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The Formalism of Quantum Mechanics

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The Formalism of Quantum Mechanics The Formalism of Quantum Mechanics In this Appendix we present only those concepts of quantum mechanics which are of great importance for the book and are not commonly well known: density matrices, the mathematical background of the Heisenberg uncertainty relations, and Feynman path integrals. AI. The density matrix The density matrix is used for the representation of a wider class of states of a quantum system than pure states represented by state vectors l'l,b). Yet all essentially quantum phenomena may be described in the language of pure states (state vectors) because, as we shall see below, a density matrix contains both quantum and classical elements (respectively state vectors and probabilities). Al.l. SUBSYSTEM STATE To arrive at the concept of a density matrix we shall consider the system of interest as a subsystem of a larger system; but first we introduce the trace of an operator. Trace of an operator An arbitrary operator B may be represented by its matrix elements: The trace of the operator is defined as the sum of its diagonal elements: (A.l) Making use of the property of any complete orthonormal basis l'l,bi) , L 1'l,b2) ('l,bi I = 1, i one may show that the value (A.l) does not depend on the choice of the basis. It is easy to derive the following very simple rule for the evaluation of an operator of the special form, B = I'P) (X I: tr(I'P)(xi) = (xl'P), 201 202 APPENDIX A and then, applying the linearity of the operation of taking the trace, Analogously, using the definition of the trace, we can show that cycling of the product of operators may be made under the sign of trace: tr(ABC ... Z) = tr(BC ... ZA), which implies the following important relation: tr(U-1 AU) = tr(A). We can now rewrite the mean value of an operator A in a state I'I/Ii) A = ('I/IIAI'I/I) = (A) as follows: A = (A)p = tr(Ap) (A.2) where the density matrix for the pure state 1'1/1) is defined as p = 1'1/1)('1/11· Reduced density matrix Let now S be a subsystem of the compound system S + E, the of basis S being {I'I/Ii)} and the basis of E being {I<I>i)}. Then the basis of the compound system S + E is {I'I/Ii) l<I>o:)} and a general state of S + E has the form Iw) = I>io: 1'1/1i) 1<1>0:) io: According to Eq. (A.2), the mean value of an operator A in this state is (A)p = (wIAlw) = trse(Alw)(wl) If, however, the operator A acts only on the subsystem S, then the trace over E may be referred only to the second factor Iw)(wl: (A)p = trs(Ap), where we have denoted p = tre(lw)(wl)· THE FORMALISM OF QUANTUM MECHANICS 203 The operator p (acting in the subsystem S) is called a reduced density matrix of this subsystem S and is equal to p = tr&(IW)(wl) = L L ciacja l'I/Ii)('I/Ijl (A.3) a tj The state of the subsystem cannot (in the general case) be described by a state vector. Instead, it should be described by a density matrix. Such a state is called mixed. A1.2. INTERPRETATION OF A DENSITY MATRIX Structure of the reduced density matrix It is easily seen that the density matrix (A.3) may be represented in the following form: where ICPa) = L cial'l/li). i Going over to the normalized vectors we have (A A) (notice that in the general case the states I'I/Ia) are not mutually orthogonal). From now on we shall denote by 'tr' the trace over the system S. The following properties of the density matrix can be easily proved: (A.5) The first property (of unit trace) is equivalent to and follows from the fact that the original state I'll) was normalized: 204 APPENDIX A In the special case where the density matrix represents a pure state, it is an idempotent operator, or a projector: Probability interpretation The properties hint that the numbers POI. might be interpreted in the purely classical sense as probabilities of alternative (i.e., excluding each other) events. This guess is valid. The state p may be interpreted as a state with incomplete knowl­ edge. We know that the system S may be in one of the pure states 11P0I.), but do not know in which of these states. Instead, we know only the probability POI. of each of them. This is sometimes called the ignorance interpretation. This interpretation is in fact a hypothesis, and it must be confirmed. The confirmation lies in the fact of the calculation of the mean value fulfilled on the basis of this hypothesis giving the correct result. Indeed, starting with our interpretation, we have to calculate the mean value of an operator A in each of the pure states 11P0I.) and find the weighted average with weights POI.. This gives the correct result: I>OI. (A)OI. = LPOI. (1P0I.1A!1P0I.) = tr(Ap) = (R)p. 01. 