wang
(Wang)
#1
polarization angleθ=α= 0, while the state|y〉corresponds toθ =α=π/2. From this
correspondence, we deduce the probabilities, and probability amplitudes,
p(|x〉→|x〉) =p(|y〉→|y〉) = 1 ⇒ 〈x|x〉=〈y|y〉= 1
p(|x〉→|y〉) =p(|y〉→|x〉) = 0 ⇒ 〈x|y〉=〈y|x〉= 0 (2.13)
Here〈| 〉denotes the Hermitian inner product in the two-dimensional complexvector space
H(we shall define this Dirac notation more carefully soon). Thus, thestates|x〉and|y〉
form an orthonormal basis forH.
The polarizer-birefringent plate experiment, and the linear superposition principle, show
that a photon with arbitrary polarization angleθ corresponds to a state inHwhich is a
linear combination of the states|x〉and|y〉,
|θ〉=ax|x〉+ay|y〉 (2.14)
for complex coefficients ax anday. As we have seen earlier, conservation of probability
requires the relation|ax|^2 +|ay|^2 = 1. Actually, the coefficientsaxandayare nothing but
the probability amplitudes to find the photon|θ〉in either state|x〉or state|y〉,ax=〈x|θ〉
anday=〈y|θ〉. Conservation of probability thus leads to
|〈x|θ〉|^2 +|〈y|θ〉|^2 = 1 (2.15)
so that the state|θ〉has also unit norm,〈θ|θ〉= 1. It follows that we have a formula for〈θ|,
for any values ofaxanday, given by
〈θ|=a∗x〈x|+a∗y〈y| (2.16)
where∗denotes complex conjugation.
We are now in a position to give the mathematical formulation for all the polarizer-
analyzer-birefringent plate experiments described earlier. The polarizer and analyzer prepare
a photon in a definite state, given respectively by the state vectors
|θ〉 = cosθ|x〉+ sinθ|y〉
|α〉 = cosα|x〉+ sinα|y〉 (2.17)
The probability amplitude to observe the|θ〉 photon in the states|x〉and |y〉after the
birefringent plate, and|α〉after the analyzer are given respectively by
ax = 〈x|θ〉= cosθ
ay = 〈y|θ〉= sinθ
atot = 〈α|θ〉= cos(θ−α) (2.18)
