3.6 Quantizing the Electromagnetic Field
3.6.4 Interaction of Matter and Radiation
The electromagnetic field, as we have mentioned before, is the first example in which one noticed the existence of quanta which, in turn, arose from the study of the interaction between matter and radiation. While this is probably not strictly a quantum field theoretic interaction — we will deal with a non-relativistic quantum mechanical system coupled to a quantized electromagnetic field — we will describe this briefly because of its historical and practical importance.
To do this, we have to study the coupling between (i) quantized elec-tromagnetic fields and (ii) a charged particle (say, an electron in an atom) described by standard quantum mechanics. A charged particle in quantum mechanics will have its own dynamical variable x and momentum p obey-ing standard commutation rules. Dependobey-ing on the nature of the system, such a charged particle can exist in different (basis) quantum states, each of which will be labeled by the eigenvalues of a complete set of commut-ing variables. (For example, the quantum state |nlm of an electron in a hydrogen atom is usually labeled by three quantum numbers n, l and m.) For the sake of simplicity, let us assume that the quantum states of the charged particle are labeled by the energy eigenvalues |E; the formalism can be easily generalized when more labels are needed to specify the quan-tum state. The coupling between the electromagnetic field and the atomic system is described by the Hamiltonian
Hint=−
d3x J· A (3.247)
where to the lowest order,70in the non relativistic limit, J = qv ∼= (q/m)p
= (q/m)(−i∇). We are interested in the transitions caused between the quantum states of the electromagnetic field and matter due to this inter-action.
Let the initial state of the system be |Ei,{nkα} where Ei represents the initial energy of the matter state and the set of integers{nkα} denote the quantum state of the electromagnetic field. We now turn on the inter-action Hamiltonian Hint. Because of the coupling, the system can make a transition to a final state|Ef,{nkα} where Ef represents the final energy of the matter state and a new set of integers{nkα} denote the final quan-tum state of the electromagnetic field. To the lowest order in perturbation theory, the probability amplitude for this process is governed by the matrix element
Q≡ Ef,{nkα}|Hint|Ei,{nkα}. (3.248) To see the nature of this matrix element, let us substitute the expansion of the vector potential written in the form
A(t, x) =
α=1,2
d3k (2π)3
akαAkα+ a†kαA∗kα
;
Akα =
1 2ωk
1/2
kαe−ikx. (3.249)
70The canonical momentum in the presence of A will have an extra piece (qA/c) which will lead to a term pro-portional to q2A2 in J · A. This quadratic term in qA is ignored in the lowest order.
into the interaction Hamiltonian; then Hint becomes the sum of two terms Hint = −
d3x J· A (3.250)
= −
d3x J.
α=1,2
d3k (2π)3
akαAkα+ a†kαA∗kα
≡ Hab+ Hem.
Since the creation and annihilation operators can only change the energy eigenstate of the oscillator by one step, it is clear that the probability amplitude Q in Eq. (3.248) will be non zero only if the set of integers characterizing the initial and final states differ by unity for some oscilla-tor labeled by kα. In other words, the lowest order transition amplitudes describe either the emission or the absorption of a single photon with a def-inite momentum and polarization. Since the creation operator a†kαchanges the integer nkα to (nkα+ 1), the term proportional to a†kα governs the emission; similarly, the term proportional to akαgoverns the absorption of the photon. Let us work out the amplitude for emission in some detail.
