1. Historical review -- Blackbody radiation and Planck's Formula -- The photoelectric effect -- The Bohr theory of the atom -- De Broglie's hypothesis -- 2. Basic developments -- The Schrödinger equation -- The probability interpretation and normalization -- Expansion of the wave function and finding coefficients -- The phase of wavefunction -- Operators in quantum mechanics -- Momentum and the uncertainty principle -- The conservation of probability -- 3. The time-independent Schrödinger Equation -- The free particle -- Bound states and 1-d scattering -- Parity -- Ehrenfest theorem -- 4. An introduction to Hilbert Space -- 5. The mathematical structure of quantum mechanics I -- Linear vector spaces -- Basis vectors -- Expanding a vector in terms of a basis -- Orthonormal sets and the Gram-Schmidt procedure -- Dirac algebra with Bras and Kets -- 6. The mathematical structure of quantum mechanics II -- Linear operators -- Eigenvalues and eigenvectors -- The Hermitian conjugate of an operator -- The commutator -- 7. The mathematical structure of quantum mechanics III -- Change of basis and unitary transformations -- Similarity transformations -- Finding a similarity transformation for a Matrix C -- The generalized uncertainty relation -- Projection operators -- Functions of operators -- Generalization to continuous spaces -- 8. The foundations of quantum mechanics -- Postulates of quantum mechanics -- Spectral decomposition -- Projective measurements -- The completeness relation -- Completely specifying a state with a CSCO -- The Heisenberg versus Schrödinger pictures -- Describing composite systems in quantum mechanics -- The matrix representation of a tensor product -- The tensor product of state vectors -- The density operator -- The density operator for a completely mixed state -- A brief introduction to the Bloch vector -- 9. The harmonic oscillator -- The solution of the harmonic oscillator in the position representation -- The operator method for the harmonic oscillator -- Number states of the harmonic oscillator -- more on the action of the raising and lowering operators -- 10. Angular momentum -- Commutation relations -- Uncertainty relations -- Generalized angular momentum and the ladder operators -- Matrix representations of angular momentum -- Coordinate representation of orbital angular momentum and the spherical harmonics -- 11. Spin-1/2 systems -- The Stern-Gerlach experiment -- Ladder operators -- Unitary transformations -- The outer product representation of the spin operators -- The Pauli Matrices -- