- Suppose the Dirac wavefunctions are normalized in the same way
as Klein-Gordon wavefunctions:

Use the Dirac equation to show that these wavefunctions differ from the ones normalized via

by a factor of .

- Show that the current density for a free-particle wavefunction
agrees with the corresponding nonrelativistic expression in the proper
limit.
- Re-derive current conservation using the Lorentz invariant form
of the Dirac equation.
- Show that

for a general Lorentz transformation, where is a matrix which is a function of the parameters of the Lorentz transformation and operates upon the four components of the column vector of the wavefunction satisfying the Dirac equation.

- Suppose is the Dirac wavefunction in a frame . What is the wavefunction in a frame which is obtained from by a rotation about the -axis through ?
- If a spin one-half particle at rest has its spin in the -direction, what is the probability that its spin will be observed to be in a direction from the -axis.
- Consider a Lorentz transformation to a frame moving in the -direction with speed . Obtain the transformation matrix .
- Use the results of part (c) to get the wavefunction of a spin one-half particle whose spin is in the -direction and whose velocity is in the -direction. Assume the wavefunction in the reset frame is and apply the boost of part (c).
- Obtain the transformation matrix for the boost of part (c) followed by the rotation of part (a).
- In this new frame in which the particle has negative helicity and a velocity from the -axis, calculate the probability that it will be observed to have its spin in the new -direction.

- Show that commutes with for proper Lorentz
transformations, , but anti-commutes with for improper Lorentz
transformations, , and hence

- Show how the bilinears
transform
under a general Lorentz transformation for
, and
.
- Given a free-particle spinor , construct for
and
, in terms of
by means of a Lorentz transformation.
- Prove the completeness relation

- Derive the following completeness relations

- The operators

are projection operators which are said to identify states of definite chirality (handedness).

- Show that , are legitimate projection operators in
that

- Demonstrate that in the limit of high energy - or
equivalently in the massless limit - that the Dirac spinors for
positive helicity (right-handed) and negative helicity (left-handed)
states of momentum are given by

where are spinors such that

Note: In order to conveniently deal with massless particles, it is important to use the normalization . The appropriate Dirac spinors can then be found by multiplying the usual forms by the factor . Demonstrate this.

- Show that

so that the chirality operator is equivalent to the helicity operator in this limit.

- Show that , are legitimate projection operators in
that
- Examine in detail the influence of the transformation
on the eigenfunctions
of an electron at rest with negative energy.
- Find out how
,
, and
and
transform under time
reversal.
- If and are two arbitrary solutions of the free Dirac
equation, prove the Gordon decomposition

It expresses the Dirac current as the sum of a convection current similar to the nonrelativistic one, and a spin current.

- Calculate the current

for the general wave packet which contains both positive and negative energy plane waves.

- At time the following wave packet with Gaussian density
distribution is defined as

Determine the wave packet at time developed from the above. Consider the intensity of the negative energy solutions in the wave packet. What does one learn in general about the applicability of the one-particle interpretation of the Dirac equation?

- Solve the Dirac equation for an attractive square well potential of
depth and radius , after determining the continuity
conditions at .
Obtain an explicit expression for the minimum with given
that just binds a particle of mass .
- The Dirac equation describing the interaction of a proton or neutron
with an applied external electromagnetic field has an additional term

involving the so-called anomalous magnetic moment. (For the proton, of course, and for the neutron .)

- Verify that the choice

corresponds to the observed magnetic moments of these particles, and

- Show that the additional interaction disturbs neither the Lorentz covariance of the equation nor the hermiticity of the Hamiltonian.

- Verify that the choice
- Suppose that the electron had a static electric dipole moment
analogous to its magnetic moment.
- Show that this could be accommodated by modifying the Dirac equation
to become

- Demonstrate that this equation is covariant but not invariant under a parity transformation.

- Show that this could be accommodated by modifying the Dirac equation
to become
- Consider a positive energy spin-1/2 particle at rest.
Suppose that at we apply an external (classical) vector
potential

which corresponds to an electric field of the form

Show that for there exists a finite probability of finding the particle in a negative energy state if such negative energy states are assumed to be originally empty. In particular, work out quantitatively the two cases: and and comment.

- As we saw in the previous section, a rapidly varying electric field
can lead to the creation of particle-antiparticle pairs.
Calculate to lowest order the probability per unit volume per unit
time of producing fermion pairs in the presence of an external
electric field

and show that

Suggestion: Utilize normalized plane wave solutions of the Dirac equation

and simple first order perturbation theory

with as in the Klein-Gordon case.

- Find the energy levels of a Dirac particle in a one-dimensional box of
depth and width .
- In order that be a symmetry operation of the Dirac theory,
the rules of interpretation of the wave function
must be the same as those of .
This means that observables composed of forms bilinear in
and
must have the same
interpretation (within a sign, appropriate to the observable) as those
of .
- Verify that this is so for the current:

and also

- Repeat these calculations for the charge-conjugation
transformation .
In particular, show

and interpret using the hole theory.

- Verify that this is so for the current:
- Obtain explicit representations for the matrix elements of the
operators
, ,
between spinors and .
Analyse in detail the case that
and
.
That is, computer
.
- In the Majorana representation all the are pure imaginary.
Find an explicit form of the Majorana representation.
- Find the exact energy eigenvalues and eigenfunctions for an electron
in a uniform magnetic field.
- Compute
- , ,
- Simplify , , .

- For a Dirac electron of mass in an attractive electrostatic
potential

- Find the energy levels.
- Solve the problem of scattering of such an electron with momentum off this potential.

- Show that the Dirac equation can not be invariant under spatial
rotations if are numbers and is a scaler.
- Show that the orbital angular momentum operator does not commute with
the Dirac Hamiltonian.
- Prove that

- Show that
commutes with the Dirac
Hamiltonian.
- Show that at high energies

where is the electron spinor. That is, show that in the extreme relativistic limit, the chirality operator () is equal to the helicity operator; and so, for example, corresponds to an electron of negative helicity.

- For a massive fermion, show that handedness is not a good quantum
number.
That is, show that does not commute with the Hamiltonian.
However, verify that helicity is conserved but is frame dependent.
In particular, show that the helicity is reversed by ``overtaking''
the particle concerned.
- For an electron of momentum
,
calculate the helicity eigenspinor.

2004-03-18