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# Exercises

1. Let be a set of Cartesian position operators, and let be the corresponding momentum operators. Demonstrate that

where , and , are functions that can be expanded as power series.

2. Assuming that the potential is complex, demonstrate that the Schrödinger time-dependent wave equation, (3.55), can be transformed to give

where

and

3. Consider one-dimensional quantum harmonic oscillator whose Hamiltonian is

where and are conjugate position and momentum operators, respectively, and , are positive constants.

1. Demonstrate that the expectation value of , for a general state, is positive definite.

2. Let

Deduce that

3. Suppose that is an eigenket of the Hamiltonian whose corresponding energy is : that is,

Demonstrate that

Hence, deduce that the allowed values of are

where . Here, and are termed ladder operators. To be more exact, is termed a lowering operator (because it lowers the energy quantum number, , by unity), whereas is termed a raising operator (because it raises the energy quantum number by unity).

4. Let be a properly normalized (i.e., ) energy eigenket corresponding to the eigenvalue . Show that the kets can be defined such that

Hence, deduce that

5. Let the be the wavefunctions of the properly normalized energy eigenkets. Given that

deduce that

where . Hence, show that

4. Consider the one-dimensional quantum harmonic oscillator discussed in Exercise 3. Let be a properly normalized energy eigenket belonging to the eigenvalue . Show that

1. Hence, deduce that

for the th eigenstate.

5. Consider the one-dimensional quantum harmonic oscillator discussed in the previous two exercises. Let be a properly normalized eigenket of the lowering operator, , corresponding to the eigenvalue , where can be any complex number. The corresponding state is known as a coherent state.
1. Demonstrate that

where the expectation values are relative to the coherent state. Hence, deduce that

In other words, a coherent state is characterized by the minimum possible uncertainty in position and momentum.

2. If is the properly normalized energy eigenket belonging to the energy eigenvalue then show that

where

3. Show that the expectation value of the energy for the coherent state is

4. Putting in time dependence, so that

where , demonstrate that remains an eigenket of , but that the eigenvalue evolves in time as

Hence, deduce that

5. Writing

where is real and positive, show that

Of course, these expressions are analogous to those of a classical harmonic oscillator of amplitude and angular frequency . This suggests that a coherent state of a quantum harmonic oscillator is the state that most closely imitates the behavior of a classical oscillator.

6. Show that the properly normalized wavefunction corresponding to the state takes the form

 (3.111)

where is the properly normalized, stationary, ground-state wavefunction.

6. Consider a particle in one dimension whose Hamiltonian is

By calculating , demonstrate that

where is a properly normalized energy eigenket corresponding to the eigenvalue , and the sum is over all eigenkets.

7. Consider a particle in one dimension whose Hamiltonian is

Suppose that the potential is periodic, such that

for all . Deduce that

where is the displacement operator defined in Exercise 6. Hence, show that the wavefunction of an energy eigenstate has the general form

where is a real parameter, and for all . This result is known as Bloch's theorem.

8. Consider the one-dimensional quantum harmonic oscillator discussed in Exercise 3. Show that the Heisenberg equations of motion of the ladder operators, and , are

respectively. Hence, deduce that the momentum and position operators evolve in time as

respectively, in the Heisenberg picture.

9. Consider a particle in one dimension whose Hamiltonian is

Suppose that the particle is in a stationary bound state. Using the time-independent Schrödinger equation, prove that

and

Here, is the energy eigenvalue. [Hint: You may assume, without loss of generality, that the stationary wavefunction is real.] Hence, prove the Virial theorem,

10. Consider a particle of mass and charge moving in the - plane in the presence of the uniform perpendicular magnetic field . Demonstrate that the Hamiltonian of the system can be written

where , and

Hence, deduce that the possible energy eigenstates of the particle are

where is a non-negative integer. These energy levels are known as Landau levels.

11. Show that the time-dependent Schrödinger equation

where , , and , can be written

Hence, deduce that if the so-called Coloumb gauge [49],

is adopted then the equation simplifies to

Demonstrate that this equation is associated with a probability conservation law of the form

where

and

Finally, show that and are invariant under a gauge transformation.

Next: Orbital Angular Momentum Up: Quantum Dynamics Previous: Flux Quantization and the
Richard Fitzpatrick 2016-01-22