Mathematical Formalism of Quantum Mechanics

Define a Hilbert space and verify that specific function spaces (e.g., L²) satisfy its axioms.

Use Dirac bra-ket notation to express states, inner products, and operator actions, and relate bras to elements of the dual space.

Classify linear operators as Hermitian, unitary, or projection operators and derive their key properties.

Apply the spectral theorem to decompose a Hermitian operator in terms of its eigenvalues and eigenstates for both discrete and continuous spectra.

Prove that two Hermitian operators can be simultaneously diagonalized if and only if they commute.

Relate the position and momentum representations via Fourier transform and interpret the Dirac delta as a distribution in a rigged Hilbert space.

Construct composite state spaces using the tensor product and distinguish entangled from separable states.

Construct the density matrix for pure and mixed states and compute expectation values and measurement probabilities using the trace formalism.