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

$$\psi^{k}(\mathbf r).$$

(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:


$$\psi^{k}(\mathbf r) = \sum_k a_m^k \Phi_m(\mathbf r).$$

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

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

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

The expression

$$\langle f \rangle^k$$

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

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

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

Then, the average over the ensemble is

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


This can be rewritten as

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

where the quantity

$$\rho_{mn} = \sum_k \frac{1}{N} a_m^{k} a_n^{k*}$$

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.