Quantum Mechanics Entropy and a Quantum Version of the H-Theorem

Entropy is a fundamental concept which emerged along with other ideas during the development of thermodynamics and statistical mechanics Landau and Lifshitz (1978); Lieb and Yngvason (1999). Entropy has developed foremost out of phenomenological thermodynamical considerations such as the second law of thermodynamics in which it plays a prominent role Wehrl (1978). With the intense interest in the investigation of the physics of matter at the atomic and subatomic quantum levels, it may well be asked whether this concept can emerge out of the study of systems at a more fundamental level. In fact, it may be argued that a correct definition is only possible in the framework of quantum mechanics, whereas in classical mechanics, entropy can only be introduced in a rather limited and artificial way. Entropy relates macroscopic and microscopic aspects of nature, and ultimately determines the behavior of macroscopic systems. It is the intention here to present an introduction to this subject in a readable manner from the quantum point of view. There are many reasons for undertaking this. The intense interest in irreversible thermodynamics Grössing (2008), the statistical mechanics of astrophysical objects Padmanabhan (1990); Pathria (1977), quantum gravity and entropy of black holes Peres & al. (2004), testing quantum mechanics Ballentine (1970) and applications to condensed matter and quantum optics Haroche & al. (2006); Raimond & al. (2001) are just a few areas which are directly or indirectly touched on here. Let us begin by introducing the concept of entropy from the quantum mechanical perspective, realizing that the purpose is to focus on quantum mechanics in particular. Quantum mechanics makes a clear distinction between observables and states. Observables such as position and momentum are mathematically described by self-adjoint operators in a Hilbert space. States, which are generally mixed, can be described by a density matrix, which is designated by ρ throughout. This operator ρ is Hermitean, has trace one and yields the expectation value of an observable A in the state ρ through the definition


Introduction
Entropy is a fundamental concept which emerged along with other ideas during the development of thermodynamics and statistical mechanics Landau and Lifshitz (1978); Lieb and Yngvason (1999).Entropy has developed foremost out of phenomenological thermodynamical considerations such as the second law of thermodynamics in which it plays a prominent role Wehrl (1978).With the intense interest in the investigation of the physics of matter at the atomic and subatomic quantum levels, it may well be asked whether this concept can emerge out of the study of systems at a more fundamental level.In fact, it may be argued that a correct definition is only possible in the framework of quantum mechanics, whereas in classical mechanics, entropy can only be introduced in a rather limited and artificial way.Entropy relates macroscopic and microscopic aspects of nature, and ultimately determines the behavior of macroscopic systems.It is the intention here to present an introduction to this subject in a readable manner from the quantum point of view.There are many reasons for undertaking this.The intense interest in irreversible thermodynamics Grössing (2008), the statistical mechanics of astrophysical objects Padmanabhan (1990); Pathria (1977), quantum gravity and entropy of black holes Peres & al. (2004), testing quantum mechanics Ballentine (1970) and applications to condensed matter and quantum optics Haroche & al. (2006); Raimond & al. (2001) are just a few areas which are directly or indirectly touched on here.Let us begin by introducing the concept of entropy from the quantum mechanical perspective, realizing that the purpose is to focus on quantum mechanics in particular.Quantum mechanics makes a clear distinction between observables and states.Observables such as position and momentum are mathematically described by self-adjoint operators in a Hilbert space.States, which are generally mixed, can be described by a density matrix, which is designated by ρ throughout.This operator ρ is Hermitean, has trace one and yields the expectation value of an observable A in the state ρ through the definition A = Tr (ρA). (1.1) Entropy is not an observable, so there does not exist an operator with the property that its expectation value in some state would be the entropy.In fact, entropy is a function of state.If where |k is a normalized eigenvector corresponding to the eigenvalue p k and |k k| is a projection operator onto |k with p k ≥ 0 and ∑ k p k = 1.Here the coefficients are positive probabilities and not complex amplitudes as in a quantum mechanical superposition.Substituting (1.4) into (1.2) finally yields, S(ρ)=− ∑ k p k log(p k ). (1.5) There is a more combinatorial approach Wehrl (1978).This will come up again subsequently when ensembles take the place of a density operator.If N measurements are performed, one will obtain as a result that for large N, the system is found p 1 N times in |1 , p 2 N times in state |2 and so on, all having the same weight.By straightforward counting, there results When N → ∞, Stirling's formula can be applied to the logarithm of (1.6) so the entropy is (n j log n j − n j )=−N ∑ j p j log(p j ). (1.7) Dividing both sides of (1.7) by N, then as N → ∞ (1.5) is recovered.It should also be noted that (1.5) is of exactly the same form as Shannon entropy, which can be thought of as a measure of unavailable information.
Of course, another way to look at this is to consider N copies of the same Hilbert space, or system, in which there are microstates |1 ⊗|2 ••• such that |1 occurs p 1 N times, |2 occurs p 2 N times, and so forth.Again (1.6) is the result, and according to Boltzmann's equation, one obtains log(W N ) for the entropy as in (1.5).In (1.5), S is maximum when all the p j are equal to 1/N.By invoking the constraint ∑ k p k = 1, (1.5) takes the form where p N = 1 − ∑ N−1 k=1 p k , and all other p k are considered to be independent variables.Differentiating S in (1.8), it is found that This vanishes of course when p k = p N = N −1 and this solution is the only extremum of S.
To summarize, entropy is a measure of the amount of chaos or lack of information about a system.When one has complete information, that is, a pure state, the entropy is zero.Otherwise, it is greater than zero, and it is bigger the more microstates exist and the smaller their statistical weight.

