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Next: Replacement of Sums over All Up: Chapter 11 Previous: Chapter 11

Choice of Basis Vectors

Choice of Basis Vectors

The previous Lecture exhibited formal aspects of quantum theory. Consider now the application of that theory to statistic mechanics. We have previously written the ensemble average of a variable as


where the sum passes over all states of the system, and where tex2html_wrap_inline763 is the value of A in the state j. To give this expression a quantum mechanical form, recall that to make a quantum observation we replace all observables A with their corresponding operators tex2html_wrap_inline771 . If one measures A for a system in state tex2html_wrap_inline775 , unless tex2html_wrap_inline775 is an eigenstate of tex2html_wrap_inline771 one gets different values for A in different measurements. If a large series of replicas of a given system tex2html_wrap_inline783 are examined, the quantum average value of A is tex2html_wrap_inline787 . Applying these principles to eq. 11.1 replaces E and A with their operator forms, so for a quantum system


the sum on j passing over all states of the system.

Equation 11.2 is significant because it embodies a double average, one average being quantum mechanical and the other statistical mechanical:

1) The quantum averages are indicated by the terms tex2html_wrap_inline795 and tex2html_wrap_inline797 . The first term gives the expectation value of tex2html_wrap_inline799 in the state tex2html_wrap_inline801 , while the second term gives the expectation value of tex2html_wrap_inline803 in the same state. If the states tex2html_wrap_inline801 were eigenstates of tex2html_wrap_inline807 and tex2html_wrap_inline771 , the quantum average would have a particularly simple form, namely eq. 11.1. If the tex2html_wrap_inline801 were not eigenstates of tex2html_wrap_inline807 and tex2html_wrap_inline771 , taking the expectation values would require expanding the tex2html_wrap_inline801 in appropriate eigenstates, as shown in eqs. 10-31 and 10-32.

2) The statistico-mechanical average is indicated by the sums on j, which reflect averages over all states of the system, the various terms being given their canonical statistical weights.

Throughout the treatment of quantum systems, all ensemble averages will have this double nature. In each case, there will be a thermal average over different states of the system. Within each term of the thermal average, there will be a quantum average to obtain the expectation values of tex2html_wrap_inline821 and tex2html_wrap_inline823 within that state. The sequence of taking the averages is significant, the quantum average being taken separately for each state, the various quantum averages then being combined to yield a thermal average. This order of taking the two averages is consistent with the thus-far established image of an ensemble average. To average tex2html_wrap_inline825 within the classical canonical ensemble, the function A is evaluated separately in each element of the ensemble, a thermally weighted average of A from different states of the ensemble then being taken. In the quantum canonical ensemble, evaluating tex2html_wrap_inline771 in each element of the ensemble requires computing its expectation value. After the expectation value has been evaluated separately for each element of the quantum ensemble, the thermally weighted statistico-mechanical average of tex2html_wrap_inline771 is taken, as in the non-quantum case.

In Lectures 8 and 9, statistical mechanics was applied to quantized systems: the rigid rotor, the quantized-spin in an external field, and the harmonic oscillator. The partition function was written as a sum over the energy eigenstates of the system. Energies computed quantum-mechanically, and a discrete form of eq. 11.1, then gave values for Q, <E>, and other quantities.

A first question is whether or not we were justified in using the list of energy eigenstates of the system as a correct list of all states of the system. There are variables which do not commute with tex2html_wrap_inline807 . If we had written the sum of eq. 11.1 while using some other set of eigenstates, instead of using energy eigenstates, would we have obtained the same value for Q? If we get different values for Q when we use different sets of basis vectors, how d0 we know which eigenstates we should use to take an average over the canonical ensemble? Fortunately, it turns out that the choice of eigenstates is irrelevant. The terms tex2html_wrap_inline795 and tex2html_wrap_inline797 can be viewed as matrix elements of tex2html_wrap_inline849 and tex2html_wrap_inline803 . The sums over states of eq. 11.1 are traces of matrices. The trace of a matrix is invariant to the choice of basis vectors. While, e. g. , tex2html_wrap_inline797 changes when the basis vectors tex2html_wrap_inline801 are changed, the sums in eq. 11.1 are independent of the choice of basis vectors. Any set of basis vectors which is a complete orthogonal set of states is equally acceptable for evaluating Q or <A>, because eq. 11.2 gives the same value for tex2html_wrap_inline861 regardless of the choice of basis vector.

The independence of ensemble averages and partition functions from the choice of basis vectors has practical applications in statistical mechanics. To obtain the thermodynamic properties of a quantum system, one only needs to evaluate its partition function


Q is easily evaluated if the tex2html_wrap_inline775 are energy eigenstates, because in that case tex2html_wrap_inline867 and


Suppose one has a system for which a set of basis vectors (a ``complete set of states'') have been found, but those basis states tex2html_wrap_inline869 are not the energy eigenstates. Q can be computed from eq. 11.3 while using the basis vectors tex2html_wrap_inline869 as the states of the system. The individual matrix elements will be more tedious to evaluate, because tex2html_wrap_inline875 involve the quantum average of tex2html_wrap_inline807 over the mixed (with respect to energy eigenstates) state tex2html_wrap_inline869 . However, use of the tex2html_wrap_inline869 rather than the energy eigenstates tex2html_wrap_inline883 will not affect the sum Q of the diagonal matrix elements of tex2html_wrap_inline803 .

As a demonstration of the independence of Q and the choice of basis vectors, consider the isolated spin system of Lecture 8. For a single spin the energy was tex2html_wrap_inline891 for tex2html_wrap_inline893 , the partition function being written


This Q may be rewritten as a double quantum and thermal average, using the notation of the previous Lecture. Eq. 11.5 is


for tex2html_wrap_inline897 and tex2html_wrap_inline899 . Eqs. 11.5 and 11.6 are written in terms of energy eigenstates. An alternative to energy eigenstates are the helicity eigenstates



in terms of which the partition function is


It may be confirmed by direct computation that Q from eq. 11.9 agrees with Q in eq. 11.5.

A technical issue arises in the quantum evaluation of ensemble averages. There is no physical guarantee that A and tex2html_wrap_inline803 commute, in which case tex2html_wrap_inline909 and tex2html_wrap_inline911 need not be equal. This issue arises already in conventional quantum mechanics, in the evaluation of such averages as tex2html_wrap_inline913 . The orthodox resolution for calculating the expectation value of non-commuting variables is to evaluate the symmetrized product tex2html_wrap_inline915 . [When I last investigted the question, no one had proposed an experiment to test the orthodox resolution.]

next up previous
Next: Replacement of Sums over All Up: Chapter 11 Previous: Chapter 11

Nicholas V Sushkin
Sun Jun 30 15:55:07 EDT 1996