# 3.P: Exercises

- Let be a set of Cartesian position operators, and let be the corresponding momentum operators. Demonstrate that
- Assuming that the potential is complex, demonstrate that the Schrödinger time-dependent wave equation, (274), can be transformed to give
- Consider one-dimensional quantum harmonic oscillator whose Hamiltonian is
- Demonstrate that the expectation value of , for a general state, is positive definite.
- Let
- Suppose that is an eigenket of the Hamiltonian whose corresponding energy is : i.e.,
- Let be a properly normalized (i.e., ) energy eigenket corresponding to the eigenvalue . Show that the kets can be defined such that
- Let the be the wavefunctions of the properly normalized energy eigenkets. Given that

- Demonstrate that the expectation value of , for a general state, is positive definite.
- Consider the one-dimensional quantum harmonic oscillator discussed in Exercise 3. Let be a properly normalized energy eigenket belonging to the eigenvalue . Show that
- Consider a particle in one dimension whose Hamiltonian is
- Consider a particle in one dimension whose Hamiltonian is
*Bloch theorem*. - Consider the one-dimensional quantum harmonic oscillator discussed in Exercise 3. Show that the Heisenberg equations of motion of the ladder operators and are
- Consider a one-dimensional stationary bound state. Using the time-independent Schrödinger equation, prove that
*Virial theorem*,

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)