QuantumPhysics.dvi

whose general solution is a linear combination ofeiωt ande−iωt. Combining now the solution for
all equations onC, we have

C(t,k) =

1

√

2 ω

∑

α=1, 2

(

εα(k)aα(k)e−iωt+ε′α(k)a′α(k)e+iωt

)

(19.21)

whereaα(k) andα′α(k) are arbitrary complex coefficients. The overall normalization factor 1/

√

2 ω
is not fixed by the Maxwell’s equations, but has been introduced for later convenience. By con-
struction, this expression automatically solves both equations in (19.15). It remains to enforce the
complex conjugation condition. To this end, we compute,

C(t,−k)∗=

1

√

2 ω

∑

α=1, 2

(

εα(−k)∗aα(−k)∗e+iωt+ε′α(−k)∗a′α(−k)∗e−iωt

)

(19.22)

Equating withC(t,k), as required by (19.18) requires thatε′α(k)a′α(k) =εα(−k)∗aα(−k)∗for all
k. Next, we make use of the fact that the basisε′α(k) may be related to the basisεα(bk), as long
as both obey the equations of (19.19). Thus, we have,

ε′α(k) = εα(−k)∗

aα(k)′ = aα(−k)∗ (19.23)

Putting all together, we have,

C(t,k) =

1

√

2 ω

∑

α=1, 2

(

εα(k)aα(k)e−iωt+εα(−k)∗aα(−k)∗e+iωt

)

(19.24)

To complete our solution, we substitute this result into theFourrier transform of (19.16),

A(t,x) =

∫

d^3 k

(2π)^3

1

√

2 ω

∑

α=1, 2

(

εα(k)aα(k)e−ik·x+εα(k)∗aα(k)∗e+ik·x

)

(19.25)

Here, we have started to use relativity notation for the inner product of the two 4-vectorskandx,

k·x≡ωt−k·x (19.26)

The expression (19.25) provides a complete solution for thevector potential to Maxwell’s equations
in transverse gauge.

19.3 The Hamiltonian in terms of radiation oscillators

In the absence of electric charge density and current, the Maxwell Hamiltonian is simply,

H=

∫

d^3 x

(

1

2

E^2 +

1

2

B^2

)

(19.27)