Derive the current conservation equation from the Dirac equation.
Start with the Dirac equation for a free particle: [ (i\hbar \gamma^\mu \partial_\mu - mc)\psi = 0 ] where ( \gamma^\mu ) are the Dirac gamma matrices, ( \psi ) is the Dirac spinor, ( ________ ___ __________ ____ ____ ___ ________ _________.
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Consider an operator for which
. Show that the expectation value of
in a parity eigenstate is zero.
Write down the eigenkets for
with
.
A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with . At t > 0, an electric field
acts on the particle where
is a constant. If the particle is in the ground state when t < 0, calculate the probability that it will be in the first excited state for t > 0.
Determine the first and second order perturbation correction to the ground state energy eigenvalue of the one-dimensional infinite potential well of width L () with the perturbation:
.
Explain how the expression for the energy levels obtained by solving Klein Gordon equation for a Coulomb field differs from the results derived from Schrödinger equation. Why is this solution not able to explain the fine-structure splitting of the energy levels?
Determine the wave function and energy of the ground state and first excited state for a system of two identical bosons in 1D simple harmonic oscillator.
Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: . You may use:
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