By Ralph Decker Bennett

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The term in Eq. 3-10 proportional to nh, the pumber of ~ h o t o n sa l r u r s s emitted. is called stimulated emission. The term that remains when nbm= 0 . I s We consider spontaneous emission first. Stimulated emission may be treated together with absorption. . To f the excited state of an atom against spontaneous emission of a photon, we set n,, = 0 and sum Eq. 3-10 over all of the k's *and a's that the emitted photon can have. That is, Now we let the volume of the box in which the electromagnetic field is quantized become infinite.

The interaction i s obtained by the prescription Making this replacement in Eq, 4-29 and adding the Hamiltonians give the totaI Harniltonian 'H p -t- Hrad + where is the particle Hamiltonian, is the Hamiltonian of the radiation field, and Interaction of Quantized Fields 55 is the interaction Hamiltonian. As before, we can divide HI into a part H' proportional to A and a part H" proportional to Aa. Expanding A and y in terms of a,, and 6, gives (5-6a) HI = H' H" + where + M(-k,, GI? 4 2 , g2, n, n')a$,az2,,) M(k, a, n, n') = ( ~ ~ ~ ~ d 3 T W ~ imc [ - l i e i k ' x u k ~(5-6d) a ~ ] y m .

According to Eq. 4-3, y(x, t ) is a linear combination of the destruction operators b,. We may interpret it as an operator which destroys a particle at the position x at the time t . Similarly, y+(x, t) is a linear combination of the creation operators b:. We may interpret it as an operator which creates a particle at the position x at the time t. The commutation relations for y and yrt. may be found from those for b, and b:. Thus = ):yn(x)y,l(x') = S(X n - x') (4-26) where Eqs. 4-7 and 4-1 1 and the completeness relation for the set of functions y, has been used.