Pre-quantum mechanics (a mathematician's perspective)
Camrud - Lecture notes - Pre-quantum mechanics (a mathematician's perspective).pdfThis notes should serve as an alternative to a traditional linear algebra course, intended for students in the physical sciences who plan on completing a quantum mechanics course. Most notably, one semester of calculus is a necessary prerequisite.
Topics covered include:
Sets, functions, injectivity and surjectivity
The complex numbers and algebra
Indexing, sequences
Summation
Vectors, vector spaces
Bases, vector representations
Inner products, norms, orthogonality
Orthonormal bases, the Gram-Schmidt method
Integration by parts
Taylor series
Fourier series
Linear maps and their inverses
Matrices, matrix multiplication
Inverse matrices
Determinants
Eigenvalues and eigenvectors
Diagonalization
Matrix functions, in particular the matrix exponential
Systems of 1st order, linear, homogeneous, constant coefficient ODEs
2nd order, linear, homogeneous, constant coefficient ODEs
Unitary and orthogonal maps
Hermitian and symmetric maps
Translating linear algebra into quantum mechanics
A quantum two-state system (polarization of light)
The particle in a 1-dimensional box problem and solution
Video lectures describing and constructing the notes may be found on my YouTube channel (see below).