Quantum Mechanics
Determine the settlement approach (approximation) for the
harmonic oscillator system due to the relativistic...
Quantum Mechanics
Determine the settlement approach (approximation) for the
harmonic oscillator system due to the relativistic term using the
perturbation method in order 2 correction.
QUANTUM MECHANICS-upper level
In the harmonic oscillator problem, the normalized wave
functions for the ground and first excited states are ψ0 and ψ1.
Using these functions, at some point t, a wave function u = Aψ0 +
Bψ1 is constructed, where A and B are real numbers.
(a) Show that the average value of x in the u state is generally
non-zero.
(b) What condition A and B must satisfy if we want the function u
to be normalized?
(c)...
Demonstrate that the WKB approximation yields the energy levels
of the linear harmonic oscillator and compute the WKB approximation
for the energy eigenfunctions for the n=0 and n=1 state and compute
with the exact stationary state solutions.
The solution of the Schrödinger's Equation for the
quantum-mechanical harmonic oscillator includes the Hermite
polynomials in the wavefunctions. (In the following questions be
sure to define all symbols.) Please make sure your writing is
legible
(a) Write the differential equation for which the Hermite
polynomials are the solution.
(b) State the recursion relation for the Hermite polynomials and
be sure to define all symbols.
(c) Write the mathematical expression for the orthogonality of
the Hermite polynomials and be sure to...
Explain why in the case of the quantum harmonic oscillator the wave
function can cross the potential barrier and why does the same not
happen in the case of the infinite potential well?
Explain in detail
1. (a) Describe in your own words and mathematically the
harmonic oscillator approximation to molecular vibration. (b) What
does this approximation lack in terms of representing real
molecular bonds? (c) When is the approximation most valid and when
are higher order approximations necessary?
Discuss TWO of the following approximation methods in quantum
mechanics. Explain the types of problems to which the approximation
methods can be applied, how the basic theory is developed, and how
they are applied in practice. Include some mathematical details in
your answer.
(i) The Hartree theory of atoms and ions.
(ii) The variational method.
(iii) Non degenerate and degenerate perturbation theory.
The simple harmonic oscillator (SHO) is probably the
single most important approximation for
describing small displacements about stable equilibrium positions.
Why is it that the SHO approximation actually works? What are the
limitations of the SHO?