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Basic properties of entropy
There are several very important properties of entropy function (1.5) which follow from simple mathematical considerations and are worth introducing at this point Peres (1995).
The first point to make is that the function S(p) is a concave function of its arguments p =( p 1 , ••• , p N ).For any two probability distributions {p j } and {q j }, and any λ ∈ [0, 1], S defined in (1.5) satisfies the following inequality This can be proved by differentiating S twice with respect to λ to obtain, This is a sufficient condition for a function to be concave.Equality holds only when p j = q j , for all j.The physical meaning of inequality (2.1) is that mixing different probability distributions can only increase uniformity.
If N is the maximum number of different outcomes obtainable in a test of a given quantum system, then any test that has exactly N different outcomes is called a maximal test, called T here.Suppose the probabilities p m for the outcomes of a maximal test T which can be performed on that system are given.It can be shown that this entropy never decreases if it is elected to perform a different maximal test.The other test may be performed either instead of T, or after it, if test T is repeatable.
To prove this statement, suppose the probabilities for test T are {p m } and those for a subsequent test are related to the {p m } by means of a doubly stochastic matrix P μm .This is a matrix which satisfies ∑ μ P μm = 1 and ∑ m P μm = 1.In this event, are the probabilities for the subsequent test.The new entropy is shown to satisfy the inequality S(q) ≥ S(p).To prove this statement, form the difference of these entropies based on (1.5), ).
In the second line, ∑ μ P μm = 1 has been substituted to get this result.Using the inequality log x ≥ 1 − x −1 , where equality holds when x = 1, and the fact that S has a negative sign, it follows that

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Theoretical Concepts of Quantum Mechanics www.intechopen.com The equality sign holds if and only if P μm is a permutation matrix, so the sets are identical.After a given preparation whose result is represented by a density matrix ρ, different tests correspond to different sets of probabilities, and therefore to different entropies.The entropy of a preparation can be defined as the lowest value attained by (1.5) for any complete test performed after that preparation.The optimal test which minimizes S is shown to be the one that corresponds to the orthonormal basis v μ given by the eigenvectors of the density matrix In this basis, ρ is diagonal and the eigenvalues w μ satisfy 0 ≤ w μ ≤ 1 and ∑ μ w μ = 1.
A basic postulate of quantum mechanics asserts that the density matrix ρ completely specifies the statistical properties of physical systems that were subjected to a given preparation.All the statistical predictions that can be obtained from (1.1) for an operator are the same as if we had an ordinary classical mixture, with a fraction w μ of the systems with certainty in the state v μ .Therefore, if the maximal test corresponding to the basis v μ is designed to be repeatable, the probabilities w μ remain unchanged and entropy S remains constant.The choice of any other test can only increase the entropy, as in the preceding result.This proves that the optimal test, which minimizes the entropy, is the one corresponding to the basis that diagonalizes the density matrix. (2.5) The matrix P μm is doubly stochastic, so as in (2.3) we have Equality holds in (2.6) if and only if σ = ρ.Inequality (2.6) can be used to prove a subadditivity inequality.Consider a composite system, with density matrix ρ, then the reduced density matrices of the subsystems are called ρ 1 and ρ 2 .Then matrices ρ, ρ 1 and ρ 2 satisfy, S(ρ) ≤ S(ρ 1 )+S(ρ 2 ). (2.7) This inequality implies that a pair of correlated systems involves more information than the two systems separately.
Consider now the relative entropy It has just been shown that relative entropy is nonnegative, so it follows from this that Tr (ρ log ρ) ≥ Tr (ρ log ρ 1 )+Tr (ρ log ρ 2 ).

