0. Useful formulas (from the appendix of Sakurai)
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1.a. The Heisenberg picture
1.b. The Schrödinger picture, particle in a constant magnetic (B) field and full solutions, degeneracy function, several solutions to time-independent Schrödinger Equation (S.E.) - namely in gaussian (2D and 3D) form, and Landau Levels
2. The Radial Gauge, L.L.L. in complex coordinates, raising and lowering operators (a, a†), and The Magnetic Monopole: The Dirac String & Wu-Yang Monopole
3. The Symmetry Gauge, Theory of Angular Momentum: Recap of J.J. Sakurai 1.6 and cover on material from 3.1 and 3.2, Translations and translational invanriance, Translation Operator J, Time Evolution Operator and Energy Conservation, Rotation operators, Generators of Rotations about each axis (Angular Momentum about each axis), Unitary Rotation Operator in Hilbert space as an nxn matrix & an example in the spin-1/2 case, and a refresher on Pauli Matricees
4. Recap of Symmetries, Transformations of Operators Under Rotations, Groups, Algebras, & Representations (& examples), Lie Groups (& examples), Lie Algebras (& examples), Representation Theory, and Orbital Angular Momentum.
5. Dimensionless Units and Transformations Under Rotations, Eigenvalues & Eigenstates of Angular Momenta, J-operators (ladder operators), Irreducible Representation of the Total Angular Momentum, Matrix Elements of J, Representations of the Rotation Operator
6. Representations of the Rotation Operator, Euler Angles, Touch on Sakurai Section 3.8, and Orbital Momentum & Spherical Harmonics
7. Properties of Spherical Harmonics summary, the Rodrigues Formula, Addition of Angular Momentum J, and Hilbert Space Addition & Formal Theory
8. Continuation of Formal theory of Rotations, Useful properties of rotations between 2 bases, Clebsch-Gordan Coefficients and their properties, examples of a spin 1/2 particle, Tensor Operators, defining scalars, vectors, and tensors in the language of transformations under a rotation operator, Examples (including of a Dyadic), Spherical Tensors of Rank l
9. Continuation of Spherical Tensors of rank k, Infinitesimal version, examples of spherical tensors using vector operators and dyadic, matrix elements of tensors, the m-selection rule, Wigner-Eckart Theorem, and Introduction to Density Operators
10. Continuation of Density Operators, including examples of finding expectation values and examples of systems with specific states (e.g. 2 spin-1/2 particles), Ensembles, Ensemble Averages, diagonalizatoin properties of the density matrix, pure vs random ensembles, mixed ensembles, time evolution of the density operator, and introduction to entropy
11. The density matrix and it's relation to entropy & statistical mechanics, Entanglement and relation to density matrix, Schmidt Decomposition, Purification of states, time evolution of the density matrix and measurements, Quantum Channel, the Kraus Operator, and Weak Measurement
12. Bell's Inequality, Discrete symmetries in quantum mechanics, Parity and it's properties, Discussion of J operator under parity, and a summary of examples of operator names, rotations, parities, and classifications
13. Discrete Symmetries on Wavefunctions, examples with harmonic oscillator, 3D free particle, symmetrical double-well potential, left & right handedness for symmetrical and asymmetrical states, and an example of the Ammonia molecule
14. Parity Selection Rule, Rotation Selection Rule, Other Discrete Symmetries, Time Reversal Symmetry and it's properties, relation of time reversal to parity (similarities, differences, etc.), representation of time reversal as a unitary and complex conjugate operator, Time Reversal Operator, anti-unitarity of the time reversal operator and commuting with H, and examples and properties
15. Checking how commutation relations behave under time-reversal, time reversal on wave functions, Theorem about Energy Eigenstate of a spinless particle on time-reversal invariance, time reversal operator in momentum and position representation, & discussion of time-reversal in different phase conventions, time reversal for a spin-1/2 system, for a spin-less state, & generalizations to j integer and j half-integer state representations, and expectation value of an operator under time reversal
16. Kramer's Degeneracy, examples of how the E field cannot lift Kramer's Degeneracy and how the B field can lift Kramer's Degeneracy (breaking time reversal symmetry), examples of spin-1/2 and spin 17/2 systems, Time-Independent Perturbation theory (nondegenerate Case), Example of a 2-level system & finding eigen energies pertubativelly, and the projector
17. Non-degenerate perturbation theory (continued), energy splitting, applicability and examples to hydrogen atom with single, spinless electron, Quadratic Stark Effect, and Polarizability
18. Time-Independent Perturbation Theory - The Degenerate Case, summary of steps for degenerate perturbation theory (time independent), the secular equation, example of degenerate time independent perturbation theory applied to the Linear Stark Effect, Basics of Hydrogen-Like atoms to see how perturbation theory is applied to fine structures with the example of a hydrogen atom
19. Hydrogen-like Atoms: Fine Structure & the Zeeman Effect, Spin Orbit Interaction & Fine Structure, Lande's Interval Rule, Example with a Sodium (Na) atom, Order of Magnitude of Fine Structure Splitting, The Zeeman Effect, First Order Shifts, The Paschen-Back Limit, Summary of Quantum #'s in Weak and Strong B-Fields, and an example with the level scheme of a proton-electron system in a B field
20. Time-Dependent Potentials, the Interaction Picture (AKA Dirac Picture), Summary Table of Interaction vs Heisenberg vs Schrodinger pictures, General Differential Equation for Time-Dependent Perturbation Theory
21. Example of a 2-level system, Dyson Series, Transition Probabilities, Constant Perturbation, the Density of States within the energy interval, Transition Rate, and Fermi's Golden Rule & it's physical interpretation
22. Harmonic Perturbation, Energy Shift & Decay Width, Decay Amplitude, and Resonant Level
23. WKB Approximation, k(x) and kappa(x), applicability of the AKB approximation, turning points, the general form of the WKB formula, Connection Formulas (overview), introduction to Bound States, and the Bohr-Sommerfeld Condition
24. Continuation of Bound States, Transmission through a barrier, and Reflection above a barrier
25. Identical Particles, the Permutation Operator, Symmetrizer & Anti-symmetrizer, Symmetrization Postulate, Composite systems, Pauli Exclusion Principle, example with a 2 electron system
26. Scattering Theory and Born Approximation (overview), Applicability of the Born Approximation, and Notes on a specific Born Approximation Problem
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