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.


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 $\hat \rho$.

  • $\begingroup$ Thank you, I realized that I overcomplicated this problem. In the proper basis we should expect the matrix to be diagonal with entries $f$ and $1 - f$. Either way our matrix will be positive-definite and its trace should always equal 1. $\endgroup$ – phoko Jun 21 '18 at 21:19
  • $\begingroup$ Yep, that's exactly it. $\endgroup$ – Emilio Pisanty Jun 21 '18 at 21:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.