Momentum of photons

Since the momentum of photons, h/A, is small compared with the crystal momentum, hla (a is the lattice constant), the momentum of electrons should be conserved during the absorption of photons. The absorption coefficient a hv) for a given photon energy is proportional to the probability, P, for transition from the initial to the final state and to the density of electrons in the initial state as well as to the density of empty final states. On this basis, a relation between absorption coefficient a and photon energy ph can be derived [2, 4]. For a direct band-band transition, for which the momentum remains constant (see Fig. 1.7), it has been obtained for a parabolic energy structure (near the absorption edge) ... 

Fig. 4.1 Orientation of angular (spin) momentum of photon with momentum hk, with right (a) and left (b) polarization.

Much of solid-state physics is carried out in reciprocal space, sometimes called Fourier space. The reasons for this will become more apparent as we progress. One of the more obvious reasons is that it is easier to represent the momentum of photons and phonons as well as particles such as electrons and neutrons and thus their interactions in reciprocal space. Recall that the momentum of either a massless particle such as a photon or a particle with mass can be written as... 

Intrinsic angular momentum of photons. In a transition which obeys the selection rule of equation (5.13),... 

Next we establish the connection of the previous formalism with the Fo< space description of photons. From the interpretation of a>(k) as the number operator for photons of momentum k polarization A, and of cA(k) and cA(k) as destruction and creation operators for... 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

Experiments had shown that a beam of light shining on an object exerts a pressure, and this, in turn, implies that a photon has momentum. Quantitative measurements of the pressure exerted by light showed that a simple equation relates the momentum of light (p) to its energy E — c As we have already described, light energy also is... 

If we increase the accuracy with which the position of the electron is determined by decreasing the wavelength of the light that is used to observe the electron, then the photon has a greater momentum, since p = hiA. The photon can then transfer a larger amount of momentum to the electron, and so the uncertainty in the momentum of the electron increases. Thus any reduction in the uncertainty in the position of the electron is accompanied by an increase in the uncertainty in the momentum of the electron, in accordance with the uncertainty principle relationship. We may summarize by saying that there is no way of accurately measuring simultaneously both the position and velocity of an electron the more closely we attempt to measure its position, the more we disturb its motion and the less accurately therefore we are able to define its velocity. 

To illustrate some of these principles the angular momentum of a photon will be examined [56]. Suppose a beam of circularly polarized light falls on a perfectly black absorbing surface, which not only heats up (E = hv) but also acquires a torque, on account of the angular momentum it absorbs. Circular polarization means that the probability of an elementary observation 0(P ) = The ratio of energy/torque = w(= 2m/), the angular frequency of... 

Classically, a circularly polarized light beam with angular frequency w(= 2nv) transfers angular momentum at a rate of E/w, where E is the rate of energy transfer. Considered as a beam of photons, E = Nhui/2-n, so that the angular momentum of each photon is h/2n = h. 

The second term s may be called the operator for spin angular momentum of the photon. However, the separation of the angular momentum of the photon into an orbital and a spin part has restricted physical meaning. Firstly, the usual definition of spin as the angular momentum of a particle at rest is inapplicable to the photon since its rest mass is zero. More importantly, it will be seen that states with definite values of orbital and spin angular momenta do not satisfy the condition of transversality. 

The phonons are not stationary modes, but traveling waves extending through the whole crystal. The momentum of a phonon can be assigned as equal to /iq, in analogy with the momentum of a photon, though it is not strictly defined, as the phonon can be described equivalently in an extended Brillouin zone (see Fig. 2.1), corresponding to a different value of the wavevector q. 

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