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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*,

** Next:** Orbital Angular Momentum
** Up:** Quantum Dynamics
** Previous:** Schrödinger Wave Equation
Richard Fitzpatrick
2013-04-08