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to classical theory, an atom cannot have a minimum-energy state even when only the Coulomb interaction is included in the Hamiltonian. It is the quantum commutation relations of the atomic operators that prevent the electron from colliding with the proton by giving it enough kinetic energy to overcome the Coulomb attraction every time the electron is localized within a certain neighborhood of the proton [409]. When the coupling with the radiation field is included to account for atomic emission, the atomic commutation relations are violated unless the field commutation relations are included [131, 556].
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EMISSION IN FREE SPACE
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The canonical variables we have used to describe this system are not unique. They are related to other possible sets of canonical variables by unitary transformations; the quantum analog of classical contact transformations. We will perform such a transformation on these canonical variables to obtain a more convenient set of variables. When written in terms of this new set, the Hamiltonian will acquire a form more amenable to physical interpretation. We will then be able to use physical insight to make some approximations that will greatly simplify the problem. Let us consider the unitary transformation T that takes pinto p - (q/c)A (0), that is, TpTt = p- ~A(O). (6.34)
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This is a translation in momentum space of -(q/c)A (0). We will now show that the generator of infinitesimal translations in one dynamical variable is the canonical conjugate variable [243]. By applying the infinitesimal translations successively, we will obtain an expression for a general finite translation that will yield T as a particular case. Let Tv. (a) be a unitary transformation that produces a translation in the space of a given dynamical variable u:
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(6.35)
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We can write the unitary transformation Tv. (0') corresponding to an infinitesimal translation 0 as (6.36) Tv.(O') = 1 + iO'X, where X is a Hermitian operator which is the generator of infinitesimal translations. From (6.37) Tv.(O')uTJ(c5) = u + c5 we obtain the following commutation relation between X and u
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[X,uj = -i.
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Equation (6.38) implies that
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(6.38)
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X=-"h v ,
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where v is the canonical conjugate variable corresponding to u:
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(6.39)
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[v,uj = in.
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A finite translation Tv. (a) is given by
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(6.40)
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Tv.(a) =
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J~= (1- ~~v)N
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(6.41)
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= exp (
SPONTANEOUS EMISSION
In the case where u =
and a = -(q/c)Aj (0), we obtain =exp
T [-~Aj(O)]
[!~Aj(O)rj],
(6.42)
The complete translation T can be achieved by translating each component of p successively:
[-~Aj(O)]
= exp
[!~ A(O) . r] .
(6.43)
The Hamiltonian (6.1) acquires the following form 5 when we apply the unitary transformation T:
THTt
:~ + VCoul + Htrans -
i3k ([II(k)
+ IIt(k)]
. rq
(6.44)
+ q2 {[r . I (k)J2 + [r . 2(k)J} ).
We notice that II in the new representation is no longer associated with the electric field but to the displacement field in this representation:
'D'(k) = T'D(k)Tt
= T (=
II(k) - 2~ { I (k) [ I (k) . rJ
+ 2(k) [ 2(k)J} ) Tt
(6.45)
-II(k).
So equation (6.44) can be written as
H' = :m
+ VCoul + H trans -
D'(O) . d
+ cdip.
(6.46)
where d is the atomic dipole -qr,
(6.47)
and cdip is the dipole self-energy,
(6.48)
The self-energy apparently diverges, but in fact, we must restrict the integral in (6.48) to long wavelengths compared to an angstrom in order to be consistent with the dipole approximation. This term leads to another sort of mass renormalization. Because we are not interested in discussing renormalization, we simply ignore (6.48).
5Frornthe factthat T can bewrittenasexp (iq/cn) d3 k r A(k) - h.c.l and the factthat A. andII. are canonical conjugate variables. a straightlorward generalization ofthe resufts presented in the preceding paragraph yields T II. Tt = II. - qT We also notice that T affects neither A nor r.
EMISSION IN FREE SPACE
The problem can be simplified even further if we assume that there are only two atomic levels involved. This is the famous two-level atom approximation. Within this approximation, the Hamiltonian (6.46) becomes
li 20"z
d kliw
~ [a!(k)a!(k) +~]