Describe the principle of guided wave propagation in a rectangular waveguide. Derive the expression for the cut-off frequency of modes in terms of waveguide dimensions.
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.
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
.
Show that if the electric field of the incident wave is normal to the plane of incidence, the electric fields of the reflected and transmitted waves are also normal to the plane of incidence.
See Answer →Determine the WKB approximation for the bound state energy of a particle of mass m in the potential:
A plane electromagnetic wave of frequency is normally incident on a medium with relative permittivity
. Calculate the wavelength of the wave in the medium.
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.