- #1
ARasmussen
- 2
- 0
Homework Statement
A particle in the infinite potential well in the region 0 < x < L is in the state
[tex]\psi(x) = \begin{cases}
Nx(x-L) & \text{ if } 0<x<L \\
0 & \text{ if } otherwise
\end{cases}[/tex]
a) Determine the value of N so that the state is properly normalised
b) What is the probability that a measurement of the energy yields the ground-state energy of the
well?
c) What is the expectation value for the Hamiltonian operator for this state?
Homework Equations
[tex]\int_{0}^{L}\left | \psi(x) \right |^{2} dx = 1[/tex]
[tex]prob(E_1) = \int_{0}^{L}\left | <\! E_1|\psi(x)\! > \right |^{2} dx[/tex]
[tex]<E_1|=\frac{\hbar^{2}\pi^{2}}{2mL^{2}}[/tex]
The Attempt at a Solution
For part a, I used the first equation to solve for N, and I got [tex]\sqrt{\frac{30}{L^{5}}}[/tex]. Part b is where I began to get confused.
Given the equations above for prob([tex]E_1[/tex]), and [tex]<E_1|[/tex], I'm unable to figure out how to find the probability that the energy state is in the ground state.
Any hints?
Thanks