will the universe without man be understood?
In order to make some sense here, we must keep an open mind about the possibility that for sufficiently complex systems, amplitudes become probabili-ties....”
In order to provide deeper insight on the meaning of Heisenberg’s uncertainty princi-ple, let us link it to Noether’s theorems, already discussed in section 2.8.1. The central point of Noether’s theorems lies in the existence of an invariant physical quantity for each continuous symmetry group in a physical theory. Heisenberg’s uncertainty relation, pre-sented in section 6.1, sets a bound on the accuracy with which we can access, by means of physical measurements, such symmetry / invariant dual or conjugate pairs. This point is further analyzed by Bohr:
“...we admire Planck’s happy intuition in coining the term ‘quantum of ac-tion’ which directly indicates a renunciation of the action principle, the central position of which in the classical description of nature he himself has empha-sized on more than one occasion. This principle symbolizes, as it were, the peculiar reciprocal symmetry relation between the space-time description and the laws of conservation of energy and momentum, the great fruitfulness of which, already in classical physics, depends upon the fact that one may exten-sively apply them without following the course of the phenomena in space and time.” (p.94 or 210).
“Indeed, the inevitability of using, for atomic phenomena, a mode of de-scription which is fundamentally statistical arises from a closer investigation of the information which we are able to obtain by direct measurement of these phenomena and the meaning we may ascribe, in this connection, to the appli-cation of the fundamental physical concepts...
Such considerations lead immediately to the reciprocal uncertainty relations set up by Heisenberg and applied by him as the basis of a thorough investigation of the logical consistency of quantum mechanics.” (p.113-114 or 247-248).
In the articleSpace-Time Continuity and Atomic Physics, Bohr (1935, p.370) further explores the relation between quantization and our use of probabilistic language:
“With the forgoing analysis we have described the new point of view brought