Mathematical Prerequisites
To succeed in this quantum mechanics course, you'll need a solid foundation in several mathematical areas. This page outlines the essential prerequisites and provides resources for self-study.
Essential Prerequisites
1. Linear Algebra
Linear algebra is the language of quantum mechanics. You must be comfortable with:
- • Vectors and vector spaces
- • Matrices and matrix operations
- • Determinants and traces
- • Eigenvalues and eigenvectors
- • Diagonalization of matrices
- • Inner products and orthogonality
- • Linear transformations
2. Calculus
Single and multivariable calculus:
- • Derivatives and integrals
- • Partial derivatives
- • Multiple integrals
- • Taylor and Fourier series
- • Gradient, divergence, curl
- • Coordinate systems (Cartesian, spherical, cylindrical)
3. Differential Equations
Ordinary and partial differential equations:
- • First and second-order ODEs
- • Series solutions
- • Separation of variables
- • Boundary value problems
- • Special functions (Bessel, Legendre, Hermite, Laguerre)
4. Complex Analysis
Complex numbers are fundamental:
- • Complex numbers and complex plane
- • Complex functions
- • Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$
- • Complex conjugation
- • Analytic functions (helpful but not essential)
Recommended Background
Classical Mechanics
Understanding classical mechanics helps build intuition:
- • Newton's laws
- • Lagrangian formalism
- • Hamiltonian formalism
- • Conservation laws
Basic Physics
- • Energy, momentum, angular momentum
- • Wave phenomena
- • Electromagnetism (Maxwell's equations)
- • Special relativity (helpful for later chapters)
Suggested Study Path
If you're starting from scratch:
- Weeks 1-4: Review linear algebra (vectors, matrices, eigenvalues)
- Weeks 5-8: Brush up on calculus (derivatives, integrals, Taylor series)
- Weeks 9-12: Study differential equations (ODEs, separation of variables)
- Weeks 13-14: Complex numbers and Euler's formula
- Week 15+: Begin Part I of this course!
If you have most prerequisites:
You can dive straight into Part I: Mathematical Foundations, reviewing specific topics as needed.
Self-Assessment Quiz
Can you answer these questions? If so, you're ready to begin!
Linear Algebra:
Find the eigenvalues of $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$
Calculus:
Compute $\int_{-\infty}^{\infty} e^{-x^2} dx$
Differential Equations:
Solve $\frac{d^2y}{dx^2} + \omega^2 y = 0$
Complex Numbers:
What is $e^{i\pi}$?