Quantum Qual Problems

1.) Given a quantum mechanical particle of mass [math]\displaystyle{ M }[/math] confined inside a box of sides [math]\displaystyle{ a,b,c }[/math]. The particle is allowed to move freely between [math]\displaystyle{ 0 <x < a, 0< y< b }[/math] and [math]\displaystyle{ 0<z<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]\displaystyle{ a=b=c }[/math] and find the first 6 eigenenergies of the problem in terms of the box side length ([math]\displaystyle{ a }[/math]), the particle mass ([math]\displaystyle{ 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]\displaystyle{ [- \frac{h^2}{2m}\Delta^2 + V]W(x,y,z)=E W(x,y,z) }[/math]