QuantumPhysics.dvi

The instruments of the polarizer, analyzer (which, recall, is just a polarizer) and the bire-

fringent plate have mathematical interpretations aslinear operators onH. Quite literally,

the analyzer transforms photons in the state |θ〉to photons in state |α〉with probability

amplitude〈θ|α〉. Mathematically, this corresponds to a projection of the state|θ〉onto the

state|α〉, and may be represented by the operator,

Pα=|α〉〈α| (2.19)

This is a projection operator because we have Pα^2 = Pα, and applyingPα to |θ〉yields

|α〉〈α|θ〉. The birefringent plate transforms photons in state|θ〉into two separate beams,

one corresponding toPx|θ〉, the other toPy|θ〉. Recombining the two beams with the second

birefringent plate, subject to the linear superposition principle of the photon states, yields

|θ〉 → (Px+Py)|θ〉 (2.20)

but in view ofPxPy= 0, the sum of the two projection operators is the unit matrixIinH,

namelyPx+Py=I. Indeed, the beam emerging from the system of two birefringent plates

is really the same beam that went into that system.

2.5 The Stern-Gerlach experiment on electron spin

A number of other two-state quantum systems are often used tobring the key principles of

quantum physics to the fore. Feynman discusses the famous two-slit interference experiment

on electrons. Here, we shall concentrate on the Stern-Gerlach (SG) experiment in which the

two states are those of the spin 1/2 degree of freedom of an electron (carried by a silver

atom in the SG experiment). We choose this case because earlier we illustrated the photon

behavior of the classical electro-magnetic wave, while in SG, we illustrate the wave behavior

of a classical particle. The SG also differs from the photon case in that the electron spin is

itself already a purely quantum phenomenon with no classical counterpart.

The physical system of interest is the spin and magnetic moment of the electron. Angular

momentum and magnetic moments of course also arise in classical mechanics, but they are

both invariably associated with orbital motion. As far as we know, the electron is a point-like

particle, with no room for internal orbital angular momentum. It is apurely quantum effect

that the electron can nonetheless have non-vanishing angular momentumS, and magnetic

momentm. Both quantities are actually related by

m=g

( e

2 mc

)

S (2.21)

Here,eis the unit of electric charge,mis the mass, andgis the so-calledg-factor, which

equals 2.00 for electrons, 5.58 for protons, and− 3 .82 for neutrons.