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CRC Press eBooks

Probability, Entropy, and Quantum

Wojciech H. Zurek

Published Mar 8, 2018• 0 citations
Publisher

Abstract

The second law of thermodynamics imposes severe constraints on the formal structure of quantum theory. In particular, that law would be violated if it were possible to distinguish non-orthogonal states or if SchrOdinger's equation were nonlinear.
DOI
10.1201/9780429502880-9

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The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.
For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.
Given an unbiased die, the probability is ⅙ for a result. This probability value may be obtained directly by imposing the condition of a lack of bias and noting there are six possible results. Alternatively, one may maximize Shannon’s entropy subject to the constraint Sum over i p(i)=1 where i runs from 1 to 6. Maximization of Shannon’s entropy seems to remove as much bias as possible with the constraint indicating what bias remains. For the die case the constraint has no bias, but for an average energy constraint Sum over i ei p(i) there is, namely ei. In previous notes we have argued that the constraint used in the maximization of entropy is “special”. Here we argue that it represents a conserved quantity if interactions occur. Thus there may be many constraints in a system, but choosing a conserved quantity in an interaction is rarer, but seems to be linked with equilibration. Finally, we suggest that maximization of entropy subject to a conserved quantity constraint removes as much bias as possible and it is this maximized entropy-probability which is subject to the second law of thermodynamics. It is possible, however, to have probabilities and to create entropies which are not based on maximization of entropy. For example, Shannon’s entropy may be applied to the spatial density of various n-slit interactions for a particle with constant momentum p. In this case, even though there is no maximization of entropy, the emerging particle has the same p magnitude (hence same uncertainty within a wavelength), but there is the added uncertainty of its direction. As a result, it seems that entropy increases in accordance with the second law. A free particle with momentum p, however, has a two dimensional probability distribution in space exp(ipx) characterized by a wavelength hbar/p. This may be taken as related to the entropy of the free particle, but a free particle may undergo spontaneous interactions which either increase or decrease p, hence increase or decrease hbar/p and entropy. In other words, the second law does not apply to such an entropy, but this entropy is associated with an unusual constraint. This conclusion seems to hold when one considers two body scattering. Thus we suggest that there are various kinds of entropy. Some are linked with maximization of entropy subject to a conserved quantity constraint (and are associated with equilibration) and others are not. The latter do not necessarily obey the second law of thermodynamics, we argue.