2 Two-state quantum systems
The configuration of a quantum system is described in terms of a quantum state. The
bound states of the electron in a Hydrogen atom, organized in various shells, all correspond
to different quantum states which an electron can occupy, for example. Here, we shall start
the description of quantum states by specializing to the simplest non-trivial systems which
have only 2 quantum states. All the fundamental principles of quantum mechanics and of
quantum measurement can be dealt with simply and concisely, on the basis of this simple
example.
We shall look at two different systems: the polarization of light, and the spin 1/2 degree
of freedom of the electron. Assuming the wave vectorkof light to be fixed, the photon can
be in precisely two polarization states; similarly, with the electron momentumpfixed, the
electron has only the spin 1/2 degree of freedom left. As we shall see below, each system
will have just two states.
2.1 Polarization of light
Consider light propagating along thezdirection, with wave numberk=|k|and frequency
ω =ck. Classically, light is an electro-magnetic wave, whose electric and magnetic fields
are transverse tok, and thus lie in thexy plane. Apolarizeris a planar material that,
when inserted into a beam^1 and orthogonally to it, transmits only light whose electric field
is along the direction of the polarizer. Light whose polarization direction is perpendicular
to the polarization direction is absorbed by the polarizer and converted into heat.
We shall denote this polarization direction by the angleθwith thexaxis. The electric
fieldE= (Ex,Ey,0) of this wave is linearly polarized, and has
Ex = E 0 cosθcos(ωt−kz)
Ey = E 0 sinθcos(ωt−kz) (2.1)
Ananalyzeris a second polarizer, but whose polarization direction isα, depicted in figure 1.
The electric field of the light emerging from the analyzer is the result of projecting the electric
field that emerges from the polarizer onto the polarization directionαof the analyzer,
Ex = E 0 cosαcos(θ−α) cos(ωt−kz)
Ey = E 0 sinαcos(θ−α) cos(ωt−kz) (2.2)
If the light intensity emerging from the polarizer wasN, then the light intensity emerging
from the analyzer will beNcos^2 (θ−a).
(^1) Light in a beam is strictly speaking a wave packet, whose wave vectorhas a small dispersion aroundk.