Mathematical Toolkit for QM

Phase 1: The Mathematical Toolkit

Syllabus Goal

Master linear vector spaces, linear operators, and Fourier transformation in the context of QM.

Hilbert Space

Infinite-dimensional vector space with an inner product.

• Dirac Notation:
  • Ket $|\psi\rangle$: Vector.
  • Bra $\langle\psi|$: Dual vector.
  • Inner Product $\langle\phi|\psi\rangle$: Complex number.

Linear Operators

$$ \hat{A} |\psi\rangle = |\phi\rangle $$

• Hermitian Operators: Observables (Real eigenvalues).
• Commutator: $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$.