Fig18(b).- Calculation of the frequencies
There are three distinct regimes in which the frequencies ofthe quantum oscillators need to be
computed. The momenta parallel to the plates are always of the following form,
kx=2 πnx
L
ky=2 πny
L
nx,ny∈Z (19.89)Along the third direction, we distinguish three different regimes, so that the frequency has three
different branches,
ω(i)=√
kx^2 +k^2 y+ (kz(i))^2 i= 1, 2 , 3 (19.90)The three regimes are as follows,
- ω > ωc: the frequencies are the same as in the absence of the plates,since the plates are
acting as transparent objects,
kz(1)=2 πnz
L
nz∈Z- ω < ωc& between the two plates.
k(2)z =
πnz
a0 ≤nz∈Z- ω < ωc& outside the two plates.
k(3)z =πnz
L−a
0 ≤nz∈Z- Summing up the contributions of all frequencies
Finally, we are not interested in the total energy, but rather in the enrgy in the presence of the
plates minus the energy in the absence of the plates. Thus, when the frequencies exceedωc, all
contributions to the enrgy cancel since the plates act as transparent bodies. Thus, we have
E(a)−E 0 =∑
nx,ny,nz(
1
2
ω(2)(nx,ny,nz) +1
2
ω(3)(nx,ny,nz)−1
2
ω(1)(nx,ny,nz))
(19.91)Whennz 6 = 0, two polarization modes haveE~with 2 components along the plates, while fornz= 0,
nx,ny 6 = 0, there is only one. Therefore, we isolatenz= 0,
E(a)−E 0 =1
2
∑
nx,nyω(nx,ny,0) (19.92)+
∑
nx,ny∑∞nz=1(
ω(2)(nx,ny,nz) +ω(3)(nx,ny,nz)− 2 ω(1)(nx,ny,nz))