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.

You are watching: Express the energy e of the particle in terms of the wave number k of the particle.

## Introduction

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 api/deki/files/9038/Psi_n1n2.JPG?revision=1&size=bestfit&width=361&height=210" />

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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Important Facts to Learn from the Particle in the Box

The energy of a particle is quantized. This means it can only take on discrete energy values. The lowest possible energy for a particle is NOT zero (even at 0 K). This means the particle always has some kinetic energy. The square of the wavefunction is related to the probability of finding the particle in a specific position for a given energy level. The probability changes with increasing energy of the particle and depends on the position in the box you are attempting to define the energy for In classical physics, the probability of finding the particle is independent of the energy and the same at all points in the box

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