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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, (274), 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 : i.e., Demonstrate that    Hence, deduce that the allowed values of are where .

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        Hence, deduce that for the th eigenstate.

5. 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.

6. 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 4. 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 the Bloch theorem.

7. 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.

8. Consider a one-dimensional stationary bound state. Using the time-independent Schrödinger equation, prove that and [Hint: You can assume, without loss of generality, that the stationary wavefunction is real.] Hence, prove the Virial theorem,    Next: Orbital Angular Momentum Up: Quantum Dynamics Previous: Schrödinger Wave Equation
Richard Fitzpatrick 2013-04-08