Entanglement and entropy
The superposition principle applied to composite systems leads to the introduction of the concept of entanglement Mintet & al. (2005); Raimond & al. (2001), and provides an important application for the density matrix.A very simple composite object is a bipartite quantum system S which is composed of two parts A and B.
where the α iμ are complex amplitudes.The states |ψ S contain information not only about the results of measurements on A and B separately, but also on correlations between these measurements.In an entangled state, each part loses its quantum identity.The quantum content of the global state is intricately interwoven between the parts.Often it is the case that there is interest in carrying out measurements on one part without looking at another part.For example, what is the probability of finding a result when measuring observable O A attached to subsystem A, without worrying about B. The complete wave function |ψ S can be used to predict the experimental outcomes of the measurement of O A ⊗ 1 B .This can also be done by introducing the density operator ρ S of a system described by the quantum state |ψ S , which is just the projector It has the same information content as |ψ S , and for all predictions on S, all quantum rules can be expressed in such a fashion; for example, the expectation values of an observable O S of S is found by (1.1).The probability of finding the system in |i after a measurement corresponding to the operator ρ i = |i i| is given by | i|ψ S | 2 in the quantum description and Tr (ρ i ρ S ) in terms of the density matrix.
The density operator approach is very advantageous for describing one subsystem, A, without looking at B. A partial density operator ρ A can be determined which has all the predictive information about A alone, by tracing ρ S over the subspace of B Thus, the probability of finding A in state |j A is found by computing the expectation value of the projector ρ j = |j A j A |, which is π j = Tr (ρ A ρ j ).Predictions on A can be done without considering B. The information content of ρ A is smaller than in ρ S , since correlations between A and B are omitted.To say that A and B are entangled is equivalent to saying that ρ A and ρ B are not projectors on a quantum state.There is however a basis in H A in which ρ A is diagonal.Let us call it |j A , so that ρ A is given by The | jB are mirroring in H B the basis of orthonormal states in H A in which ρ A is diagonal.These mirror states are also orthogonal to each other as can be seen by expressing the fact that ρ is diagonal At this point, the mirror state can be normalized by means of the transformation | ĵB = |j B / λ j giving rise to the Schmidt expansion, (3.7) The sum over a basis of product mirror states exhibits clearly the entanglement between A and B. The symmetry of this expression shows that ρ A and ρ B have the same eigenvalues.Any pure entangled state of a bipartite system can be expressed in this way.Now a measure of the degree of entanglement can be defined using the density matrix.As the λ j become more spread out over many non-zero values, more information is lost by concentrating on one system and disregarding correlations between A and B. This loss of mutual information can be linked to the degree of entanglement.This information loss could be measured by calculating the von Neumann entropy of A or B from (1.5) This is the entropy of entanglement S e = S A = S B , and it expresses quantitatively the degree of disorder in our knowledge of the partial density matrices of the two parts of the entangled system S.
If the system is separable, then one λ j is non-zero and S e = 0, so maximum information on the states of both parts obtains.As soon as two λ j are non-zero, S e becomes strictly positive and A and B are entangled.The maximum entropy, hence maximum entanglement obtains when the λ j are equally distributed among the A and B subspaces.It is maximal and equal to log N A , when ρ A is proportional to 1 A , that is ρ A = 1 A /N A .In a maximally entangled state, local measurements performed on one part of the system are not predictable at all.What can be predicted are the correlations between the measurements performed on both parts.For example, consider a bipartite system in which one part has dimension two.There are only two λ-values in the Schmidt expansion, and satisfy λ 1 + λ 2 = 1.Then from (1.5), the entropy when The degree of entanglement is equal to zero when λ 1 = 0 or 1 and passes through a maximum at λ 1 = 1/2 at which S e = 1.The degree of entanglement measured by the von Neumann entropy is invariant under local unitary transformations acting on A or B separately, a direct consequence of the invariance of the spectrum of the partial density operators.

