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.
Quantization of the radiation field
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
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
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:
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:
Parity change: NO
Absorption of a photon
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