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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, (3.55), 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
: that is,
Demonstrate that
Hence, deduce that the allowed values of
are
where
. Here,
and
are termed ladder operators. To be more exact,
is termed a lowering operator (because it lowers the energy quantum number,
, by unity), whereas
is termed a raising operator (because it raises the energy quantum number by unity).
 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 the onedimensional quantum harmonic oscillator discussed in the previous
two exercises. Let
be a properly normalized eigenket of the lowering operator,
,
corresponding to the eigenvalue
, where
can be any complex number. The
corresponding state is known as a coherent state.
 Demonstrate that
where the expectation values are relative to the coherent state.
Hence, deduce that
In other words, a coherent state is characterized by the minimum possible uncertainty in position and momentum.
 If
is the properly normalized energy eigenket belonging to the energy eigenvalue
then show that
where
 Show that the expectation value of the energy for the coherent state is
 Putting in time dependence, so that
where
,
demonstrate that
remains an eigenket of
, but that the eigenvalue
evolves in time as
Hence, deduce that
 Writing
where
is real and positive, show that
Of course, these expressions are analogous to those of a classical harmonic oscillator
of amplitude
and angular frequency
. This suggests that a coherent state
of a quantum harmonic oscillator is the state that most closely imitates the behavior of
a classical oscillator.
 Show that the properly normalized wavefunction corresponding to the state
takes the form

(3.111) 
where
is the properly normalized, stationary, groundstate wavefunction.
 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 6. 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 Bloch's 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 particle in one dimension whose Hamiltonian is
Suppose that the particle is in a stationary bound state. Using the timeindependent Schrödinger equation,
prove that
and
Here,
is the energy eigenvalue.
[Hint: You may assume, without loss of generality, that the stationary wavefunction is real.] Hence, prove the Virial
theorem,
 Consider a particle of mass
and charge
moving in the

plane in the presence of
the uniform perpendicular magnetic field
. Demonstrate that the Hamiltonian of the
system can be written
where
, and
In addition, show that
Hence, deduce that the possible energy eigenstates of the particle are
where
is a nonnegative integer. These energy levels are known as Landau levels.
 Show that the timedependent Schrödinger equation
where
,
, and
,
can be written
Hence, deduce that if the socalled Coloumb gauge [49],
is adopted then the equation simplifies to
Demonstrate that this equation is associated with a probability conservation law of the form
where
and
Finally, show that
and
are invariant under a gauge transformation.
Next: Orbital Angular Momentum
Up: Quantum Dynamics
Previous: Flux Quantization and the
Richard Fitzpatrick
20160122