16 Mixtures and Statistical Entropy
In this section, the notion of statistically mixed ensembles, of pure and mixed states, of coherent and
incoherent superpositions are introduced and the corresponding formalism of thedensity operator
or equivalently, of thedensity matrixis developed.
16.1 Polarized versus unpolarized beams
In chapter 2, we introduced the interference properties of quantum mechanics using the physical
examples of the polarization states of light or of a spin 1/2 particle. In any of these set-ups,
the first piece of apparatus used was always a polarizer, which was responsible for filtering a
definite polarization out of an unpolarized beam. We have then obtained a complete mathematical
description of the polarized beam in terms of vectors in a Hilbert space of states. In particular,
apolarized beamof light should be viewed as an ensemble of photons which are all in the same
quantum state. Similarly, a polarized beam of Silver atoms in the Stern-Gerlach experiment should
be viewed as an ensemble of spins which are all in the same quantum state. In each case, the
quantum state of all particles in the beam are identical to one another. The phases of the various
individual photons are all the same, and the ensemble of states in the beam is said to becoherent.
What we have not yet given is a mathematical description for abeam which is unpolarized
(or partially polarized). The defining property of an unpolarized beam of light is as follows. We
consider a beam propagating in thez-direction, and insert a (perfect) polarizer transverse tothe
beam at an angleθwith respect to thex-axis. Next, we measure the probabilityPθfor observing
photons as a function of the angleθ. (Recall that for a beam polarized in thex-direction, we have
Pθ= cos^2 θ.) The beam isupolarizedprovided the probabilityPθisindependent of the angleθ.
Analogously, the defining property of anunpolarized beam of spin 1/2 particlesis as follows.
Take a beam propagating in thex-direction, and insert a Stern-Gerlach apparatus orientedin a
directionnwheren^2 = 1. Next, we measure the probability Pn for observing the spins in the
quantum staten·S= + ̄h/2 as a function ofn. (Recall that for a beam of spins withSz= + ̄h/2,
this probability isn^2 z.) The beam isunpolarizedprovided the probability for observingn·S= + ̄h/ 2
isPn= 1/ 2 for any anglen.
An unpolarized beam cannot be described mathematically as astate in Hilbert space. We can
easily see why for the spin 1/2 case, for example. If the unpolarized beam did correspond to a state
Ψ〉in Hilbert space, then measuring the probability for observing the spins in a quantum state with
n·S= + ̄h/2 would amount to
Pn=|〈n|Ψ〉|^2 (16.1)where|n〉is the + ̄h/2 eigenstate ofn·S. If|Ψ〉is a state in this same 2-dimensional Hilbert space,
then we may decompose it onto the basis|±〉as well,|Ψ〉=a|+〉+b|−〉, with|a|^2 +|b|^2 = 1. Using
the result of Problem 1 in Problem set 2 of 221A, every unit vectorncorresponds to a unitary
rotationU(n) in Hilbert space, n·S =U(n)SzU†(n), so that the eigenstate|n〉 ofn·Swith
eigenvalue + ̄h/2 is concretely given by|n〉=U(n)|+〉. Parametrizingnby Euler angles,θ,ψ, we