AMATH 473 COURSE SUMMARY
History of Quantum Theory [Ch. 1]
- Planck and Thermal [Blackbody] Radiation
- Einstein and the Photoelectric Effect:
- Bohr and Atomic Spectral Line Emissions
- De Broglie and Matter Waves:
- Brief Intro to Relativistic Quantum Mechanics and Quantum Field Theory
Classical Hamiltonian Mechanics [Ch. 2]
- Abstraction Level Map:
Hamiltonians à Differential Equations of Motion à Functions à Variables à Number Predictions
- Hamiltonian:
- Poisson Bracket Properties:
- Hamilton’s Equation:
- Conservation of Quantities: f conserved if - ie Energy conserved if
- Classical Representation:
Quantum Hamiltonian Mechanics [Ch. 3]
- Hamiltonian:
- Poisson Bracket Properties: (same as above, only with hats)
- Commutator:
- Difference in Q.M.: instead of
- Canonical Commutation Relations:
- Hamilton’s Equation of Motion:
- Heisenberg Equation of Motion:
- Observables (Hermitian):
- Example - Matrix Representation:
- Raising/Lowering Operators:
- Free Particle Solutions:
Hilbert Spaces & Dirac Notation [Ch. 3]
- Complex Vector Space: H :
- Dirac Notation: “ket” vectors H & “bra” maps H *
- Hermitian Conjugate: H H *
- Inner Product: H :
with equality iff
- Unitary/Pre-Hilbert Space: - complex vector space + inner product defined
- Norm & Normalization:
- Distance:
- Hilbert Space: - pre-hilbert space + all vectors have a finite norm
- Hilbert Basis: s.t. H
- Separable Hilbert Space: - hilbert bases are countable
- Vector Coefficients of a State:
- Wave Function of a State:
- Resolution of the Identity:
- Observable Expression:
- Domain of an Observable: H so that H
- Expectation Value:
- Uncertainty:
- Uncertainty Relations:
- Position/Momentum:
- Energy/Time:
Time Evolution of Quantum Systems [Ch. 4]
- Ehrenfest Theorem: QM EOM = CM EOM iff is a polynomial in x & p of degree 2
- Time-Evolution Operator:
- Time Evolution of Observables:
- Calculating: - no t-dependence:
- with t-dependence: for T = time-orderer
- Unitary Property:
- Pictures of Time Evolution:
- Heisenberg Picture: As above, given , solve for & find all
- Schrödinger Picture:
for no t-dep.
- Dirac Picture:
Eigenspaces & Continuous Bases
- Eigenvalues & Eigenvectors: & system in for
- Compatible Observables: max # in system = 3N
- State Collapse
- Observable/Symmetric Operator:
- Sharp Observable/Self-Adjoint: diagonalizable, eigenvectors Hilbert basis
- Continuous Eigenbasis: such that
H
- Wave Function of a State:
- Identity Map:
- Observable Expression:
- Inner Product:
- Position Eigenbasis:
- Matrix Elements:
- Operators:
- Schrödinger Equation:
- Momentum Eigenbasis:
- Matrix Elements:
- Operators:
- Change of Basis between Position & Momentum:
Probability & Feynman Path Integrals
- Green’s Functions:
- Green’s Method: with
- Binary/Projective Observable:
- Example: If , then
- Probability Amplitude: for finding state at time t1
- Continuous Base Prob.: If , then
- Connection:
- Unobserved Probability: Probability amplitudes obey standard sum & product rules for events.
- Lagrangian:
- Feynman Path Integral:
where
Entanglement
- Mixed State: for Hilbert basis
- Pure State:
- von Neumann Entropy: = “degree of ignorance”
- Decoherence: Maximization of
- Heat Bath:
- Tensor Product of Hilbert Spaces: H (C) = H (A) H (B)
- Vectors: All linear combos of H (A), H (B)
- Rules:
- Basis & Identity:
- Operators:
- Partial Trace:
- Unentangled Distinguishable Systems:
- Entangled Systems:
- Boson: H where
- Fermion: H where
Angular Momentum & Spin
- Translation Operator:
- Invariance: If and then is conserved
- Rotation Operator:
- Cyclic Tensor: for (ijk)=(123)…; -1 for (ijk)=(321)…; 0 otherwise
- Angular Momentum CCRs:
- Commuting Pair:
- Eigenvalues: where for
where for
- Boson:
- Fermion:
- Orbital Angular Mom.:
- Spin Angular Mom.: H (tot) = H (orb) H (spin)
- Electrons, Nucleons, etc: ,
Philosophy of Probabilistic Nature
- Hidden Variables: Can system be described completely by
- EPR Hypothesis: such that (disproven by Bell)
- Nonlocality Generalization
-- Ryan Newson, F2004