The emission process, in which the quantum system makes the transi-tion from|Ei to |Ef and the electromagnetic field goes from a state |nkα to|nkα+ 1, during the time interval (0, T ), is governed by the amplitude
A =
T 0
dtEf|nkα+ 1|Hem|nkα|Ei
= −
T 0
dt
d3xEf|J · A∗kα|Ei (nkα+ 1)1/2 (3.251)
where we have used the fact that nkα+ 1|A|nkα = A∗kα(nkα+ 1)1/2. Using the expansion for Akα in Eq. (3.249) and the fact that the energy eigenstates have the time dependence exp (−iEt), the amplitude A can be written as
A = −
1 2ωk
1/2
(nkα+ 1)1/2 (3.252)
×
T 0
dt
d3x φ∗f(x)
J.kαe−ik.x
φi(x) e−i(Ei−Ef−ω)t
where φi(x) , φf(x) denote the wave functions of the two states. Denoting the matrix element
d3xφ∗f(x)
Je−ik·x
φi(x) = q m
d3xφ∗f(x) p e−ik.xφi(x) (3.253) (which is determined by the system emitting the photon) by the symbol Mf i, the probability of transition|A|2 becomes
|A|2= P (T ) =
1 2ωk
|kα· Mf i|2(nkα+ 1)|F (T )|2 (3.254)
where
|F (T )|2=
T 0
dt e−i(Ei−Ef−ω)t
2=
sin(RT/2) R/2
2
(3.255)
withR ≡ (Ei− Ef− ω). In the limit of T → ∞, for any smooth function S(ω), we have the result
∞
0
dω S(ω)sin2[(ω− ν)T/2]
[(ω− ν)/2]2 2T S(ν)
∞
−∞
sin2η
η2 dη = 2π T S(ν) (3.256) which shows that,
lim
T→∞
sin2[(ω− ν)T/2]
[(ω− ν)/2]2 → 2π T δD(ω− ν) (3.257) in a formal sense. Hence Eq. (3.254) becomes, as T → ∞,
P (T ) ∼= T
π ωk
kα· Mfi 2(nkα+ 1) δD(Ei− Ef− ω). (3.258)
The corresponding rate of transition is P = P (T )/T , which gives a finite rate for the emission of photons:71
P ≡ dP dt =
1 2ωk
(nkα+ 1)|kα· Mf i|22πδD(Ei− Ef− ω) . (3.259) This expression gives the rate for emission of a photon with a specific wave vector k and polarization α. The delta function, δD(Ei−Ef−ω) expresses conservation of energy and shows that the probability is non zero only if the energy difference between the states Ei− Ef is equal to the energy of the emitted photonω.
Usually, we will be interested in the probability for emission of a photon in a frequency range ω, ω + dω and in a direction defined by the solid angle element dΩ. To obtain this quantity, we have to multiply the rate of transition by the density of states available for the photon in this range.
The density of states (for unit volume) is given by dN
dωdΩ = d3k (2π)3
1
dωdΩ = 1 (2π)3
k2dkdΩ dωdΩ = k2
(2π)3 = ω2k
(2π)3. (3.260) Hence
dP dtdωdΩ
emi
= dP dt
dN dωdΩ =dP
dt ω2k (2π)3
=
1 2ω
(nkα+ 1)|kα· Mf i|22πδD(ωk− ωf i) ω2k (2π)3
∝ ωk(nkα+ 1)|kα· Mf i|2δD(ωk− ωf i) (3.261) withωfi≡ Ei− Ef.
The analysis for the absorption rate of photons is identical except that only the annihilation operator akαcontributes. Sincenkα− 1|akα|nkα = n1/2kα, we get nkαrather than (nkα+ 1) in the final result:
dP dΩdtdω
abs
∝ ωknkα|kα· Mf i|2δD(ωk− ωf i) . (3.262)
The probabilities for absorption and emission differ only in their de-pendence on nkα. The probability for absorption scales in proportion to
71The integration over the infinite range of t implies, in practice, an inte-gration over a range (0, T ) with ωT 1. If the energy levels have a char-acteristic width Δω ω, then the above analysis is valid for ω−1 T (Δω)−1.
72An intuitive way of understanding stimulated emission is as follows. Con-sider an atom making a transition from the ground state|G to an excited state
|E absorbing a single photon out of n photons present in the initial state, leaving behind a (n− 1) photon state.
