whose energy and momentum is readily computed, and we have,
H|ψ〉=E|ψ〉 E=|k 1 |+···+|kn|
P|ψ〉=p|ψ〉 p=k 1 +···+kn (19.62)Energy and momentum are simply the sums of thenone-particle energy and momentum values, a
superposition relation characteristic of non-interacting, orfreeparticles. It is instructive to calculate
the relativistic invariant mass of a system of two free photons, for example. We find,
m^2 = (|k 1 |+|k 2 |)^2 −(k 1 +k 2 )^2 = 2|k 1 ||k 2 |(1−cosθ) (19.63)whereθis the angle betweenk 1 andk 2. For generic values ofθ, we havem^26 = 0, signaling that they
form a genuine two-state system. Forθ= 0, we have collinear photons, withm^2 = 0. It may seem
that the state with two parallel photons or with one photon with exactly the same total energy are
really one and the same thing. But this is not so, because there exists in fact an operator whose
eigenvalues distinguish between these two cases. This is the photon number operator, defined by
N=
∫ d (^3) k′
(2π)^3
∑
α
a†α(k)aα(k) (19.64)
Its eigenvalue on a general state is given by,
N|k 1 ,α 1 ;···;kn,αn〉=n|k 1 ,α 1 ;···;kn,αn〉 (19.65)
and thus distinguished between states with identical energy and momentum but different numbers
of photons. Note that, for free photons, we have
[N,H] = [N,P] = 0 (19.66)
so that the number of free photons is conserved during time evolution. Of course, once photons
interact, such as with charges particles or dipoles, then the number of photons will no longer remain
conserved andHwill no longer commute withN.
Finally, we have seen how the operatora†α(k)creates a photonwith momentumkand polariza-
tionα. But similarly, the operatoraα(k)annihilates a photonwith momentumkand polarization
αout of any state. If the state contains no such photon, then the operatoraα(k) produces 0. More
generally, we have
aα(k)|k 1 ,α 1 ;···;kn,αn〉=
∑n
i=1
(2π)^3 δα,αiδ(3)(k−ki)|k 1 ,α 1 ;···;k̂i,αi;···;kn,αn〉 (19.67)
where the wide hat overki,αigives the instruction of deleting the entryki,αiin the state descrip-
tion.