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:
\rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \},
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 a(t)^{k}_m a(t)^{k*}_n. 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

<br /> \psi^{k}(\mathbf r).<br />

(Different copies have different wave functions).


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


<br /> \psi^{k}(\mathbf r) = \sum_k a_m^k \Phi_m(\mathbf r).<br />

The numbers a^k are complex expansion coefficients.

(Such expansion is possible if the set of functions \Phi_m 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 \rho_{mn} that appears in the calculation of average value of some quantity f, say energy, over the ensemble.

The average over the ensemble is the weighted sum

<br /> <br /> \langle \langle f \rangle \rangle = \sum_k p_k \langle f \rangle^k,<br /> <br />

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

The expression

<br /> <br /> \langle f \rangle^k<br />

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

<br /> \langle \psi^k | \hat f | \psi^k \rangle = \sum_{m,n} a_m^{k} a_n^{k*} f_{nm}. <br />

where f_{nm} = \langle \Phi_n|\hat f|\Phi_m\rangle.

Then, the average over the ensemble is

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


This can be rewritten as

<br /> <br /> \sum_m \left( \rho_{mn}f_{nm} \right)<br /> <br />

where the quantity

<br /> \rho_{mn} = \sum_k \frac{1}{N} a_m^{k} a_n^{k*} <br />

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.