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Theoretical Concepts of Quantum Mechanics www.intechopen.com Consider the case of a two-level system with states |0 and |1 , where the density matrix is a two-by-two hermitean matrix given by ρ A = ρ 00 ρ 01 ρ 10 ρ 11 . (3.10) The entropy can be calculated for this system.Its positive diagonal terms are the probabilities of finding the system in |0 or |1 and they sum to one.The nondiagonal terms satisfy ρ 01 = ρ * 10 and are zero for a statistical mixture of |0 and |1 .Since ρ A is a positive operator is satisfied, and the upper bound is reached for pure states.
The density matrix ρ A can be expanded with real coefficients onto the operator basis made up of the identity matrix I and the Pauli matrices where R =(u, v, w) is three-dimensional and σ =(σ x , σ y , σ z ).The components of R are linked to the elements of the density matrix as follows The modulus R of R satisfies R ≤ 1, equality holding only for pure states.This follows from Tr (ρ 2 A ) ≤ 1.If nonlinear functions of an observable A are defined as ). (3.12) To each density matrix ρ A , the end of the vector R can be located on the surface of a sphere.The surface of the sphere R = 1 is the set of pure states with S = 0.The statistical mixtures correspond to inside the sphere R < 1.The closer the point to the center, the larger the von Neumann entropy.The center of the sphere corresponds to the totally unpolarized maximum entropy state.Any mixed state can be represented in an infinite number of ways as a statistical mixture of two pure states, since any P with its end inside the sphere can be expressed as a vector sum of a P 1 and P 2 whose ends are at the intersection of the sphere with an arbitrary line passing by the extreme end of P, so one can write P = λP 1 +(1 − λ)P 2 for 0 < λ < 1.The density matrix which is a linear function of P is then a weighted sum of the projectors on the pure states |u 1 and |u 2 corresponding to P 1 and P 2 , Thus, there exists an ambiguity of representation of the density operator which, if P = 0, can be lifted by including the condition that |u 1 and |u 2 be orthogonal.Before finishing, it is worth discussing the following application, which seems to have very important ramifications.A violation of the second law arises if nonlinear modifications are

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Quantum Mechanics Entropy and a Quantum Version of the H-Theorem www.intechopen.comintroduced into Schrödinger's equation Weinberg (1989).A nonlinear Schrödinger equation does not violate the superposition principle in the following sense.The principle asserts that the pure states of a physical system can be represented by rays in a complex linear space, but does not demand that the time evolution obeys a linear equation.Nonlinear variants of Schrödinger's equation can be created with the property that if u(0) evolves to u(t) and v(0) to v(t), the pure state represented by u(0)+v(0) does not evolve into u(t)+v(t), but into some other pure state.
The idea here is to show that such a nonlinear evolution violates the second law of thermodynamics.This is provided the other postulates of quantum mechanics remain as they are, and that the equivalence of the von Neumann entropy to ordinary entropy is maintained.Consider a mixture of quantum systems which are represented by a density matrix where 0 < λ < 1 and Π u , Π v are projection operators on the pure states u and v.In matrix form the density matrix is represented as The eigenvalues are found by solving the polynomial det(ρ − w1)=0 for the eigenvalues w.

