Lesson: Free Particles and Wave Packets

Introduction

In classical physics, particles are described as point-like objects with well-defined positions and momenta. However, in quantum mechanics, particles behave like both particles and waves. A free particle is a particle that is not subject to any forces, such as an electron in a vacuum.

Wave Properties of Particles

The wave-particle duality of quantum mechanics means that particles have wave-like properties. The wavelength of a free particle is related to its momentum by the de Broglie equation:

λ = h/p

where:

• λ is the wavelength
• h is Planck's constant
• p is the momentum

Wave Packets

A wave packet is a mathematical function that describes the wave properties of a particle. It is a localised wave that represents the particle's position and momentum. As the particle moves, the wave packet spreads out.

The Uncertainty Principle

The uncertainty principle states that it is impossible to know both the position and momentum of a particle with perfect accuracy. This is because the act of measuring one quantity inevitably affects the other.

Applications of Free Particles and Wave Packets

Free particles and wave packets are fundamental concepts in quantum mechanics. They have applications in a wide range of fields, including:

• Quantum computing
• Quantum optics
• Semiconductor physics

Learning Resources

• MIT OpenCourseWare: Quantum Mechanics I
• Caltech: Quantum Mechanics I
• Stanford University: Quantum Mechanics

Exercises

1. Calculate the wavelength of an electron with a momentum of 10^-24 kg m/s.
2. Explain the uncertainty principle and its implications for the measurement of particle properties.
3. Describe an application of wave packets in quantum computing.

Assessment

Students will be assessed on their understanding of the following concepts:

• Wave-particle duality
• The de Broglie equation
• Wave packets
• The uncertainty principle
• Applications of free particles and wave packets