Consider an operator for which
. Show that the expectation value of
in a parity eigenstate is zero.
Let (\hat{O}) be an operator and (\hat{\pi}) the parity operator. The parity operator satisfies (\hat{\pi}^\dagger \hat{O} \hat{\pi} = -\hat{O}). Let (|\psi\rangle) be a parity eigenstate, so that (\hat{\pi}|\psi\rangle = \lambda |\psi\rangle), where (\lambda = \pm 1). The expectation value of (\hat{O}) in (|\psi\rangle) is: [ \langle \psi | \hat{O} | \psi \rangle ] Insert (\hat{\pi}^\dagger \hat{\pi} = I) (the identity) around (\hat{O}): [ \langle \psi | \hat{O} | \psi ____ __________ ___ _______ ________ ______ ______ _______ ___.
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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
.
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
.
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 wave function and energy of the ground state and first excited state for a system of two identical bosons in 1D simple harmonic oscillator.
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:
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?