# Chapter 8: The Free Particle

Our study of one-dimensional quantum mechanics begins with what you may think is a simple problem: that of a free particle. The Schrödinger equation for this system is as simple as it gets, after all, *V*(*x*) = 0. However, it is not so simple to construct a *particle-like* solution should we wish to compare quantum mechanics to classical mechanics. In order to do this we must add together an infinite number of individual solutions to the free Schrödinger equation (energy eigenfunctions) to get a (Gaussian) wave packet that in many ways behaves like a classical free particle.

# Table of Contents

## Sections

- Section 8.1: Classical Free Particles and Wave Packets.
- Section 8.2: The Quantum-mechanical Free-particle Solution.
- Section 8.3: Exploring the Addition of Complex Waves.
- Section 8.4: Exploring the Construction of a Packet.
- Section 8.5: Towards a Wave Packet Solution.
- Section 8.6: The Quantum-mechanical Wave Packet Solution.
- Section 8.7: Exploring Fourier Transforms by Matching.
- Section 8.8: Exploring Wave Packets with Classical Analogies.

## Problems

- Problem 8.1: Write expressions for the three wave functions.
- Problem 8.2: Rank the wave functions according to momentum and energy.
- Problem 8.3: Which plane waves have the same mass?
- Problem 8.4: Analyze the properties of a free-particle Gaussian wave packet.
- Problem 8.5: Rank the wave packets according to momentum, kinetic energy, and total energy.
- Problem 8.6: Analyze the properties of an accelerating Gaussian wave packet.

## Alternate Visualizations

- Section 8.2: The Quantum-mechanical Free-particle Solution.
- Section 8.3: Exploring the Addition of Complex Waves.
- Section 8.4: Exploring the Construction of a Packet.
- Section 8.5: Towards a Wave Packet Solution.
- Section 8.6: The Quantum-mechanical Wave Packet Solution.
- Section 8.7: Exploring Fourier Transforms by Matching.
- Section 8.8: Exploring Wave Packets with Classical Analogies.
- Problem 8.2: Rank the wave functions according to momentum and energy.
- Problem 8.3: Which plane waves have the same mass?
- Problem 8.4: Analyze the properties of a free-particle Gaussian wave packet.
- Problem 8.5: Rank the wave packets according to momentum, kinetic energy, and total energy.
- Problem 8.6: Analyze the properties of an accelerating Gaussian wave packet.