Introduction to Quantum Electrodynamics, страница 2

 Non-relativistic       \                                                            

          theory                \                                                                                            F=1

                           \               /                                       F=0

                                          \______________/                

                                                                                      Lamb             Hyperfine                              

                                       Fine structure               shift                 splitting                                                                                                                                   

                                            splitting

Quantization of  the free electromagnetic   

                             field

Classical free electromagnetic fields are described by the Maxwell equations

     ,                 ,

        ,                            .

     and   are conveniently expressed in terms of the scalar and vector potentials:

     ,                  .

The potentials   and   are not unique quantities. They admit  the following transformation

      ,                  ,                               

which is known as a gauge transformation.

In what follows, we will use the Colomb gauge:

                           .

In this gauge,   can be chosen to vanish:

                                 .   

Then,  the  second couple of the Maxwell equations takes the form

                  ,

while the first couple is satisfied automatically if the representation in terms of the vector and scalar potentials is employed.

     Radiation field as a collection of harmonic   

                             oscillators

We assume the periodic  boundary conditions for  enclosed in a box with  . Remembering the reality of  , we can write the Fourier decomposition in the form     

                       ,     

where    are the polarization vectors and . The polarization vectors obey the transversality condition

       

the ortogonality condition  and the completeness condition

              

Substituting the Fourier decomposition of  

into the Maxwell equation, we obtain

         

or

 .

This means that the radiation field can be regarded as a collection of independent harmonic oscillators.

The energy of the radiation field:

.

Substituting        and        into this equation and using the Fourier decomposition of   given above, we obtain

.

Thus, the energy of the radiation field is the sum of the energies of the harmonic oscillators.

So far, we considered classical fields. To pass to the quantum theory of the radiation field, we must quantize the radiation harmonic oscillators according to the procedure we learnt from non-relativistic quantum mechanics.

                   Harmonic oscillator

In classical physics, the harmonic oscillator is described

by the Hamiltonian

                      .

In variables

              ,

it takes the form

                            .

To pass to the quantum theory, we replace the classical quantities  and  by  so-called annihilation () and creation () operators  for a quant of energy   .

In the usual (Schroedinger)  representation, the operators do not depend on time. The time dependence is determined by the wave function.   The operators  and  obey the following commutation relations

The classical Hamiltonian  is replaced by the energy operator

                      .

Using the commutation relations, we obtain

                         .

A state  defined by