QM Basics

Basics of Quantum Mechanics

Phase 2: Quantum Mechanics Core

Syllabus Goal

Physical basis, uncertainty principle, and Schrödinger equation.

Postulates

  1. State is represented by $|\psi\rangle$.
  2. Observables are Hermitian operators.
  3. Measurement yields eigenvalues.
  4. Probability is $|\langle a_n | \psi \rangle|^2$.
  5. Time evolution: $i\hbar \frac{\partial}{\partial t}|\psi\rangle = \hat{H}|\psi\rangle$.

Uncertainty Principle

$$ \sigma_x \sigma_p \ge \frac{\hbar}{2} $$ Cannot simultaneously know position and momentum with arbitrary precision.

Wave Function Constraints

Single-valued, continuous, normalizable ($\int |\psi|^2 dx = 1$).