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.)

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

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)

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

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

• Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have
  

$w(x) = A\sin(mx)$
$w(y) = C\sin(ky)$
$w(z) = E\sin(qz)$

Also, w(a)=0 which gives $A\sin(ma)=0, m=\frac{\pi n_x}{a}$. For y component $C\sin(kb)=0, k=\frac{\pi n_y}{b}$ and for z $E\sin(qc)=0, q=\frac{\pi n_z}{c}$

A, C and E are normalization constants

$\frac{1}{A^2}=\int\sin^2 (\pi nx/a) dx = \frac{a}{2}$, limits are from 0 to a.

The eigenfunction for each component will be

$w(x) = \sqrt{\frac{2}{a}} \sin(\pi n_x x/a)$
$w(y) = \sqrt{\frac{2}{b}} \sin(\pi n_y y/b)$
$w(z) = \sqrt{\frac{2}{c}} \sin(\pi n_z z/c)$

The eigenenergies

$E_{n_x} = \frac{\pi^2 \hbar^2 {n_x}^2}{2Ma^2}$, $E_{n_y} = \frac{\pi^2 \hbar^2 {n_y}^2}{2Mb^2}$, $E_{n_z} = \frac{\pi^2 \hbar^2 {n_z}^2}{2Mc^2}$
Total energy is sum of these energies.

• $E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}$, where $n^2=n_x ^2 + n_y ^2 + n_z ^2$