Solve schroedinger's equation for a three dimensional harmonic
oscillator and obtain its eigen values and eigen...
Solve schroedinger's equation for a three dimensional harmonic
oscillator and obtain its eigen values and eigen functions.Are the
energy levels degenerate? Explain what is the minimum uncertainty
in its location in the lowest state.
1)
a) Establish schrodinger equation,for a linear harmonic
oscillator and solve it to obtain its eigen values and eigen
functions.
b) calculate the probability of finding a simple harmonic
oscillator within the classical limits if the oscillator in its
normal state.
Consider a three-dimensional isotropic harmonic oscillator for
which the Hamiltonian is given by H = p2 2m+ 1/2mω2r2. Use the
variational method with the trial function u(r) = 1πa2 3/4
exp(−r2/2a2) and obtain E. Minimizing E with respect to a2, show
that the upper bound for the ground-state energy reproduces the
exact result for the energy given by a =(mω and Ea = 32ω.
Substitute the above value of a in the trial function and show that
it also reproduces...
consider Three-Dimensional harmonic oscillator with the same
frequencies along all three directions. a) determine the wave
function and the energy of the ground state. b) how many quantum
numbers are needed to describe the state of oscillation? c) the
degeneracy of the first excited state. express the wave function
involved in the schrodinger equation as a product given by x, y, z
and separate the variables.
Consider a system of three non-interacting particles confined by a
one-dimensional harmonic oscillator potential and in thermal
equilibrium with a total energy of 7/2 ħw.
(a) what are the possible occupation numbers for this system
if the particles are bosons.
(b) what is the most probable energy for a boson picked at
random from this system.
Suppose a particle of mass m and charge q is in a
one-dimensional harmonic oscillator potential with natural
frequency ω0. For times t > 0 a time-dependent
potential of the form
V1(x) = εxcos(ωt) is turned on. Assume the system starts
in an initial state|n>.
1. Find the transition probability from initial state |n> to
a state |n'> with
n' ≠ n.
2. Find the transition rate (probability per unit time) for the
transition |n>→|n'>.