Consider the following one-dimension simple harmonic oscillator Hamiltonian operator$
Use a trial wave function with a variational parameter
to estimate the upper bound to the ground state energy.
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:
.
Calculate the matrix elements for J2 for a system of two spin half particles.
See Answer →Using boundary conditions on the electric and magnetic fields, derive the Fresnel reflection and transmission coefficients for a plane wave incident normally on a boundary between two dielectric media.
See Answer →Write down the eigenkets for
with
.
Derive the electromagnetic wave equation for and
fields in vacuum starting from Maxwell's equations. Discuss the general properties of plane electromagnetic waves, including their speed and polarisation.
Define the Poynting vector. Starting from Maxwell's equations, derive the Poynting theorem and explain its physical interpretation in terms of energy conservation in electromagnetic fields.
See Answer →Write Maxwell's equations in both differential and integral forms. Explain the physical significance of the displacement current in Maxwell's modification of Ampere's law.
See Answer →Define the action of the permutation operator for a system of two particles 1 and 2 and two states
and
. Show that
and determine the eigenvalues of
.
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.
See Answer → Consider an operator for which
. Show that the expectation value of
in a parity eigenstate is zero.
Explain the significance of the pseudogap phase in cuprates.
See Answer →Write the space translation operator in quantum mechanics for a finite translation a along the x direction. Calculate the commutator
. You may use the Baker-Campbell-Hausdorff formula:
The critical temperature of lead (Pb) of average atomic mass is
. Calculate the critical temperature of a specimen of lead isotope with mass
using the normal isotope effect.
Show that for the superconducting transition in the absence of magnetic field, there is a discontinuity in the specific heat of a superconductor at Tc which can be written as:
For a superconducting specimen, the critical fields are and
at
and
respectively. Calculate the critical temperature and critical field at
.
For the hydrogen molecule which has two hydrogen atoms each with one electron occupying the 1s energy level, write the two particle wave functions for the singlet and triplet states. Determine the eigenvalues of the effective Hamiltonian
Discuss the concept of thermodynamic fluctuations in the canonical ensemble. Define energy fluctuations and derive an expression for the mean square fluctuation of energy. Using a classical ideal gas, show that the relative fluctuation in energy varies as and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems.
For Chromium () vapour at
with a number density of atoms
calculate:
i) The Larmor diamagnetic susceptibility assuming the atomic radius to be 1.2 A.
ii) The Curie paramagnetic susceptibility.
See Answer →Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.
See Answer →