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CRC Press eBooks

Quantum Density Matrix

Eric R. Bittner

Published Jul 21, 2009• 0 citations

Abstract

Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as 〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this deﬁnition we can write ρˆ = ∑ mn c∗ncm |m〉〈n| (6.3) = ∑ mn ρmn|m〉〈n| (6.4) where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace 〈A(t)〉 = ∑ mn Amnρmn = Tr [ ˆAρ(t)] (6.5) where Tr [ ˆAρ(t)] denotes the trace operation: Tr [ ˆAρ(t)] = ∑ nn′ If ρ is diagonal, then Tr [ ˆAρ(t)] = ∑ Annρnn = ∑ 〈n|A|n〉Pn where Pn = ρnn is the statistical probability of ﬁnding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑ Pn = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.

DOI: 10.1201/b15835-9

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Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as 〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this deﬁnition we can write ρˆ = ∑ mn c∗ncm |m〉〈n| (6.3) = ∑ mn ρmn|m〉〈n| (6.4) where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace 〈A(t)〉 = ∑ mn Amnρmn = Tr [ ˆAρ(t)] (6.5) where Tr [ ˆAρ(t)] denotes the trace operation: Tr [ ˆAρ(t)] = ∑ nn′ If ρ is diagonal, then Tr [ ˆAρ(t)] = ∑ Annρnn = ∑ 〈n|A|n〉Pn where Pn = ρnn is the statistical probability of ﬁnding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑ Pn = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.

Here Tr is the trace operator. The discussion in [1] goes on by arguing that R must be Hermitian (have real eigenvalues) and that ρ ought to be positive with trace 1. An important special case is when ρ is one-dimensional: ρ = |ψ〉〈ψ| for a vector |ψ〉. Then the state is pure, and is equivalently specified by the vector |ψ〉. In general the formula (1) is a consequence of Born’s formula: The probability of observing a pure state |φ〉 when the system is prepared in a pure state |ψ〉 is given by |〈φ|ψ〉|2.

Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as

tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle }

where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states

\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|

where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and

pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1

In quantum mechanics, a physical observable is described by a (linear) operator Q on a Hilbert space H and a (pure) state is described by a unit vector Ψ in H. The expectation value (the average of measured values for a large number of repeated measurements) of the observable Q in the State Ψ is given by 1.1.1 ψ( Q )=( ψQψ ) ]] <![CDATA[

\psi \left( Q \right) = \left( {\psi ,Q\psi } \right)

More general states called mixtures are described by a positive operator ρ of trace 1, called a density matrix, for which the expectation value is 1.1.2 ψ( Q )=trρQ ]] <![CDATA[

\psi \left( Q \right) = tr\rho Q

and is at most countable sum of states of the form (1.1.1). ((1.1.1) is a special case where ρ is a one-dimensional projection operator.)

Introduction to the density operator Motivation and definition It is often inconvenient, for one reason or another, to keep track of all the variables of a complex system. For example, in a many-body system it would be impracticable to consider the co-ordinates of each particle. Furthermore, such information is actually not of much interest. Thus in a spin system the behaviour of the individual magnetic moments is unimportant; the components of the total magnetisation are the variables of primary interest. Very generally, for a system with ∼ 10 23 co-ordinates, one is unlikely to need more than, say, ten variables to describe its observable properties. However, it is quite clear that this reduced amount of information is no longer sufficient to write down a wavefunction and, therefore, it is no longer possible to calculate the evolution of such a system using the usual methods of quantum mechanics. In Chapter 5 we saw that it was possible, by the introduction of probabilistic arguments, to calculate the evolution of the total magnetisation of a spin system. The calculations were performed using the machinery of quantum mechanics and meaningful and useful results were thereby obtained. In this chapter we will look at things from a rather different point of view. We will not, initially, be concerned with the expectation values of certain specified observables, rather we direct our attention to the general description of the ‘state’ of the system and the way it evolves with time – possibly towards thermal equilibrium.