# Homework Help: Rigid Walled Box (Quantum Mechanics)

1. Oct 27, 2014

### robotpie3000

1. The problem statement, all variables and given/known data

I have just started my undergraduate quantum mechanics lectures and I am currently stuck in this question:

A rigid-walled box that extends from -L to L is divided into three sections by rigid
interior walls at -x to +x, where x<L . Each section contains one particle in its ground
state.
(a) What is the total energy of the system as a function of x?
(b) Sketch E(x) versus x.
(c) At what value of x is E(x) a minimum?

2. Relevant equations

En=(n2π2ħ2)/(2mL2)
EΨ(x,t) = iħ(∂Ψ(x,t))/(∂t)

3. The attempt at a solution

a) The energy of the system as a function of x should obey Schrodinger's Equation, so EΨ(x,t) = iħ(∂Ψ(x,t))/(∂t), but I'm not sure how to find a solution to Ψ(x,t) that may help me move forward.

b) After reading my textbook for a while, I have decided to draw the total energy as a straight line parallel to the horizontal x axis.

c) I was thinking since there are 3 particles within the rigid walled box, the minimum energy of the system would be E3=(9π2ħ2)/(2m(2L2)), since the length of the box is 2L.

Help is much appreciated!!!!

2. Oct 27, 2014

### Staff: Mentor

No, you have three rigid walled boxes (bounded by the "rigid interior walls"), each containing one particle.

3. Oct 29, 2014

### robotpie3000

There are three regions: [-L, -x],[-x, x] and [x, L]. So the length of each region would be L-x, 2x and L-x respectively. If we let a1=1/(L-x)2, a2=1/(2x)2, and a3=1/(L-x)2, and use the inequality relationship ((a12 + a22 + a32)/n)1/2 ≥ (a1 + a2 + a3)/n, we can show that ([1/(L-x)2] + [1/(2x)2] + [1/(L-x)2])/3 ≥ 9/4L2.

E is a minimum if
L-x = 2x, which means x=L/3, and Emin=3*[(π2ħ2)/(2m)]*[27/(4L2)]=(27π2ħ2)/(8mL2).