Ensemble methods in quantum mechanics
In classical mechanics, one relinquishes the idea of a description of the microscopic mechanical states of trillions of microscopic interacting particles by instead computing averages over a virtual ensemble of systems which replicate the real system.Quantum theory is faced with a similar problem, and the remedy takes the form of the Gibbs ensemble.This last section will take a slightly different track and discusses ensemble theory in quantum mechanics.Two of the main results will be to produce a quantum version of the H-Theorem, and to show how the quantum mechanical canonical ensemble can be formulated.An astronomic number of states, or of microstates, is usually compatible with a given set of macroscopic parameters defining a macrostate of a thermophysical system.Consequently, a virtual quantum mechanical ensemble of systems is invoked, which is representative of the real physical system.The logical connection between a physical system and ensemble is made by requiring the time average of a mechanical property G of a system in thermodynamic equilibrium equal its ensemble average calculated with respect to an ensemble made up of N * → ∞ systems representing the actual system The ensemble average G is the ordinary mean of G over all the systems of the ensemble.If N * r systems are in a state with eigenvalue G r corresponding to G, where the sum is over all allowed states.Adopt as a basic set the states ψ jrm••• uniquely identifiable by the quantum numbers j, r, m, ••• referring to a set of compatible properties.A particular system of the ensemble will not permanently be in one of these states ψ jrm••• , as there exists only a probability to find a system in any one.Let us compress the basic states to read ψ jr if we let r stand for the entire collection of quantum numbers r, m, •••.These cannot strictly be eigenstates of the total energy, since a system occupying a particular eigenstate of its total Hamiltonian H at any one moment will remain in this state forever.The state of the real system, which the ensemble is to represent, is a superposition of eigenstates belonging to the same or different values of the energy.To obtain an ensemble where the individual members are to change, we suppose the basic set ψ jr is made up of eigenstates of the unperturbed Hamiltonian H 0 .Assume it is possible to write such that H 1 is a small perturbation added to the unperturbed Hamiltonian H 0 , and vary with the physical system considered.

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Quantum Mechanics Entropy and a Quantum Version of the H-Theorem www.intechopen.comSuppose E 0 j are the eigenvalues of the unperturbed H 0 and ψ 0 jr the eigenstates corresponding to them, where r again denotes a set of compatible quantum numbers.Introducing H 1 now changes the energy eigenvalues and energy eigenfunctions by an amount E 1 jr and ψ 1 jr , which should be very small compared with the unperturbed values.It is precisely the eigenstates ψ 0 jr of H 0 rather than H that are used as basic states for the construction of the ensemble.Since these for the most part will appear in what follows, we continue to omit the superscript for both the eigenfunctions ψ jr and eigenvalues E jr whenever the situation indicates that unperturbed quantities are intended.A perturbed system finding itself initially in any one of the unperturbed states ψ jr does not remain indefinitely in this state, but will continually undergo transitions to other unperturbed states ψ ks due to the action of the perturbation H 1 .In analogy with a classical system, a quantum ensemble is described by the number of systems N * jr in each state ψ jr .The probability P jr of finding a system, selected at random from the ensemble, in the state ψ jr is clearly This is the change in occupation number over (0, t).Dividing (4.7) by N * and using (4.4) gives A stationary ensemble or one in statistical equilibrium defined as N * ks (t)=N * ks (0) for all ks holds when N * jr (0)=N * ks (0), at least when A jr ks (t) = 0.The contribution to the right side of (4.8) comes from an extremely narrow interval ΔE = 2h/t centered at E j = E k , as indicated by 480 Theoretical Concepts of Quantum Mechanics www.intechopen.comperturbation theory.In this interval, it can be assumed P jr (0) depends on the j-index weakly enough that we can use P ks (0) in their place, so the term in brackets in (4.8) does not depend on j.The energy spectrum is very nearly continuous for a thermophysical system, so the sum over j can be approximated by an integral over E. This implies an approximation of the form (4.9) The quantities W (k) sr are time independent provided H 1 is time independent.Consequently, they are nonnegative and depend only on the displayed indices.Substituting (4.9) and P jr (0)=P kr (0) into (4.8)gives 1 t (4.10) In the limit when t becomes arbitrarily small, (4.10) can be approximated by expanding about t = 0 on the left to give the final result for the time rate of change of the probability P ks , This equation was first derived by W. Pauli, and will lead to a quantum version of the H-Theorem next.It signifies that of the N * P kr (0) systems occupying state kr at t = 0, N * P kr (0)W (k) sr will, per unit time, go over to ks.Thus, the W ks are interpreted as transition probabilities per unit time that the system will go from state kr to ks.They must satisfy sr ≥ 0 and the symmetry conditions sr .This is also referred to as the principle of microscopic reversibility.

