Quantum Infinite Potential Well - Probability of Ground-State Energy

[ARasmussen]

Homework Statement

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?

Homework Equations

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

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

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.

 

[paranormal]

for your question!

For part b, you can use the second equation to find the probability that a measurement of the energy yields the ground-state energy. The <E_1| term can be substituted with the given value of <E_1|, and then <\! E_1|\psi(x)\! > can be substituted with the given \psi(x) function. The resulting integral can then be solved to find the probability.

For part c, the expectation value for the Hamiltonian operator can be found using the formula <\psi|H|\psi>, where H is the Hamiltonian operator. In this case, the Hamiltonian operator for the infinite potential well is given by H = -\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}. You can substitute this into the formula and use the given \psi(x) function to solve for the expectation value.