- Show that in the nonrelativistic limit, the free-particle
Klein-Gordon equation becomes the free-particle Schrödinger
- The Lorentz-invariant step function
is defined by
It is said that this step function is Lorentz invariant because it
only distinguishes between the past and the future, which is a Lorentz
Show that this step function is Lorentz invariant.
Hint: I believe this is true only if is restricted to be
a time-like vector.
- Use explicit plane-wave solutions,
establish the following normalization and orthogonality relationships:
- Derive the conserved current for a scalar field interacting
with an electromagnetic field with minimal coupling.
- Show that
is satisfied for a solution to the Klein-Gordon equation,
, that is a superposition of negative energy plane-wave
- Show that
reduces to the proper nonrelativistic expression in the nonrelativistic
- The positive and negative energy two-component solutions of the
Klein-Gordon equation in the Schrödinger form were define as
Show that the
- Show that in the nonrelativistic limit
- By completeness, any wavepacket can be expanded in terms of a linear
combination of positive and negative energy solutions:
are defined as
are general scalar functions of the
magnitude of .
Derive the normalization requirement for
to obtain experessions for
- Derive the expectation value of the position operator for a wavepacket
that is a mixture of positive and negative energy components:
Show that your result contains a piece that represents the expect
uniform velocity motion of the wavepacket and a piece that represents
a rapid wiggling (Zitterbewegung) of the position of the particle about
its central location, due to the interference of positive and negative
- Solve the coulomb potential problem.
In lectures we obtained
Look for solutions that are finite at and and show that
where is or a positive integer and is the
non-negative solution of
Show the energy can be expanded in powers of and to order
is the total quantum number and can take on
positive integer values.
Identify the rest energy, the energy in the nonrelativistic theory and
the fine-structure energy.
Calculate the spread of the fine-structure levels for a given .
(Note: they are much larger than obeserved experimentally in the
- The homogeneous Green function can be written as
Show that without loss of generality .
- Show that
- Show that
- Prove that
and thus show that
- Find the solution of the Klein-Gordon equation for the
-meson in a Coulomb potential and discusss the energy
The pion has a mass
MeV and spin 0.
- Starting with a positive energy wavepacket only, show that there is a
minimum width to the wavepacket.
- Sarting with a positive energy wavepacket only, show that they can not
be localized within a distance smaller than the Compton wavelength.
- We have seen that the energy levels of a mesonic atom are given by
- Show that the ground state energy for any mesonic atom heavier than
Explain what this complex energy means.
- Mesonic atoms have been well studied at places like Los Alamos and it
has been found that the ground states of atoms as heavy as lead
() or uranium () are quite stable.
How do you reconcile this fact with the result obtained above?
Be as quantitative as you can.
- A rapidly varying electric field can lead to the creation of
Calculate to lowest order in the probability per unit volume
per unit time of producing such pairs in the presence of an external
and show that
Suggestion: Use as an interaction potential the usual form
Utilize normalized plane waves solutions of the Klein-Gordon equation
and simple first order perturbation theory
- Consider a free charge Klein-Gordon particle of mass and
charge immersed in a uniform magnetic field in the
Using the gauge
show that motion
is quantized with energy
- Develop the wave equation for the nonrelativistic Hamiltonian developed
and show that you get positive- and negative-energy solutions.
Show that the Schrödinger equation is not Lorentz invariant.