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A. Non-relativistic Quantum Mechanics
Absorption and emission of photons: For electromagnetic radiation, far
from its sources, the vector potential satisfies the wave equation
∂
2
A
∇
2
A −
= 0.
(A.29)
∂t 2
Solution:
A(x, t) = A 0 exp(−iωt + ik · x) + A 0
∗ exp(+iωt − ik · x).
(A.30)
With gauge condition ∇ · A = 0 we have
k · A 0 = 0
(A.31)
and there are two independent polarization vectors for photons.
Treat the interaction in first-order perturbation theory:
V ˆ (x, t) = (iq/m)A(x, t) · ∇.
(A.32)
Thus
A 0 exp(−iωt + ik · x) ≡ absorption of photon of energy ω
A
∗
0 exp(+iωt + ik · x) ≡ emission of photon of energy ω. (A.33)
A. Non-relativistic Quantum Mechanics
Absorption and emission of photons: For electromagnetic radiation, far
from its sources, the vector potential satisfies the wave equation
∂
2
A
∇
2
A −
= 0.
(A.29)
∂t 2
Solution:
A(x, t) = A 0 exp(−iωt + ik · x) + A 0
∗ exp(+iωt − ik · x).
(A.30)
With gauge condition ∇ · A = 0 we have
k · A 0 = 0
(A.31)
and there are two independent polarization vectors for photons.
Treat the interaction in first-order perturbation theory:
V ˆ (x, t) = (iq/m)A(x, t) · ∇.
(A.32)
Thus
A 0 exp(−iωt + ik · x) ≡ absorption of photon of energy ω
A
∗
0 exp(+iωt + ik · x) ≡ emission of photon of energy ω. (A.33)