Physics 739, Problem sheet 5

1. [From G. Baym, Lectures on Quantum Mechanics, (Benjamin, 1969).](a) Suppose that a system in an energy eigenstate |i〉 at t = 0 is acted on by an

external perturbation V̂ (t). Let θ̂ be the operator corresponding some observableof the system. Show that the expectation value of θ̂ at time t is given to first orderin V̂ by

1

ih̄

∫ t

0

〈i| [θ̂(t), V̂ (t′)] |i〉 dt′

plus the zero order term. The operators in this expression are in the interactionrepresentation.

(b) Calculate the time-dependent dipole moment induced in a one-dimensionalharmonic oscillator by an electric field E cosωt turned on at t = 0; repeat for thecase that the electric field is turned on slowly.

2. Answer question 5.22 from your textbook Modern Quantum Mechanics, 2ndedition, by J.J. Sakurai and J. Napolitano, but in addition please derive the result

〈n′|x̂|n〉 =√h̄/(2mω0)(

√n+ 1δn′,n+1 +

√nδn′,n−1)

that is used in that question. Note that in the first edition this is problem 5.22.

3. Answer question 5.23 from your textbook Modern Quantum Mechanics, 2ndedition, by J.J. Sakurai and J. Napolitano. Note that in the first edition this isproblem 5.23.

4. Answer question 5.30 from your textbook Modern Quantum Mechanics, 2ndedition, by J.J. Sakurai and J. Napolitano. Note that in the first edition this isproblem 5.30.

5. Answer question 5.38 from your textbook Modern Quantum Mechanics, 2ndedition, by J.J. Sakurai and J. Napolitano. Note that in the first edition this isproblem 5.35.

6. Answer question 3.17 from your textbook Modern Quantum Mechanics, 2ndedition, by J.J. Sakurai and J. Napolitano. Note that in the first edition this isproblem 3.15.

7. Answer question 3.28 from your textbook Modern Quantum Mechanics, 2nd

1

edition, by J.J. Sakurai and J. Napolitano. Note that in the first edition this isproblem 3.24.

2