A particle in a 1-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape.

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The particle in a box problem is a common application of a quantum mechanical model to a simplified system consisting of a particle moving horizontally within an infinitely deep well from which it cannot escape. The solutions to the problem give possible values of E and \(\psi\) that the particle can possess. E represents allowed energy values and \(\psi(x)\) is a wavefunction, which when squared gives us the probability of locating the particle at a certain position within the box at a given energy level.

To solve the problem for a particle in a 1-dimensional box, we must follow our Big, Big recipe for Quantum Mechanics:

Define the Potential Energy, V Solve the Schrödinger Equation Define the wavefunction Define the allowed energies

Step 1: Define the Potential Energy V

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The probability of finding a particle a certain spot in the box is determined by squaring \(\psi\). The probability distribution for a particle in a box at the \(n=1\) and \(n=2\) energy levels looks like this:

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