Quantum Qual Problems

1.) Given a quantum mechanical particle of mass [math]M[/math] confined inside a box of sides [math]a,b,c[/math]. The particle is allowed to move freely between [math]0 \lt x \lt a, 0\lt y\lt b [/math] and [math]0\lt z\lt c[/math].

• Use the time-independent Schrodinger equation for this problem to obtain the general form for the eigenfunctions of the particle

• Now apply boundary conditions to obtain the specific eigenfunctions and eigenenergies for this specific problem.

• Assume [math]a=b=c[/math] and find the first 6 eigenenergies of the problem in terms of the box side length ([math]a[/math]), the particle mass ([math]M[/math]) and standard constants. What are their quantum number? Make a sketch of the eigenvalue spectrum, a table listing these eigenenergies and the quantum numbers of all the states that correspond to them.

Solution:

2.) [math] - \frac{h^2}{2m}\Delta^2 + V [/math]