1. Particle in a box: Consider the time-independent Schr¨odinger equation:
iℏ
∂ψ
∂t = −
ℏ
2
2m
∂
2ψ
∂x2
+ V (x)ψ
which is the underlying equation of quantum mechanics where V (x) is a given
potential.
(a) Let ψ = u(x) exp(−iEt/ℏ) and derive the time-independent Schr¨odinger equation. (Note that E here corresponds to energy).(b) Show that the resulting eigenvalue problem is of Sturm-Liouville typ(c) Consider the potential
V (x) = 0 |x| < L
∞ elsewhere
which implies u(L) = u(−L) = 0. Calculate the normalized eigenfunctions
and eigenvalues.(d) If an electron jumps from the third state to the ground state, what is the
frequency of the emitted photon? Recall that E = ℏω.(e) If the box is cut in half, then u(0) = u(L) = 0. What are the resulting
eigenfunctions and eigenvalues?2. Find the Green’s function (fundamental solution) for each of the following problems,
and express the solution u in terms of the Green’s function.
(a) u
′′ + c
2u = f(x) with u(0) = u(L) = 0.(b) u
′′ − c
2u = f(x) with u(0) = u(L) = 0.3. Calculate the solution of the Sturm-Liouville problem using the Green’s function
approach.
Lu = −[p(x)ux]x + q(x)u = f(x) 0 ≤ x ≤ L
with
α1u(0) + β1u
′
(0) = 0 and α2u(L) + β2u
′
(L) = 0
Advanced, AMATH, Differential, Equations, Homework, solved
[SOLVED] Amath 568 advanced differential equations homework 3
$25
File Name: Amath_568_advanced_differential_equations_homework_3.zip
File Size: 489.84 KB
Reviews
There are no reviews yet.