Quantum Infinite Potential Well - Probability of Ground-State Energy

[ARasmussen]

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

A particle in the infinite potential well in the region 0 < x < L is in the state

\psi(x) = \begin{cases}
Nx(x-L) & \text{ if } 0<x<L \\
0 & \text{ if } otherwise
\end{cases}

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?


2. Relevant equations

\int_{0}^{L}\left | \psi(x) \right |^{2} dx = 1
prob(E_1) = \int_{0}^{L}\left | <\! E_1|\psi(x)\! > \right |^{2} dx

<E_1|=\frac{\hbar^{2}\pi^{2}}{2mL^{2}}

3. The attempt at a solution

For part a, I used the first equation to solve for N, and I got \sqrt{\frac{30}{L^{5}}}. Part b is where I began to get confused.

Given the equations above for prob(E_1), and <E_1|, I'm unable to figure out how to find the probability that the energy state is in the ground state.

Any hints?

Thanks

[vela]

You found the normalization constant correctly, but the last two of your equations aren't correct. The energy E1 of the ground state |\phi_1\rangle is equal to E_1 = \hbar^2\pi^2/2mL^2.

The amplitude that the particle is in the ground state |\phi_1\rangle is given by

\langle \phi_1 | \psi \rangle = \int \phi_1^*(x)\psi(x)\,dx

where \phi_1(x) is the wave function of the ground state. The probability P that the particle is in the ground state is equal to the modulus of the amplitude squared: P = |\langle \phi_1 | \psi \rangle|^2.

[ARasmussen]

Thanks for the reply. How would I go about finding the wave function of the ground state?

[vela]

The infinite square well problem is likely already solved in your textbook or your notes, and you can just look up what the eigenstate wave functions are. If not, you need to solve the Schrödinger equation with the appropriate potential and boundary conditions.