Consider an operator for which
. Show that the expectation value of
in a parity eigenstate is zero.
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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.
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:
.
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
.
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 WKB approximation for the bound state energy of a particle of mass m in the potential:
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.