8/13/2019 Quantum.35

1/1

20   ELECTROMAGNETIC RADIATION AND MATTER

is linearly polarized along the  x-axis is represented by  a†1(k)

0, in analogy

with a plane wave in classical electromagnetism. Likewise a photon linearlypolarized along the y-axis is represented by  a

†2(k)

0. Here the unit vectors

ε1  and   ε2  are chosen along the  x-axis and the  y-axis respectively and theyform with   k/ωk   a right-handed triplet of (real) unit vectors in the order(ε1, ε2, k/ωk). When discussing linearly polarized photons, we can and doomit the complex conjugate symbol over   ελ(k). Such photons are calledtransverse photons.

To represent circularly polarized photons we introduce

aR(k) = a1(k)+ ia2(k)

aL(k

)=

a1(k

)−

ia2(k

) (1.68)

We will show that a right-handed (RH) photon is created byk,R

=

a†R(k)

0

 and that a left-handed (LH) photon is created by

k, L

= a

†L(k)

0

.

Creation operators in the expression for  A(x, t ) are accompanied by a factorexp(−ikx). If we fix the position  x  and let  t  vary we get a factor exp(iωt ).We separate the real and imaginary parts of the resulting expression

a†R(k)e

iωt 0

= (a1 + ia2)

†eiωt 0

= (a†1 − ia†2)(cosωt + i sinωt )

0

= (a†1 cosωt + a

†2 sinωt )

0

+ i(a

†1 sinωt − a

†2 cosωt )

0

  (1.69)

Consider first the real part of Equation (1.69). We plot in Figure 1.1 onan   x, y   coordinate system the   coefficient   of   a

†1   plotted along the   x-axis

and the  coefficient  of  a†2  plotted along the  y-axis (the respective directions

of polarization related to   a†1   and   a†2). It is seen that the point defined

by these two coefficients lies on a circle and that the point moves in a

counter-clockwise direction (in the direction from the  x-axis to the  y-axis)around the circle as time increases. The latter statement can be checkedby substituting  ωt = 0 (A in Figure 1.1) and  ωt = π/2 (B in Figure 1.1) inEquation (1.69) and following the position of the point along the circle fromA to B. The same counter-clockwise direction of movement of the point isspecified by the imaginary part of Equation (1.69). This is consistent withthe subscript R of the coefficient aR. The photon moves in the direction of  k,in the figure out of the paper toward the reader, and the counter-clockwisemovement of the point corresponds to a RH movement relative to   k   in

the manner defined by the right-hand rule of the cross product of twovectors. In a similar manner we can show that a LH photon is representedby

k,L= a

†L(k)

0. For our discussion we have selected the term in  A that

creates a photon. One can (and should) verify that the arguments also holdfor the terms that annihilate a photon.