Interaction with External Fields
207
The angular spread of the beam is
α ~ 1.10 × 10
–5
rad
(6.126)
In travelling a distance of 100 m, the beam will spread by about
∆x ~ 0.11 cm
(6.127)
Example 7
If an excited state is replenished by decays from a higher excited state, the
decay rate is not given by the simple exponential function in Eq. (6.76).
Consider three states with energies E 0 < E 1 < E 2 and decay probabilities A 10 ,
A 20 and A 21 . Then the changes in N 1 (t) in time dt are
dN 1 (t) = – A 10 N 1 (t) dt + A 21 N 2 (t) dt
(6.128)
dN 2 (t) = – A 2 N 2 (t) dt
(6.129)
where A 2 = A 20 + A 21 . The solutions to these equations are
N 2 (t) = exp (– A 2 t) N 2 (0)
N 1 (t) = exp (– A 10 t) N 1 (0) +
21
10
2
A
A
A
−
(exp [– A 2 t] – exp [– A 10 t]) N 2 (0)
(6.130)
It is observed that if A 21 N 2 (0) > A 10 N 1 (0), N 1 (t) will increase for small
t but will eventually start decreasing.
PROBLEMS
1. Obtain the energy levels of the (1s)
2 1
S 0 , (1s) (2p)
1
P 1 , and (1s) (3d)
1
D 2
states of the helium atom in the presence of a magnetic field of strength
1 Wb/m
2
. What are the shifts in Å for the allowed transitions, if l P → S =
584.4 Å, λ D → P = 6678 Å for the unperturbed states under consideration?
2. Describe the Zeeman patterns of the
2
D 3/2 and
2
P 3/2 states. Calculate the
frequency shifts for the transitions
2
D 3/2 →
2
P 3/2 with ∆M J = 0, ∆M J = 1,
and ∆M J = – 1. If the sodium line with λ = 8195 Å corresponds to a
2
D 3/2 →
2
P 3/2 transition, what is the maximum shift in its wavelength when
a magnetic field of 1 Wb/m
2
is introduced?
3. Obtain the shifts in the frequency for
2
D 3/2 →
2
P 1/2 transitions in the presence
of a weak magnetic field.
4. What is the magnetic moment of sulphur in the ground state
3
P 2 ? In the
presence of a magnetic field of 2 Wb/m
2
, what is the resonance frequency?
5. Indicate the energy levels of
2
P and
2
D states and the allowed transitions
between them in the presence of a strong magnetic field.
207
The angular spread of the beam is
α ~ 1.10 × 10
–5
rad
(6.126)
In travelling a distance of 100 m, the beam will spread by about
∆x ~ 0.11 cm
(6.127)
Example 7
If an excited state is replenished by decays from a higher excited state, the
decay rate is not given by the simple exponential function in Eq. (6.76).
Consider three states with energies E 0 < E 1 < E 2 and decay probabilities A 10 ,
A 20 and A 21 . Then the changes in N 1 (t) in time dt are
dN 1 (t) = – A 10 N 1 (t) dt + A 21 N 2 (t) dt
(6.128)
dN 2 (t) = – A 2 N 2 (t) dt
(6.129)
where A 2 = A 20 + A 21 . The solutions to these equations are
N 2 (t) = exp (– A 2 t) N 2 (0)
N 1 (t) = exp (– A 10 t) N 1 (0) +
21
10
2
A
A
A
−
(exp [– A 2 t] – exp [– A 10 t]) N 2 (0)
(6.130)
It is observed that if A 21 N 2 (0) > A 10 N 1 (0), N 1 (t) will increase for small
t but will eventually start decreasing.
PROBLEMS
1. Obtain the energy levels of the (1s)
2 1
S 0 , (1s) (2p)
1
P 1 , and (1s) (3d)
1
D 2
states of the helium atom in the presence of a magnetic field of strength
1 Wb/m
2
. What are the shifts in Å for the allowed transitions, if l P → S =
584.4 Å, λ D → P = 6678 Å for the unperturbed states under consideration?
2. Describe the Zeeman patterns of the
2
D 3/2 and
2
P 3/2 states. Calculate the
frequency shifts for the transitions
2
D 3/2 →
2
P 3/2 with ∆M J = 0, ∆M J = 1,
and ∆M J = – 1. If the sodium line with λ = 8195 Å corresponds to a
2
D 3/2 →
2
P 3/2 transition, what is the maximum shift in its wavelength when
a magnetic field of 1 Wb/m
2
is introduced?
3. Obtain the shifts in the frequency for
2
D 3/2 →
2
P 1/2 transitions in the presence
of a weak magnetic field.
4. What is the magnetic moment of sulphur in the ground state
3
P 2 ? In the
presence of a magnetic field of 2 Wb/m
2
, what is the resonance frequency?
5. Indicate the energy levels of
2
P and
2
D states and the allowed transitions
between them in the presence of a strong magnetic field.