Dynamics of Quantized Particles and Classical Light Fields

2 DYNAMICS OF QUANTIZED PARTICLES AND CLASSICAL LIGHT FIELDS 

Consider now the transition from classical mechanics to the quantum mechanics of the particles in the presence of a classical field. (The case of quantized particles in the presence of a quantized field is discussed in Chapter 12.) 

To quantize the dynamics of the particles first requires that we express the velocities of the particles in terms of canonical momenta. In the presence of electromagnetic fields, the canonical momenta are not merely m dx-Jdt). Rather, in order to incorporate Lorentz s velocity-dependent forces into Hamilton s formulation of classical mechanics, the canonical momenta are given by [2] 

Having expressed the Hamiltonian in terms of the canonical momenta, we can readily quantize the particles dynamics. To do so we replace each particle s canonical momentum by the momentum operator in the coordinate representation, 

Here we have used the fact that, in the Coulomb gauge, V - A(ry, t)(R i]/) = A(ry, t) Vy R i/r), where R = r. xN, with N being the total number of particles. Equation (1.44) is often referred to as being in the velocity gauge. 

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