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 space translation operator for a finite translation (a) along the (x)-direction in quantum mechanics is defined as: [ \hat{T}(a) = e^{- \frac{i}{\hbar} a \hat{p}_x} ] where (\hat{p}_x) is the momentum operator along (x). _______ __________ ________ __________ ___ ______ _______.
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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
.
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:
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?
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:
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.