# Introduction to Quantum Electrodynamics, страница 3

is an eigenstate of   with eigenvalue  :

.

Using the commutation relations

one finds  that the states

are eigenstates of  with eigenvalues

These states exhaust the spectrum of . The operator

is known as the occupation number operator . Its eigenvalues  are

Explicit matrix representations of  ,  , and

consistent with the commutation relations are

They are assumed to act on a column vector represented by

where only the  entry is different from zero.

We replace   and   by the annihilation and creation operators:

and postulate the following  commutation relations

.

The operators   and   are time-independent. The time dependence is  determined by the wave function

(the Schroedinger representation).

The vector potential operator

.

The fields   and  are also replaced by operators

.

The energy operator:

.

Taking into account the commutation relation

,

we have

,

where

is the occupation number operator, which has eigenvalues

Let    is  a state with  photons of energy , momentum , and polarization  . Using the commutation relations for   and , one finds

,

,

.

Non-zero matrix elements of   :

Emission and absorption of photons by atoms

Let us consider a transition of  a non-relativistic atom from  state  to  state   via emission of a photon with energy  , momentum , and polarization  .  For the system “atom + radiation field”, this is  a transition from the state   to the state   due to the interaction

,

where   is the electron momentum operator and  is the electron spin operator.

Consider first the transition due to the term

.

According to the general prescription of quantum mechanics, the transition probability per unit time into a

wave vector element  is (the Golden Rule)

We have to evaluate

Because for a non-relativistic  atom

,

in the operator , we can replace

.

It yields

,

where

.

Transforming

,

we obtain

.

It gives

.

In the expression  , the first term  corresponds to the induced emission while the second term

corresponds to the spontaneous emission.

We note that in the approximation used the matrix element  can be represented as the matrix element of the operator

.

For this reason, this transition is known as   the electric dipole transition, E1.

The total transition probability is obtained by integration over  and summing over photon polarization. For the spontaneous emission, one finds

.

The selection rules for the electric dipole transition:

NO:

Parity change:  YES

To the next order in     ,  the M1 and E2 transitions occur. The M1 transition is determined by the operator

.

For an H-like ion, its amplitude is smaller by factor   than the E1 amplitude.

The selection rules for the magnetic dipole transition:

NO:

Parity change:  NO

###### The lifetime of  state    is  determined by

.

Absorption of a photon