theory \ F=1
\ / F=0
Fine structure shift splitting
Quantization of the free electromagnetic
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
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
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.
In classical physics, the harmonic oscillator is described
by the Hamiltonian
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
Чтобы распечатать файл, скачайте его (в формате Word).
Ссылка на скачивание - внизу страницы.