Book contents
- Frontmatter
- Contents
- List of Symbols
- Acknowledgments
- 1 A Brief History
- 2 Maxwell's Equations
- 3 Electromagnetic Waves
- 4 Special Relativity
- 5 Mechanics and Maxwell's Equations
- 6 Mechanics, Lagrangians, and the Calculus of Variations
- 7 Potentials
- 8 Lagrangians and Electromagnetic Forces
- 9 Differential Forms
- 10 The Hodge ⋆ Operator
- 11 The Electromagnetic Two-Form
- 12 Some Mathematics Needed for Quantum Mechanics
- 13 Some Quantum Mechanical Thinking
- 14 Quantum Mechanics of Harmonic Oscillators
- 15 Quantizing Maxwell's Equations
- 16 Manifolds
- 17 Vector Bundles
- 18 Connections
- 19 Curvature
- 20 Maxwell via Connections and Curvature
- 21 The Lagrangian Machine, Yang-Mills, and Other Forces
- References
- Index
- Plate Section
15 - Quantizing Maxwell's Equations
Published online by Cambridge University Press: 05 February 2015
- Frontmatter
- Contents
- List of Symbols
- Acknowledgments
- 1 A Brief History
- 2 Maxwell's Equations
- 3 Electromagnetic Waves
- 4 Special Relativity
- 5 Mechanics and Maxwell's Equations
- 6 Mechanics, Lagrangians, and the Calculus of Variations
- 7 Potentials
- 8 Lagrangians and Electromagnetic Forces
- 9 Differential Forms
- 10 The Hodge ⋆ Operator
- 11 The Electromagnetic Two-Form
- 12 Some Mathematics Needed for Quantum Mechanics
- 13 Some Quantum Mechanical Thinking
- 14 Quantum Mechanics of Harmonic Oscillators
- 15 Quantizing Maxwell's Equations
- 16 Manifolds
- 17 Vector Bundles
- 18 Connections
- 19 Curvature
- 20 Maxwell via Connections and Curvature
- 21 The Lagrangian Machine, Yang-Mills, and Other Forces
- References
- Index
- Plate Section
Summary
Summary: Our goal is to quantize Maxwell's equations, leading to a natural interpretation of light as being composed of photons. The key is that the quantization of harmonic oscillators (or, more prosaically, the mathematics of springs) is critical to understanding light. In particular, we show that the possible energies of light form a discrete set, linked to the classical frequency, giving us an interpretation for the photoelectric effect.
Our Approach
From Einstein's explanation of the photoelectric effect, light seems to be made up of photons, which in turn are somewhat particle-like. Can we start with the classical description of light as an electromagnetic wave solution to Maxwell's equations when there are no charges or currents, quantize, and then get something that can be identified with photons? That is the goal of this chapter. (As in the previous chapter, while all of this is standard, we will be following the outline given in chapter 2 of [41].)
Maxwell's equations are linear. We will concentrate on the ‘monochromatic’ solutions, those with fixed frequency and direction. We will see that the Hamiltonians of these monochromatic solutions will have the same mathematical structure as a Hamiltonian for a harmonic oscillator. We know from the last chapter how to quantize the harmonic oscillator. More importantly, we saw that there are discrete eigenvalues and eigenvectors for the corresponding Hamiltonian operator. When we quantize the Hamiltonians for the monochromatic waves in an analogous fashion, we will again have discrete eigenvalues and eigenvectors. It is these eigenvectors that we can interpret as photons.
- Type
- Chapter
- Information
- Electricity and Magnetism for MathematiciansA Guided Path from Maxwell's Equations to Yang–Mills, pp. 186 - 200Publisher: Cambridge University PressPrint publication year: 2015