Quantum Qual Problems

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

• 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 $a=b=c$ and find the first 6 eigenenergies of the problem in terms of the box side length ($a$), the particle mass ($M$) 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.)

• $[- \frac{h^2}{2m}\Delta^2 + V]W(x,y,z)=E W(x,y,z)$
  

In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z)

$\frac{d^2 w(x)}{dx^2} - c^2 w(x) = 0$(1)
The same will be for y and z.

Solution of equation (1) is following
$w(x) = A\sin(cx)+B\cos(cx)$
$w(y) = C\sin(cy)+D\cos(cy)$
$w(z) = E\sin(cz)+F\cos(cz)$