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- 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.

- Assuming that the potential is complex, demonstrate that the Schrödinger time-dependent wave equation, (274), can be transformed to give

where

and

- Consider one-dimensional quantum harmonic oscillator whose Hamiltonian is

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

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

- Let

Deduce that

- 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 .

- 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

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

deduce that

where . Hence, show that

- 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.

- 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.

- 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*.

- 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.

- 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*,