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## Problem

Consider a particle of mass trapped in an infinite square well of width :

Part 1: What are the energy eigenstates?

Part 2: What is the probability of finding each eigenstate? Verify that all probabilities sum up to 1.

Part 3: Calculate the Hamiltonian .

## Solution

Part 1

To calculate the eigenenergies of each eigenstate of an infinite square well of width , use the formula .

Part 2

We know our eigenstates are orthogonal, and a quick check shows that the wavefunction is already normalized, so we can just calculate the probabilities directly.

Part 3

Our eigenvalues are the eigenenergies calculated above, so