A quantum H-theorem
The ensemble which represents a real physical system is determined by the thermodynamic state and environment of the actual system.The virtual ensemble has constituents which must duplicate both aspects.Of great practical interest and the one considered here is the case of isolated systems.An isolated system is characterized not only by a fixed value of the energy E, but also by a definite number of particles and volume V.Under these conditions, a quantum H-theorem can be formulated Yourgrau et al. (1966).Classically the error with which the energy of the real system can be specified can be theoretically reduced to zero.However, quantum theory claims there is a residual error specified by the uncertainty relation.All members of the ensemble cannot be said then to occupy eigenstates belonging to the same energy.It must be assumed the systems are distributed over energy levels lying within a finite range, ΔE.The following restrictions on the occupation numbers of the ensemble are imposed an isolated system 12) It will be shown that the ensemble specified by (4.12) exhibits a one-directional development in time ending ultimately in equilibrium.

Quantum mechanical canonical ensemble
us devise an ensemble which is representative of a closed isothermal system of given volume, or characterized by definite values of the parameters T, V and N.This approach brings us back to one of the ways entropy was formulated in the introduction, and need not rely on the specification of a density matrix.Suppose there are N * members of the ensemble each with the same values of V and N as the real system.However, they are not completely isolated from each other, so each is surrounded by a surface that does not permit the flow of particles but is permeable to heat.The collection of systems can be packed into the form of a lattice and the entire construction immersed in a heat reservoir at temperature T until equilibrium is attained.The systems are isothermal such that each is embedded in a heat reservoir composed of the remaining N * − 1.Once the ensemble is defined, it can be asked which fraction of the N * systems occupies any particular eigenstate of the unperturbed Hamiltonian of the experimental system.Let us study the ensemble then which is regarded as a large thermophysical system having energy E * , volume V * = N * V and made up of N * N particles.The quantum states of this large supersystem belonging to energy E * are to be enumerated.The thermal interaction energy is assumed to be so small that a definite energy eigenstate can be assigned to each individual system at any time.As energy can be exchanged between constituent systems, the eigenstates accessible to them do not pertain to one value of energy.The energy eigenstates of a system are written are compatible with a given distribution N * 1 , N * 2 , •••.Not all sets of N * j are admissible.The physically relevant ones satisfy the two constraints The supersystem then consists of a number N * of fixed but arbitrary systems with a constant energy E * .The number of physically possible supersystem states is clearly given as where the summation is to be extended over all N * j satisfying constraints (4.20).According to the earlier postulate, all allowed quantum states of an isolated system are equiprobable.Consequently, from this principle all states which satisfy (4.20) occur equally often.The probability P * that a particular distribution

.23)
Substituting P * into (4.23), it can be written as Quantum Mechanics Entropy and a Quantum Version of the H-Theorem

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To obtain a more useful expression for N * k , the right-hand side can be transformed to a set of primed integers.To this end, define Using these, constraints (4.20) get transformed into Consequently, where the prime means the sum extends over all N * ′ k which satisfy constraints (4.25).Comparing (4.26) with (4.20), the right-hand side of (4.26) is exactly Dividing this by N * and taking the logarithm of both sides results in the expression, The result in (4.27) can be expanded in a Taylor series to first order if we take N * >> 1 and From the constraint N * = ∑ j N * j = N * e −α ∑ e −βE j , e α can be obtained.Replacing this back in (4.28) and exponentiating gives The result in (4.29) gives what the average distribution of systems over system states will be in a supersystem at equilibrium.The instantaneous distribution will fluctuate around this distribution.The relative fluctuations of the occupation numbers for large enough N * are negligible, so to this accuracy, N * k /N * can be equated to P k .Setting Z = ∑ j e −βE j , the instantaneous probability that an arbitrarily chosen system of this supersystem will be in system state k can be summarized as follows This distribution is the quantum version of the canonical distribution in phase space, and is referred to as the quantum mechanical canonical ensemble.The function Z so defined is called the partition function.

