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1 Introduction 2 the Density Operator Physics 125c Course Notes Density Matrix Formalism 040511 Frank Porter 1 Introduction In this note we develop an elegant and powerful formulation of quantum me- chanics, the “density matrix” formalism. This formalism provides a structure in which we can address such matters as: • We typically assume that it is permissible to work within an appropriate subspace of the Hilbert space for the universe. Is this all right? • In practice we often have situations involving statistical ensembles of states. We have not yet addressed how we might deal with this. 2 The Density Operator Suppose that we have a state space, with a denumerable orthonormal basis {|uni,n =1, 2,...}. If the system is in state |ψ(t)i at time t, we have the expansion in this basis: |ψ(t)i = X an(t)|uni. (1) n We’ll assume that |ψ(t)i is normalized, and hence: ∗ hψ(t)|ψ(t)i =1 = X X an(t)am(t)hum|uni n m 2 = X |an(t)| (2) n Suppose that we have an observable (self-adjoint operator) Q. The matrix elements of Q in this basis are: Qmn = hum|Quni = hQum|uni = hum|Q|uni. (3) The average (expectation) value of Q at time t, for the system in state |ψ(t)i is: ∗ hQi = hψ(t)|Qψ(t)i = X X am(t)an(t)Qmn. (4) n m 1 We see that hQi is an expansion quadratic in the {an} coefficients. Consider the operator |ψ(t)ihψ(t)|. It has matrix elements: ∗ hum|ψ(t)ihψ(t)|uni = am(t)an(t). (5) These matrix elements appear in the calculation of hQi. Hence, define ρ(t) ≡|ψ(t)ihψ(t)|. (6) We call this the density operator. It is a Hermitian operator, with matrix elements ∗ ρmn(t)=hum|ρ(t)uni = am(t)an(t). (7) Since ψ(t) is normalized, we also have that 2 1=X |an(t)| = X ρnn(t)=Tr[ρ(t)] . (8) n n We may now re-express the expectation value of observable Q using the density operator: ∗ ∗ hQi(t)=X X am(t)an(t)Qmn m n = X X ρnm(t)Qmn m n = X [ρ(t)Q]nn n =Tr[ρ(t)Q] . (9) The time evolution of a state is given by the Schr¨odingerequation: d i |ψ(t)i = H(t)|ψ(t)i, (10) dt where H(t) is the Hamiltonian. Thus, the time evolution of the density operator may be computed according to: d d ρ(t)= [|ψ(t)ihψ(t)|] dt dt 1 1 = H(t)|ψ(t)ihψ(t)|− |ψ(t)ihψ(t)|H(t) i i 1 = [H(t),ρ(t)] (11) i 2 Suppose we wish to know the probability, P ({q}), that a measurement of Q will yield a result in the set {q}. We compute this probability by projecting out of |ψ(t)i that protion which lies in the eigensubspace associated with observables in the set {q}. Let P{q} be the projection operator. Then: P ({q})=hψ(t)|P{q}ψ(t)i =TrhP{q}ρ(t)i . (12) We note that the density operator, unlike the state vector, has no phase ambiquity. The same state is described by |ψ(t)i and |ψ0(t)i = eiθ|ψ(t)i. Under this phase transformation, the density operator transforms as: ρ(t) → ρ0(t)=eiθ|ψ(t)ihψ(t)|e−iθ = ρ(t). (13) Furthermore, expectaion values are quadratic in |ψ(t)i, but only linear in ρ(t). For the density operators we have been considering so far, we see that: ρ2(t)=|ψ(t)ihψ(t)||ψ(t)ihψ(t)| = ρ(t). (14) That is, ρ(t) is an idempotent operator. Hence, Trρ2(t)=Trρ(t)=1. (15) Finally, notice that: 2 hun|ρ(t)uni = ρnn(t)=|an(t)| ≥ 0 ∀n. (16) Thus, for an arbitrary state |φi, hφ|ρ(t)φi≥0, as may be demonstrated by expanding |φi in the |ui basis. We conclude that ρ is a non-negative definite operator. We postulate, in quantum mechanics, that the states of a system are in one-to-one correspondence with the non-negative definite density operators of trace 1 (defined on the Hilbert space). 