The fact that this absorption probabil-ityP{|G; n → |E; n − 1} ∝ n ≡ Qn is proportional to n seems intuitively acceptable. Consider now the prob-ability P for the time reversed pro-cess|E; n − 1 → |G; n. By principle of microscopic reversibility, we expect P = P giving P ∝ n ≡ Qn. Call-ing n− 1 = m, we get P{|E; m →
|G; m + 1} = Qn = Q(m + 1). Clearly P is non zero even for m = 0 with P{|E; 0 → |G; 1} = Q which gives the probability for the spontaneous emission while Qm gives the probabil-ity for the stimulated emission. Thus, the fact that absorption probabilities are proportional to n while emission probabilities are proportional to (n+1) originates from the principle of micro-scopic reversibility.
73Make sure you understand this point. The structure of the Klein-Gordon equation for a massless parti-cle, φ = 0, is identical to Maxwell’s equations Ai = 0 in a particular gauge. The solutions for φ involve pos-itive and negative energy modes and so do the solutions for Ai. Sometimes a lot of fuss is made over negative en-ergy solutions, backward propagation in time, etc. for the φ = 0 equa-tion while you have always been deal-ing quite comfortably with theAi= 0 equation, all your life. When you quantize the systems, they have the same conceptual structure.
nkα. Clearly, if nkα = 0, this probability vanishes; this is obvious since no photons can be absorbed if there were none in the initial state to begin with. But the probability for emission is proportional to (nkα+ 1) and does not vanish even when nkα= 0. Hence there is a non-zero probability for a system at an excited state to emit a photon and come down to a lower state spontaneously. If the initial state of the electromagnetic field has a certain number of photons already present, then the probability for emission is further enhanced. The emission of a photon by an excited system when no photons were originally present is called spontaneous emission and the emission of a photon in the presence of initial photons is called stimulated emission. Both these processes exist and contribute in electromagnetic transitions.72
Your familiarity with this result should not prevent you from appre-ciating it. Spontaneous emission is a conceptually non-trivial quantum field theoretic process and arises directly through the action of the “anti-photon” creating term in Eq. (3.172). The electron in the excited state in the atom, say, can be described perfectly well by the Schrodinger equation
— until, of course, it pops down to the ground state emitting a photon.
We originally had just one particle, the electron (in an external Coulomb field) to deal with, and the Schrodinger equation is adequate. But once it creates a photon, we have to deal with at least two particles of which one is massless and fully relativistic. So, this elementary process of emission of the (anti)photon by an excited atom has the key conceptual ingredient which we started out with, while explaining the need for quantum field theory. Without a quantized description of the electromagnetic field, we cannot account for an elementary process like the emission of a photon by an excited system in a consistent manner. This can also be seen from the fact that the initial state (of an electron in the excited atomic level) has no photons, and yet a photon appears in the final state. People loosely talk of this as “vacuum fluctuations of the electromagnetic field interacting with the electron”; but the way we have developed the arguments, it should be clear that this is a field theoretic effect73arising from the fact that negative energy solutions of the field equations for Ai are included in Eq. (3.172).
Finally, let us see how these results are related to the Planck spectrum of photons in a radiation cavity. If we consider quantized electromagnetic radiation in equilibrium with matter at temperature T , then in steady state we will expect the condition NupRem = NdownRab to hold where Nup and Ndown represent the two levels which we designate as up and down andRem and Rab are the rate of emission and absorption which we have computed above. In thermal equilibrium, the population of atoms in the two energy levels will satisfy the condition Nup = Ndownexp(−βE) = Ndownexp(−βωk) where E > 0 is the difference in the energies of the two states and ωk is the frequency of the photon corresponding to this energy difference. On the other hand, we see from our expressions forRemandRab
that their ratio is given by Rem/Rab= [(nkα+ 1)/nkα]. The equilibrium
which tells you the number density of photons in a cavity, say, if the radia-tion is in equilibrium with matter atoms in the cavity held at temperature
T . We get the historically important Planck law for radiation in the cavity:
nkα= 1
eβωk− 1 (3.264)
The role played by the factors n + 1 and n in leading to this result is obvious. That, in turn, is due to the fact that electromagnetic field can be decomposed into a bunch of harmonic oscillators with the usual creation and annihilation operators.