Quantum Statistical Mechanics. I. Decoherence, Wave Function Collapse, and the von Neumann Density Matrix Phil Attard (Dated: 31 Oct., 2013, [email protected]) The probability operator is derived from first principles for an equilibrium quantum system. It is also shown that the superposition states collapse into a mixture of states giving the conventional von Neumann trace form for the quantum average. The mechanism for the collapse is found to be quite general: it results from the conservation law for a conserved, exchangeable variable (such as energy) and the entanglement of the total system wave function that necessarily follows. The relevance of the present results to the einselection mechanism for decoherence, to the quantum measurement problem, and to the classical nature of the macroscopic world are discussed. I. INTRODUCTION reservoir or measuring device. A brief summary of the conventional einselection approach to decoherence and Perhaps the most puzzling aspect of quantum mechan- a discussion of the similarities and differences with the ics is that a quantum system may exist simultaneously in present approach is given in Appendix A. a superposition of states, whereas in the classical world The present paper derives a formula for the statisti- a system can only ever be in one state at a time. The cal average of a canonical equilibrium system (i.e. a sub- precise process whereby a quantum system with its al- system that can exchange energy with a reservoir). It is lowed superposition of states passes to the observed clas- shown that due to the conservation law the wave function sical limit of pure states is the subject of ongoing de- of the total system necessarily has an entangled form. bate; Schr¨odinger’s cat is periodically resuscitated for It is this that causes the system to collapse from a su- post mortem examination. perposition of states to a mixture of pure states. Both the Maxwell-Boltzmann probability operator and the von In quantum statistical mechanics, the superposition Neumann density matrix emerge from the first principles problem disappears in the conventional formulation in derivation. terms of the von Neumann density matrix.1–4 This ex- presses the statistical aspects of the problem as an en- semble or mixture of systems, each in a pure quantum II. DERIVATION state, which is akin to the classical experience of one state at a time. It is axiomatic that Two fundamental questions naturally arise: how does one get from the underlying quantum superposition wave all microstates of an isolated system (1) function to a mixture of quantum states, and how does have equal weight. one derive the statistical probability distribution from quantum mechanical first principles? Both questions are See Appendix B for a justification. This axiom is im- addressed in this paper. plied by the standard interpretation of quantum mechan- ics: representing the system in the basis corresponding The appearance of a mixture of states from a superpo- to some operator, |ψi = ψO |ζO i, with Oˆ|ζO i = sition of states is referred to here as the collapse of the kg kg kg kg O O O∗ system or wave function. This is of course closely related Ok|ζkg i, the product ψkgψkgP is interpreted as being pro- to the traditional use of the word ‘collapse’ in reference to portional to the probability of the system being in the mi- O 2,3 the transition of a quantum system from a superposition crostate ζkg . With this, the expectation value of the op- O∗ O O∗ O of states to a single eigenstate following the application erator is simply O(ψ) = kg ψkg ψkg Ok/ kg ψkg ψkg. of an operator. The so-called measurement problem, the If the microstates did notP have equal weight,P then this emergence of the mixed state von Neumann density ma- expression for the expectation value would need to in- arXiv:1401.1786v1 [cond-mat.stat-mech] 25 Nov 2013 trix, and the classical nature of the macroscopic world clude an additional factor proportional to the weight of are all closely related phenomena. each microstate. One promising approach to the problems of wave func- Consider an isolated sub-system with Hamiltonian op- tion collapse and the classical behavior of macroscopic erator Hˆ and orthonormal energy eigenfunctions such ˆ E E objects explores the decoherence that arises from the in- that H|ζkg i = Ek|ζkgi. Here k is the principle energy teractions of a sub-system with its environment.5–12 Es- quantum number and labels the energy macrostates, and E sentially the approach says that the environment or reser- g = 1, 2,...,Nk labels the degenerate