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
.
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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
.
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
.
Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: . You may use:
$
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:
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.
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.