Next: Selection Rules
Up: TimeDependent Perturbation Theory
Previous: Spontaneous Emission
Consider an electron in a onedimensional harmonic oscillator
potential aligned along the axis. According to Sect. 5.8, the
unperturbed energy eigenvalues of the system are

(1135) 
where is the frequency of the corresponding classical
oscillator. Here, the quantum number takes the values .
Let the be the (real) properly normalized unperturbed eigenstates of the system.
Suppose that the electron is initially in an excited state: i.e., . In principle, the electron can decay to a lower energy state via
the spontaneous emission of a photon of the appropriate frequency.
Let us investigate this effect. Now, according to Eq. (1131),
the system can only make a spontaneous transition from an
energy state corresponding to the quantum number to one
corresponding to the quantum number if the associated electric dipole moment

(1136) 
is nonzero [since
for the case in hand]. However, according
to Eq. (416),

(1137) 
Since we are dealing with emission, we must have . Hence, we
obtain

(1138) 
It is clear that (in the electric dipole approximation) we can only have
spontaneous emission between states whose quantum numbers differ
by unity. Thus, the frequency of the photon emitted when the th excited
state decays is

(1139) 
Hence, we conclude that, no matter which state decays, the emitted photon always has the same frequency as the classical oscillator.
According to Eq. (1131), the decay rate of the th excited
state is given by

(1140) 
It follows that

(1141) 
The mean radiated power is simply

(1142) 
Classically, an electron in a onedimensional oscillator potential
radiates at the oscillation frequency with the mean power

(1143) 
where is the oscillator energy. It can be seen that a quantum
oscillator radiates in an almost exactly analogous manner to
the equivalent classical oscillator. The only difference is the
factor
in Eq. (1142)this is
needed to ensure that the groundstate of the quantum oscillator does not radiate.
Next: Selection Rules
Up: TimeDependent Perturbation Theory
Previous: Spontaneous Emission
Richard Fitzpatrick
20100720