energy states; ζkg voir coupled to the sub-system causes the wave function labels the microstates of the system in the energy repre- to collapse into a mixture of pure quantum states (envi- sentation. ronmental selection, or einselection). That is, the super- From the equal weight condition, the weight of the en- E posed components of the original sub-system wave func- ergy microstates can be taken to be wkg = 1 and the E E tion are rendered decoherent by the interaction with the entropy of the energy microstates is Skg = kB ln wkg = 0, 2 r s where kB is Boltzmann’s constant. Note that the same such that Ej1 = Etot − Ei1 . Assuming that the coeffi- s r value is used irrespective of the energy of the isolated sys- cient matrix is dyadic, cig,jh = cigcjh, then for all j 6= j1 r tem and whether or not this is equal to Ek. According one must have cjh = 0. But this would mean that there to the set theoretic formulation of probability for statis- is only one value of the reservoir energy, and hence that 13,14 tical mechanics, the weight of an energy macrostate Ei1 is the only possible value of the sub-system energy. E E is therefore wk = g wkg = Nk and the entropy of an Since this contradicts the fundamental definition of the energy macrostateP is just the logarithm of the number statistical system, namely that the sub-system can ex- E of microstates that it contains, Sk = kB ln Nk. This change energy with the reservoir, one concludes that the is Boltzmann’s original formula. Finally, soon will be coefficient matrix cannot be dyadic and that the wave invoked the thermodynamic definition of temperature, state of the total system must be entangled.15 namely13,14 Using the entangled form of the total wave function, and the vanishing of the coefficients for terms that vi- 1 ∂S(E) = . (2) olate energy conservation, the expectation value of an T ∂E arbitrary operator on the sub-system is Now allow the sub-system s to exchange energy with 1 (s) (r) ∗ O(ψ ) = c ′ ′ ′ ′ c a reservoir r such that the total system is isolated and tot Z′ i g ,j h ig,jh ig,iX′g′ jh,jX′h′ therefore Etot = Es + Er is constant. The Hilbert space s,E ˆ s,E r,E r,E is Htot = Hs ⊗Hr. If it is possible to factor the wave state × hζi′ g′ |O|ζig ihζj′ h′ |ζjh i of the total system as 1 (s) (r) ∗ s,E = ′ ci′g′,jhcig,jhOi′g′,ig Z ′ ′ |ψtoti = |ψsi |ψri , (3) ig,iXg Xjh 1 (s) (r) ∗ s,E = c ′ c O ′ . (7) then the wave state is said to be a separable state, or a Z′ ig ,jih ig,ji h ig ,ig product state. Here and below the direct product symbol i,g,gX′ Xh on the right hand side is suppressed. Conversely, with ′ s r Here the norm is Z ≡ hψtot|ψtoti, ji is defined such that {|ζi i} a basis for the sub-system and {|ζj i} a basis for r s Eji = Etot − Ei , and the energy conservation condition, the reservoir, the most general wave state of the total Eq. (6), has been used. system is One sees how energy conservation and entanglement s r have collapsed the original four sums over independent |ψtoti = cij |ζi i|ζj i. (4) principle energy states to a single sum over principle en- Xi,j ergy states. Specifically the principle energy states of s r the sub-system have at this stage collapsed, whilst the If the coefficient matrix is dyadic, cij = ci cj , then the (s) s s degenerate energy states of the sub-system remain in su- wave state is separable with |ψsi = i ci |ζi i, and perposition form. (r) r r |ψri = j cj |ζj i. Alternatively, if the coefficientsP can’t An expression for the reservoir entropy will be required be writtenP in dyadic form, then the wave state is insepa- for the two derivations of the statistical average that fol- 2,3 rable. This is called an entangled state. low. One has the axiomatic result given above that the Whether or not the total system can be written as reservoir entropy for a reservoir energy macrostate is the a product state or is necessarily an entangled state is logarithm of the number of degenerate states, the main difference between the present analysis and the 5–12 r r r r (r) conventional decoherence einselection approach. (See S (Ej )= kB ln Nj , where Nj ≡ . (8) Appendix A.) Xh In general conservation laws give rise to entangled This must equal the general thermodynamic result that, states. This may be seen explicitly for the present case given a total energy Etot and a sub-system energy Es, of energy exchange, with Etot = Es + Er fixed. Using the the reservoir entropy is13,14 energy eigenfunctions described above as a basis for the Es sub-system and the reservoir, the most general expansion Sr(Er)= Sr(Etot − Es) = const.