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Theoretical Concepts of Quantum Mechanics www.intechopen.com In effect, this formalism has permitted the construction of a type of measuring device.Let us show that the microscopic ideas which have led to these results immediately imply consequences at the macroscopic level.To this end, it will be established what the exact form of the connection between Z and the Helmholtz free energy F actually is.The starting point is the second part of (4.20).Putting U = E * /N * , it implies U = ∑ j P j E j .(4.31) Formula (4.31) is in agreement with the postulate maintaining that the energy U of the physical system must be identified with the ensemble average E of the energy.
Begin by considering the change dU of the energy U when the experimental system remains but undergoes an infinitesimal reversible process.Equation (4.31) implies that = ∑ j (E j dP j + P j dE j ).
(4.32) Now (4.30) can be solved for E j in the form E j = −β −1 (log Z + log P j ).Consequently, since ∑ j P j = 1, it is found that ∑ j dP j = 0. Combining these it then follows that − ∑ j E j dP j = β −1 ∑ j (log Z + log P j ) dP j = β −1 ∑ j log P j dP j = β −1 d( ∑ j P j log P j ).Consequently, F = U − TS, (4.39) implies the following result Through the construction of these ensembles at a fundamental quantum level, a formalism has been obtained which will allow us to obtain concrete predictions for many equilibrium thermodynamic properties of a system once the function Z = Z(T, V, N) is known.In fact, it follows from the thermodynamic equation

Conclusions
It has been seen that formulating the concept of entropy at the microscopic level can be closely related to studying the foundations of quantum mechanics.Doing so provides a useful formalism for exploring many complicated phenomena such as entanglement at this level.Moreover, predictions can be established which bridge a gap between the microscopic and the macroscopic realm.There are many other topics which branch out of this introduction to the subject.For example there is a great deal of interest now in the study of the quantization of nonintegrable systems Gutzwiller (1990), which has led to the field of quantum chaos.There many indications of links in this work between the areas of nonintegrability and the kind of ergodicity assumed in statistical mechanics which should be pursued.
− dW the work done on the system during the given process, we have that∑ j P j dE j = − dW,(4.34)(4.33)and (4.34), we get the result dU = −β −1 d( ∑ j P j log P j )− dW.(4.35) Comparing (4.35) with the first law dU = dQ− dW, it is asserted that β dQ = −d( ∑ j P j log P j ).(4.36)Since the right-hand side of (4.36) is an exact differential, it is concluded that β is an integrating factor for dQ.By the second law of thermodynamics, β must be proportional to T −1 and the proportionality constant must be the reciprocal of B .With β of this form, when combined with the second law dQ = TdS, we have dS = −k B d( ∑ j P j log P j ).
and a Quantum Version of the H-Theorem www.intechopen.comwhere the integrating constant C is independent of both T and V.In fact the additive property of entropy requires that C = 0.This complicated procedure has returned us in some sense to where we began with (1.5), but by a different route.To get a relation between Z and F, use (4.30), (4.31) and (4.38) to write TS = −k B T ∑ j P j log P j = k B T log Z + ∑ j P j E j = k B T log Z + U.(4.39)