01. A1.3. DECOMPOSITION OF A DENSITY MATRlX Non-orthogonal set of vectors Eq. (A.4) supplies a decomposition of a density matrix into a sum (with positive coefficients) of density matrices of pure states. If we have some density matrix, i.e., an operator p with the properties (A.5), it may always be decomposed in this way. If we do not require that the vectors !1P0I.) be mutually orthogonal, decomposition is ambiguous. This may be seen from such a simple counter-example referring to a two-dimensional space: if with arbitrary x. Despite this ambiguity, the interpretation formulated above is valid for each of the decompositions in the sense that the calculation of mean values of operators may be achieved with the help of any of the decompositions. THE FORMALISM OF QUANTUM MECHANICS' 205 We have to make an important remark in this connection. It is stated in the probability interpretation of the decomposition (A.4) that POI is the probability that the system is in the state l"pa). This statement, however, should not be confused with the answer to the question: what is the proba­ bility that the system will be found (by the corresponding measurement) in the state l"pa). NaYvely one may answer this question with the same number POI' In actuality the probability is larger. Let us first prove this and then explain the apparent paradox. How can we answer the formulated question? We have to perform the measurement with two alternative results corresponding to the two (mutually comple­ mentary) projectors: P = l"pa)("pal, Q = 1 - P. The measurement result corresponding to the projector P will mean the answer 'yes' to the above question. The probability of this result may be calculated according to the usual quantum mechanical rules (Sect. 2.3): Prob(P) = tr(pP) = LPa'I("pal"pa')1 2 • (A.6) a' The r.h.s. may be rewritten as POI + L POI' 1("pOI l"pa') 12 , 01'=/:01 which is generally larger than POI' The reason for this purely quantum paradox is that to be in some pure state and to be found in it are not the same things. If the system is in a state l"pa') not identical to l"pa), it may be found (by the above measurement) in the state l"pa) with probability 1("pal"pa')12 . Therefore, even if we know that the system is in l"pa) with probability POI' the probability that the system will be found in l"pa) is given by Eq. (A.6). Orthogonal set of vectors (eigenvectors) Among all the decompositions of the density matrix there is only one de­ composition in the orthogonal set of pure states (up to degeneracy, as we shall see). The corresponding orthogonal states are eigenstates of the oper­ ator p. Denoting them l"pn), we have From the properties (A.5) of the density matrix the following properties of its eigenvalues follow: Pn ~ 0, LPn = 1 n 206 APPENDIX A This gives, in fact, a spectral decomposition of the density matrix: P = LPn l1/7n) (1/7nl· (A.7) n The vectors l1/7n) are automatically orthogonal if the corresponding eigen­ values are different and may be chosen orthogonal for coinciding eigenvalues (in the case of degeneracy). In any case (for both a degenerate and a non­ degenerate spectrum) the decomposition into the orthogonal projectors on subspaces of different eigenvalues is unique: P= LPNPN, PNPN' = dNN,PN, N In the case of orthogonal decomposition there is no paradox. The prob­ ability of finding the pure state l1/7n) in the mixed state p is equal to the corresponding coefficient Pn of the decomposition: The spectrum of the density matrix may be continuous. In this case the sum in the decomposition must be replaced by the corresponding integral: P = Jp(c) P(c) dc. The orthogonality of projectors has now the form P(A) P(B) = 0 for An B = 0, where P(A) = Lp(c) P(c) dc. A2. Uncertainty relations Let us derive the uncertainty relations (UR) for two Hermitian operators A, B. Form an operator M = A - i)"B, where).. is a real number, and notice that the operator Mt M is positive, MtM~O. This means that its mean value in an arbitrary state 11/7) is non-negative, (A.8) THE FORMALISM OF QUANTUM MECHANICS 207 (the latter follows from the fact that the l.h.s.
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