3 3 Statistical Mixtures We may wish to consider cases where the system is in any of a number of different states, with various probabilities. The system may be in state |ψ1i with probability p1, state |ψ2i with probability p2, and so forth (more gener- ally, we could consider states over some arbitrary, possibly non-denumerable, index set). We must have 1 ≥ pi ≥ 0 for i ∈{index set}, and Pi pi =1. Note that this situation is not the same thing as supposing that we are in √ the state |ψi = p1|ψ1i+p2|ψ2i+···(or even with p1, etc.). Such statistical mixtures might occur, for example, when we prepare a similar system (an atom, say) many times. In general, we will not be able to prepare the same exact state every time, but will have some probability distribution of states. We may ask, for such a system, for the probability P ({q}) that a mea- surement of Q will yield a result in the set {q}. For each state in our mixture, we have Pn({q})=hψn|P{q}ψni =TrρnP{q} , (17) where ρn = |ψnihψn|. To determine the overall probability, we must sum over the individual probabilities, weighted by pn: P ({q})=X pnPn({q}) n = X pnTr ρnP{q} n ! =TrX pnρnP{q} n =TrρP{q} , (18) where ρ ≡ X pnρn. (19) n Now ρ is the density operator of the system, and is a simple linear combina- tion of the individual density operators. Note that ρ is the “average” of the ρn’s with respect to probability distribution pn. Let us investigate this density operator: • Since ρn are Hermitian, and pn are real, ρ is Hermitian. 4 • Trρ =Tr(Pn pnρn)=Pn pnTrρn = Pn pn =1. • ρ is non-negative-definite: hφ|ρφi = Pn pnhφ|ρnφi≥0. • Let Q be an operator with eigenvalues qn. In the current situation, hQi refers to the average of Q over the statistical mixture. We have: hQi = X qnP ({qn})=X qnTr ρP{qn} n n ! =Trρ X qnP{qn} n =Tr(ρQ), since Q = X qnP{qn}. (20) n • We may determine the time evolution of ρ.Forρn(t)=|ψn(t)ihψn(t)| we know (Eqn. 11) that dρ (t) i n =[H(t),ρ (t)] . (21) dt n Since ρ(t) is linear in the ρn, ρ(t)=Pn pnρn(t), we have dρ(t) i =[H(t),ρ(t)] . (22) dt • Now look at 2 ρ = X X pmpnρmρn m n = X X pmpn|ψmihψm|ψnihψn| m n =6 ρ, in general. (23) What about the trace of ρ2? Let |ψmi = X(am)j|uji. (24) j Then 2 ρ = X X pmpn|ψmihψm|ψnihψn| m n " # = X X p p X X(a )∗(a ) δ X X(a ) (a )∗|u ihu | m n m i n j ij m k n ` k ` m n i j k ` ∗ ∗ = X pmpn(am)i (an)i(am)k(an)` |ukihu`|. (25) m,n,i,k,` 5 Let’s take the trace of this. Notice that Tr(|ukihu`|)=δk`, so that 2 ∗ ∗ Tr(ρ )= X pmpn(am)i (an)i(am)k(an)k. (26) m,n,i,k ∗ But hψm|ψni = Pi(am)i (an)i, and thus: 2 2 Tr(ρ )=X X pmpn|hψm|ψni| m n ≤ X X pmpnhψm|ψmihψn|ψni, (Schwarz inequality) m n ≤ X pm X pn m n ≤ 1. (27) The reader is encouraged to check that equality holds if and only if the system can be in only one physical state (that is, all but one of the pn’s corresponding to independent states must be zero). Note that, if Tr(ρ2) = 1, then ρ = |ψihψ|, which is a projection operator. We encapsulate this observation into the definition: Def: A state of a physical system is called a pure state if Tr(ρ2) = 1; the density operator is a projection. Otherwise, the system is said to be in a mixed state, or simply a mixture. The diagonal matrix elements of ρ have a simple physical interpretation: ρnn = X pj(ρj)nn j = X pjhun|ψjihψj|uni j 2 = X pj|(aj)n| . (28) j This is just the probability to find the system in state |uni. Similarly, the off-diagonal elements are ∗ ρmn = X pj(aj)m(aj)n. (29) j The off-diagonal elements are called coherences. Note that it is possible to choose a basis in which ρ is diagonal (since ρ is Hermitian). In such a basis, the coherences are all zero. 6 4 Measurements, Statistical Ensembles, and Density Matrices Having developed the basic density matrix formalism, let us now revisit it, filling in some motivational aspects. First, we consider the measurement process. It is useful here to regard an experiment as a two-stage process: 1. Preparation of the system. 2. Measurement of some physical aspect(s) of the system. For example, we might prepare a system of atoms with the aid of spark gaps, magnetic fields, laser beams, etc., then make a measurement of the system by looking at the radiation emitted. The distinction between preparation and measurement is not always clear, but we’ll use this notion to guide our discussion. We may further remark that we can imagine any measurement as a sort of “counter” experiment: First, consider an experiment as a repeated prepa- ration and measurement of a system, and refer to each measurement as an “event”.
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