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# Exercises

1. Consider the two-state system examined in Section 8.3. Suppose that

where , , , and are real. Show that

where , , and

Hence, deduce that if the system is definitely in state 1 at time then the probability of finding it in state 2 at some subsequent time, , is

2. Consider an atomic nucleus of spin- and -factor placed in the magnetic field

where . Let be a properly normalized simultaneous eigenket of and , where is the nuclear spin. Thus, and , where . Furthermore, the instantaneous nuclear spin state is written

where .
1. Demonstrate that

for , where and .

2. Consider the case . Demonstrate that if and then

3. Consider the case . Demonstrate that if and then

4. Consider the case . Demonstrate that if and then

3. Derive Equation (8.45).

4. Derive Equation (8.83).

5. If

is the probability that a system initially in state at time is still in that state at some subsequent time, deduce that the mean lifetime of the state is .

6. Demonstrate that when , where is the momentum operator, and is a real function of the position operator, . Hence, show that the Hamiltonian (8.128) is Hermitian.

7. Demonstrate that

where the average is taken over all possible directions of the unit vector .

8. Demonstrate that

where is specified by Equation (8.165). [Hint: Write

where .]

9. Demonstrate that a spontaneous transition between two atomic states of zero orbital angular momentum is absolutely forbidden. (Actually, a spontaneous transition between two zero orbital angular momentum states is possible via the simultaneous emission of two photons, but takes place at a very slow rate [56,15,102].)

10. Find the selection rules for the matrix elements and to be non-zero. Here, denotes an energy eigenket of a hydrogen-like atom corresponding to the conventional quantum numbers, , , and .

11. The Hamiltonian of an electron in a hydrogen-like atom is written

where . Let denote a properly normalized energy eigenket belonging to the eigenvalue . By calculating , derive the Thomas-Reiche-Kuhn sum rule,

where

and , and the sum is over all energy eigenstates. Here, stands for either , , or . (See Exercise 6.)

1. Show that the only non-zero electric dipole matrix elements for the hydrogen atom take the values

2. Likewise, show that the only non-zero electric dipole matrix elements take the values

3. Finally, show that the only non-zero electric dipole matrix elements take the values

1. Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any state to a state of a hydrogen atom is

where is the fine structure constant. Hence, deduce that the relative natural width of the associated spectral line is

where denotes wavelength.

2. Likewise, show that the spontaneous decay rate from any state to a state is

Hence, deduce that the relative natural width of the associated spectral line is

1. Demonstrate that the net oscillator strength for transitions in a hydrogen atom is

irrespective of the polarization of the radiation.
2. Likewise, show that the net oscillator strength for transitions is

irrespective of the polarization of the radiation.

12. Taking electron spin into account, the wavefunctions of the states of a hydrogen atom are written where . Here, the are standard radial wavefunctions, and the are spin-angular functions. Likewise, the wavefunctions of the states are written , where . Finally, the wavefunctions of the states are written , where , , , . As a consequence of spin-orbit interaction, the four states lie at a slightly higher energy than the two states. Demonstrate that the spontaneous decay rates, due to electric dipole transitions, between the various and states are

where

Here, the states have been labeled by their spin-angular functions. Hence, deduce that for an ensemble of hydrogen atoms in thermal equilibrium, the spectral line associated with transitions is twice as bright as that associated with transitions.

13. Taking electron spin into account, the wavefunctions of the states of a hydrogen atom are written where . The wavefunctions of the states are specified in the previous exercise. Demonstrate that the spontaneous decay rates, due to electric dipole transitions, between the various and states are

where

Here, the states have been labeled by their spin-angular functions.

1. Consider the electric dipole transition of a hydrogen atom. Show that the angular distribution of spontaneously emitted photons is

where , and the photon's direction of motion is parallel to .
2. Consider the electric dipole transition of a hydrogen atom. Show that the angular distribution of spontaneously emitted photons is

3. Hence, show that for an ensemble of hydrogen atoms in thermal equilibrium, the angular distribution of spontaneously emitted photons associated with the electric dipole transition is

14. The properly normalized triplet and singlet state kets of a hydrogen atom can be written

and

respectively. Here, the kets on the left are kets, where and are the conventional quantum numbers that determine the overall angular momentum, and the projection of the overall angular momentum along the -axis, respectively. Finally, and are the properly normalized spin-up and spin-down states for the proton and the electron, respectively. As explained in Section 7.9, the triplet states have slightly higher energies than the singlet state, as a consequence of spin-spin coupling between the proton and the electron. Spontaneous magnetic dipole transitions between the triplet and singlet states occur via the interaction of the magnetic component of the emitted photon and the electron magnetic moment. (The magnetic moment of the proton is much smaller than that of the electron, and consequently does not play a significant role in this transition.) In the following, the initial state, , corresponds to one of the triplet states, and the final state, , corresponds to the singlet state.
1. Demonstrate that

where and are the energies of the initial and final states, and is the proton -factor.
2. Let be the magnetic dipole matrix element, where is the electron spin. Show that

if the initial state is the state,

if the initial state is the state, and

if the initial state is the state.
3. Deduce that the angular distribution of the spontaneously emitted photon is

if the initial state is either the state or the state. Show that the angular distribution is

if the initial state is the state. Here, the photon's direction of motion is parallel to .
4. Finally, show that the overall spontaneous transition rate is

15. Consider the electric quadrupole transition of a hydrogen atom.
1. Show that the angular distribution of spontaneously emitted photons can be written

2. Demonstrate that the only non-zero electric quadrupole matrix elements takes the values , , and

3. Hence, deduce that

and

Here, the spontaneously emitted photon's direction of motion is parallel to .

16. Let

where is a non-negative integer. Demonstrate that

and

Hence, deduce that

and

17. Treating as a small parameter, and neglecting terms of order , show that

18. Repeat the calculation of Section 8.15 for the case where the electric polarization vector of the incident photon is . Hence, show that, in this case, the differential photo-ionization cross-section is

Deduce that the photo-ionization cross-section for unpolarized electromagnetic radiation propagating in the -direction is

19. Consider the photo-ionization of a hydrogen atom in the state. Demonstrate that formula (8.249) is replaced by

20. Consider the inverse process to the photo-ionization of a hydrogen atom investigated in Section 8.15. In this process, which is known as radiative association, an unbound electron of energy is captured by a (stationary) proton to form a hydrogen atom in its ground state, with the emission of a photon of energy

where . Here, is (negative) hydrogen ground-state energy, and is the (positive) ground-state ionization energy. Radiative association can be thought of as a form of stimulated emission in which the initial state is unbound. Thus, according to the analysis of Section 8.9,

where is the electron momentum, is the wavevector of the emitted photon, and is the number of photon states whose angular frequencies lie between and , and whose direction of motion lie in the range of solid angles . Here, are two independent unit vectors normal to .
1. Show that

where the initial electron and final photon are both assumed to be contained in a periodic box of volume .
2. Hence, demonstrate that

where , and .

3. Finally, show that

where .

Next: Identical Particles Up: Time-Dependent Perturbation Theory Previous: Photo-Ionization
Richard Fitzpatrick 2016-01-22