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3.P: Exercises
 Last updated
 08:20, 16 Nov 2014

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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 timedependent wave equation, (274), can be transformed to give
where
and
 Consider onedimensional 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 onedimensional 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 onedimensional 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 onedimensional stationary bound state. Using the timeindependent Schrödinger equation, prove that
and
[Hint: You can assume, without loss of generality, that the stationary wavefunction is real.] Hence, prove the Virial theorem,