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 the two state problem in which the unperturbed Hamiltonian has just two eigenkets,
and
with:
;
, and E2 > E1. The system is subjected to a time-dependent perturbation:
.
Calculate the probability for the system to be in the state at time t, given that it is in the state
at
.
Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: . You may use:
$
Consider an operator for which
. Show that the expectation value of
in a parity eigenstate is zero.
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?
Derive the current conservation equation from the Dirac equation.
Calculate the matrix elements for J2 for a system of two spin half particles.
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.