The definition of the density operator in Pathria

[silverwhale]

Hello Everybody,

I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator:
[tex] \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, [/tex]
where N is the number of systems in the ensemble and the a(t)'s are expansion coefficents.
Now my question is: what does this definition mean? Especially the term [tex] a(t)^{k}_m a(t)^{k*}_n. [/tex] I do not get it.

Any help would be greatly appreciated!

 

[Jano L.]

The formula you wrote above refers to the definition of density matrix for a finite ensemble of isolated systems (beware, not for a subsystem in interaction with environment). It works as follows.


Imagine that there is an ensemble of N copies of the system. Denote the wave function describing the k-th system by

[tex]
\psi^{k}(\mathbf r).
[/tex]

(Different copies have different wave functions).


It is assumed that this function can be expressed as a discrete linear combination of some basis functions [itex]\Phi_m[/itex], which are the same for all k:


[tex]
\psi^{k}(\mathbf r) = \sum_k a_m^k \Phi_m(\mathbf r).
[/tex]

The numbers [itex]a^k[/itex] are complex expansion coefficients.

(Such expansion is possible if the set of functions [itex]\Phi_m[/itex] is complete, like for Hamiltonian eigenfunctions of harmonic oscillator. In case of hydrogen eigenfunctions, things are more complicated, due to continuous spectrum of Hamiltonian).

The density matrix is introduced usually as a quantity [itex]\rho_{mn}[/itex] that appears in the calculation of average value of some quantity [itex]f[/itex], say energy, over the ensemble.

The average over the ensemble is the weighted sum

[tex]

\langle \langle f \rangle \rangle = \sum_k p_k \langle f \rangle^k,

[/tex]

where [itex]p_k = 1/N[/itex] is the probability that the system is in state described by k-th wave function.

The expression

[tex]

\langle f \rangle^k
[/tex]

used above is the average of [itex]f[/itex] in a state described by [itex]\psi^k[/itex] function and can be expressed as

[tex]
\langle \psi^k | \hat f | \psi^k \rangle = \sum_{m,n} a_m^{k} a_n^{k*} f_{nm}.
[/tex]

where [itex]f_{nm} = \langle \Phi_n|\hat f|\Phi_m\rangle[/itex].

Then, the average over the ensemble is

[tex]

\langle \langle f \rangle\rangle = \sum_k \frac{1}{N} \sum_{m,n} a_m^{k} a_n^{k*} f_{nm}.
[/tex]


This can be rewritten as

[tex]

\sum_m \left( \rho_{mn}f_{nm} \right)

[/tex]

where the quantity

[tex]
\rho_{mn} = \sum_k \frac{1}{N} a_m^{k} a_n^{k*}
[/tex]

was named the density matrix.

 

[silverwhale]

Perfect.

Many many thanks!
If you were living in Hamburg, Germany, I would give you a bag full of cookies!
I am really thankful! :)

 

[Jano L.]

Glad to be of help.