F
dF = −SdT− pdV + μ dN, (4.41)where μ is the chemical potential per molecule, thatS = −( ∂F ∂T ) V,N = k B log Z + k B T( of these results, consider the one-dimensional harmonic oscillator which has quantum mechanical energy eigenvalues given by n == Nk B T log(2 sinh( 1 2 βhω)).
By means of F, (4.42) and (4.43) imply that μ = F/N, p = 0 and the entropy and energy areS = Nk B [ βhω e βhω − 1 − log(1 − e −βhω )], U = N[ The entropic properties of composite systems obey numerous inequalities as well.Let {v m } and {e μ } be two orthonormal basis sets for the same physical system.Let ρ = ∑ w m |v m v m | and σ = ∑ ω μ |e μ e μ | be two different density matrices.Their relative entropy S(σ|ρ) is The states of A and B belong to two separate Hilbert spaces called H A and H B which are spanned by the bases |i A and |i B , and may be discrete or continuous.If A and B are prepared independently of each other This is what the expression quantum entanglement means.The superposition principle is at the heart of the most intriguing features of the microscopic world.A quantum system may exist in a linear superposition of different eigenstates of an observable, suspended between different classical realities, as when one says a particle can be at two positions at the same time.It seems to be impossible to get a classical intuitive representation of superpositions.When the superposition principle is applied to composite systems, it leads to the concept of entanglement.Moreover, as Bell has shown, entanglement cannot be consistent with any local theory containing hidden variables.Even if the state S cannot be factorized according to the superposition principle, it can be expressed as a sum of product states |i A ⊗|μ B , which make up a basis of the global Hilbert 474 and are not coupled together at some point, S is described by the tensor product |ψ S = |ψ A ⊗|ψ B .Each subsystem is described by a well-defined wave function.Any manipulation of one part leaves the measurement prediction for the other part unchanged.System S can also be prepared by measuring joint observables, which act simultaneously on A and B.Even if S has been prepared by measuring separate observables, A and B can become coupled by means of an interaction Hamiltonian.In this instance, it is generally impossible to write the global state |ψ S as a product of partial states associated to each component of S.Theoretical Concepts of Quantum Mechanics www.intechopen.comspace, H S .Consequently, an entangled state can be expressed as .4) In (3.4), λ j are positive or zero eigenvalues which sum to one.By neglecting B, there is acquired only a statistical knowledge of state A, with a probability λ j of finding it in |j A .It is possible to express the state for S in a representation which displays the entanglement.The superposition (3.1) claims nothing as to whether the state can be factored.To put this property in evidence, choose a basis in H A , called |j A in which ρ A is diagonal.
Quantum Mechanics Entropy and a Quantum Version of the H-Theorem www.intechopen.comwhere state | jB is given by n and v will spontaneously decrease in a closed system, which is in violation of the second law of thermodynamics.To retain the law, | u(t), v(t) | 2 = | u(0), v(0) | 2 must hold for every u and v. From Wigner's theorem, the mapping v(0) → v(t) is unitary, so Schrödinger's equation must be linear if the other postulates of quantum mechanics remain fixed.
there must also exist some n for which the reverse holds,| u n (t), v(t) | 2 > | u n (0), v(0) | 2 .In this event, the entropy of a 478 Theoretical Concepts of Quantum Mechanics www.intechopen.commixture of u An ensemble can be representative of a physical system in thermodynamic equilibrium only in this context if the occupation numbers N * jr are constants.A more general picture could consider the occupation numbers as functions of time N * jr = N * jr (t).The ensemble corresponds to a system removed from equilibrium.Let us ask then how do the N * jr vary with time.Quantum mechanics claims the existence of A jr ks (t) which determine the probability of a system in state ψ jr at time zero to be in ψ ks at time t.The final state could correspond to the initial state.Since N * .4) The quantities N * jr must sum up to N * , ∑ jr N * jr = N * , jr (0) systems are in a state specified by quantum numbers jr at t = 0, A jr ks (t)N * jr (0) systems will make the transition from jr to ks during (0, t) The number of systems in ks at time t will be N * ks (t)= ∑ j ∑ 481Quantum Mechanics Entropy and a Quantum Version of the H-Theorem www.intechopen.comPauli's equation can be used to obtain the rate of change of the quantum mechanical H-function which is defined to be H (P r − P s ) log(P s ).P s )(log(P r ) − log(P s )) ≥ 0. Consequently, each term in the sum in (4.17) is either zero or positive, hence H * (t) decreases monotonically with time, Equality holds if and only if P s = P r for all pairs (r, s) such that W sr = 0. Thus H * decreases and statistical equilibrium is reached only when this condition is fulfilled.Originally enunciated by Boltzmann in a classical context, (4.18) constitutes a quantum mechanical version of the H-theorem.
* (t)= ∑ s P s log(P s ).s Ṗs log(P s )+ ∑ s Ṗs = ∑ s Ṗs log(P s ).s ∑ r W sr (4.15) Interchanging r and s and using the symmetry property W rs = W sr , this is Ḣ * (t)= ∑ r ∑ s W rs (P s − P r ) log(P r )=− ∑ r ∑ s W sr (P r − P s ) log(P r ).r ∑ s W sr (P r − P s )(log(P r ) − log(P s )).(4.17)Recalling that W sr ≥ 0 as well as the inequality (x − y)(log x − log y) ≥ 0 for each (r, s),i t follows that (P r − Only one system-state j belongs to energy eigenvalue E j .An energy eigenstate of the supersystem is completely defined once the energy eigenstate occupied by each system is specified.It is only needed to stipulate the number N * j of systems occupying every system state j.Any set of values of the occupation numbers N * 1 , N * 2 , ••• define a quantum mechanical distribution.Clearly the W * supersystem states calculated by