wang
(Wang)
#1
In our context, these are given byax= cosθ,ay= sinθ, anda′x= cosα,a′y= sinα. The
total amplitude results from linearly combining these partial amplitudes,
atot = axa′x+aya′y
= cosθcosα+ sinθsinα
= cos(θ−α) (2.9)
Thus, the probability amplitudes satisfy alinear superposition principle.
The total probabilityptotis given in terms of the probability amplitudeatotby,
ptot=|atot|^2 = cos^2 (θ−α) (2.10)
which is now in agreement with the result from electro-magnetic wavetheory.
2.3 General parametrization of the polarization of light
The polarization of light and photons described above is not the mostgeneral one. The most
general polarization (at fixed wave vectork) has the electric field given by
Ex = E 0 cosθcos(ωt−kz−δx) =E 0 Re(axe−iωt+ikz)
Ey = E 0 sinθcos(ωt−kz−δy) =E 0 Re(aye−iωt+ikz) (2.11)
By time-translation and periodicity, it is clear that only the difference in phasesδx−δy
modulo 2πis physically significant. Whenθ=±π/4, andδx−δy=±π/2, this is circular
polarization, while for general values, it is elliptical, and we have
ax = cosθ eiδx
ay = sinθ eiδy (2.12)
The complex two-component vector (ax,ay) obeys|ax|^2 +|ay|^2 = 1 and its phase is physically
immaterial, thus leaving two real parameters.
2.4 Mathematical formulation of the photon system
The combination of the polarizer and birefringent plate demonstrates that photons with
polarization angleθdecompose intox- andy-polarization for allθ, the only difference being
the relative intensity of these components. The photon states withx- and y-polarizations
will be represented mathematically by two vectors|x〉and|y〉in a two-dimensional complex
vector spaceH. In the system of a polarizer and analyzer, the state|x〉corresponds to the
