Theme 2

Hilbert Spaces and Dirac Notation

Choose an exercise to open its statement and solution.

Keywords: pre-Hilbert space · supremum norm · parallelogram identity · counterexample

\ell^2(\mathbb{N})

Keywords: shift · ell2 · isometry · unitary operator · infinite dimension

Exercise 3—Riesz representative in finite dimension

Keywords: Riesz · linear functional · inner product · antilinearity · finite dimension

Exercise 4—Algebra of creation and annihilation operators

Keywords: harmonic oscillator · creation · annihilation · commutator · occupation number · Heisenberg

Exercise 5—Adjoints and projectors

Keywords: adjoint · conjugate transpose · orthogonal projector · idempotent

Exercise 6—Truncation of the harmonic oscillator

Keywords: harmonic oscillator · truncation · Heisenberg algebra · commutator · finite dimension

Exercise 7—Spin rotation operator and unitarity

Keywords: spin 1/2 · Pauli · rotation · unitary operator · operator exponential

Exercise 8—Hermitian inner product in C3

Keywords: Hermitian inner product · norm · orthogonality · Cauchy--Schwarz inequality · antilinearity

Exercise 9—The Gram—Schmidt procedure in C3

Keywords: Gram--Schmidt · orthonormal basis · orthogonal projection · decomposition · Parseval's identity

Exercise 10—Ket-bra operators

Keywords: Dirac notation · ket-bra · rank-one operator · adjoint · trace · eigenvalues · normal operator

Exercise 11—Properties of the trace

Keywords: trace · cyclicity · change of basis · closure relation · projector · Pauli matrices

Exercise 12—Change of orthonormal basis

Keywords: change of basis · unitary matrix · change-of-basis matrix · representation of an operator · Hadamard · invariants

Exercise 13—Orthogonal projector onto a plane

Keywords: orthogonal projector · subspace · Gram--Schmidt · complementary projector · distance to a subspace · best approximation

Exercise 14—General properties of orthogonal projectors

Keywords: orthogonal projector · eigenvalues · sum of projectors · product of projectors · commutation · orthogonal subspaces

Exercise 15—Oblique projector and orthogonal projector

Keywords: projector · idempotent · oblique projector · orthogonal projector · self-adjoint operator · measurement postulate

Exercise 16—Self-adjoint operators

Keywords: self-adjoint operator · Hermitian matrix · real eigenvalues · orthogonal eigenvectors · spectral decomposition

Exercise 17—Unitary operators

Keywords: unitary operator · preservation of the inner product · eigenvalues of modulus one · orthonormal columns · exponential

Exercise 18—Householder reflection

Keywords: reflection · orthogonal reflection · unitary operator · self-adjoint operator · eigenvalues · eigenspaces · projector

Exercise 19—Plane rotations and complex diagonalisation

Keywords: rotation · orthogonal matrix · complex eigenvalues · diagonalisation · generator · Pauli matrices · circular polarisation

Exercise 20—Rotations in three-dimensional space: axis and angle

Keywords: rotation · orthogonal matrix · rotation axis · rotation angle · trace · eigenvalues · cyclic permutation

Exercise 21—Diagonalisation of a two-by-two Hermitian matrix

Keywords: diagonalisation · Hermitian matrix · two-level system · mixing angle · avoided crossing · level repulsion

Exercise 22—Diagonalisation with a degenerate eigenvalue

Keywords: diagonalisation · degenerate eigenvalue · eigenspace · orthonormal basis · spectral projectors · spectral decomposition

Exercise 23—Non-diagonalisable matrices and normal matrices

Keywords: diagonalisation · nilpotent matrix · Jordan block · normal matrix · spectral theorem · non-orthogonal eigenvectors

Exercise 24—Functions of a matrix

Keywords: functional calculus · spectral decomposition · square root · inverse · exponential · minimal polynomial · spectral projectors · time evolution

Exercise 25—Matrix exponential

Keywords: matrix exponential · nilpotent matrix · Pauli matrices · commutation · determinant · trace · unitary operator

Exercise 26—Properties of commutators

Keywords: commutator · Leibniz rule · Jacobi identity · anti-Hermitian operator · trace · canonical commutation relation · Pauli matrices

Exercise 27—Commutation and common eigenvectors in two dimensions

Keywords: commutator · common eigenvectors · nondegenerate spectrum · compatible observables · incompatible observables · Pauli matrices

Exercise 28—Simultaneous diagonalisation and a complete set of commuting observables

Keywords: simultaneous diagonalisation · commuting observables · invariant eigenspace · degeneracy · common eigenbasis · complete set of commuting observables

Exercise 29—The Hilbert—Schmidt inner product

Keywords: Hilbert--Schmidt inner product · matrix space · trace · orthonormal basis · Pauli matrices · Cauchy--Schwarz inequality

Exercise 30—Pauli decomposition of two-by-two matrices

Keywords: Pauli matrices · decomposition · Hermitian matrix · eigenvalues · rank-one projector · Bloch vector

Exercise 31—Positive operators and square roots

Keywords: positive operator · square root · singular values · operator norm · Cayley--Hamilton theorem · golden ratio

Exercise 32—Fourier basis and Parseval's identity

Keywords: Hilbert basis · Fourier series · Parseval's identity · Bessel's inequality · best approximation · infinite dimension