Density matrix of a partially polarized beam of electrons

Question

The following problem is from Huang's Statistical Mechanics (2nd edition):

8.1 Find the density matrix for a partially polarized incident beam of electrons in a scattering experiment, in which a fraction $f$ of the electrons are polarized along the direction of the beam and a fraction $1 - f$ is polarized opposite to the direction of the beam.

Earlier in the chapter, he gives

$$\rho_{m n} \equiv (\Phi_n, \rho \Phi_m) \equiv \delta_{m n} |b_n|^2 \tag{8.10}$$

where $\Phi_n$ and $b_n$ have the same meaning as in

$$\Psi = \sum_n b_n \Phi_n {,} \tag{8.7}$$

the wave function of a system (each $\Phi_n$ is a wave function for $N$ particles contained in a volume $V$ and is an eigenfunction of the Hamiltonian of the system; the $\Phi_n$ are chosen such that together they form a complete orthonormal set). He also defines the density operator, but I'll omit that here.

I am not looking for a solution, just some guidance or a hint. I am having trouble coming up with an expression for the $\Phi_n$'s. For an individual electron, we can write

$$\left[ A \begin{pmatrix} 1 \\ 0 \end{pmatrix} + B \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] e^{i \mathbf{k} \cdot \mathbf{r}}$$

Can we just use this same wave function for this problem? If so, why? I'm thinking our coefficients $A$ and $B$ should satisfy $|A|^2 + |B|^2 = f$ for the electrons polarized in the direction of the beam and $|C|^2 + |D|^2 = 1 - f$ for the electrons polarized opposite to the direction of the beam.

Another thought: $|b_1|^2 = f$ and $|b_2|^2 = 1 - f$ in $(8.7)$, so that I would expect the density matrix to be a $2 \times 2$ matrix. But I really am unsure.

This is not homework, I am trying to work through problems in QSM in order to read some papers this summer. Any help would be greatly appreciated.

Answer 1

Generally speaking:

• If you prepare a system in the pure state $|\psi\rangle$, then the density matrix of the system in that state will be $\hat \rho = |\psi\rangle\langle \psi|$.
• If you have $N$ preparation procedures which produce the system in states with density matrices $\hat\rho_1,\ldots,\hat \rho_N$ with probabilities $p_1,\ldots,p_N$, where those probabilities must obey $p_j\geq 0$ and $\sum_{j=1}^N p_j = 1,$ then the probabilistic procedure will be described by the density matrix $$\hat \rho = \sum_{j=1}^N p_j \hat \rho_j.$$

In your case you have two preparation procedures with probabilities $p_1=f$ and $p_2=1-f$, so your task is to produce appropriate descriptions of the $\hat \rho_j$ and to add them up correctly to get $\hat \rho$.