Expectation Value of a Physical Observable

Consider a physical observable F defined by its quantum mechanical operator F, and the evolution of the system is described by a Hamiltonian H with the basis vectors |Ф„), defined by the eigenvalue equation:

From quantum mechanics, the diagonal matrix element F„„ is described by the following equation

with Ф„({q Ф„), and Ф* (q) = (Ф„ |q). The matrix element F„„ in 14.4 permits to evaluate the expectation value of the physical observable F quantized as an operator F:

or

So, for a mixed state the expectation value of the physical observable F is written as:

We see that it has two averaging procedures, one is the usual quantum mechanical average procedure in 14.4 and the other is the classical averaging of multiplying the probability of being in a state by the value of being in that state as in 14.5 as well as in 14.6.

Density Matrix

In relation 14.7, it is obvious that the quantity

Since the coefficients W„ are real then p(q,q') is obviously a Hermitian operator and so can be diagonalized. Here, p{q,q') is the so-called density matrix in the coordinate representation - a fundamental quantity that is the summit in quantum statistical mechanics from where all concepts are derived as well as the concepts of thermal equilibrium and temperature T clarified. This requires the definition of the weighted function of eigenstates for any operator,

This is the so-called statistical density matrix (that is positive definite) in the coordinate representation that is the matrix element of the density operator and also determines the thermal average of the particle density of a quantum-statistical system:

In relation 14.9, the quantities and W„ are the eigenfunctions and eigenvalues of p indicative that p(q,q') corresponds to a mixture of pure states of the wave functions VJq) with the respective weights W„. In 14.10, |4'„){4(n| can be interpreted as the probability distribution of the system in the eigenstate |4,„), while W„, the normalized probability to encounter the system in the state |VF„). So, p{q,q') should be the normalized average particle density in space. The advantage of working with p{q,q') rather than the wave functions is that we can more easily treat an infinite volume that avoids the complications of the boundaries of the system. For the evaluation of the partition function Z by explicit calculation of we must take a large, but finite volume. It is instructive to note that at low temperatures, only the lowest energy state survives and p(q,q') achieves the particle distribution at the ground state. At high temperatures, quantum effects are expected to be irrelevant and we therefore expect matrix density p(q,q') to imitate that of a classical particle distribution.

The expression 14.9 applies generally to a mixed state. The physical sense of the density operator in 14.10 entails that for any Hermitian operator p we may always find such a representation, say |^Р(), for which it is diagonalized. From 14.10, we may reformulate quantum mechanics: Any system is described by the density matrix in 14.10 with

(a) |*Fj) being some complete orthonormal set of vectors and the probability W, has the following properties:

(b) For a given operator, F the quantum mechanical-statistical expectation value can be found via a trace, for any representation,

Since {'P,|f|'P,) is the expectation value of F in the state I'P,) then from a) to d) the density matrix pti in the diagonal 14* f) representation is, simply, interpreted as the probability of the system to be found in the state |4*,). Generally, deriving the density matrix from 14.10 as well as in 14.13 we do not define concretely the exact diagonal representation and at least suppose they exist. If all but one of the W, is zero, then the system is in the pure state :

and otherwise it is in the mixed state. So, the density matrix handles both pure and mixed states.

 
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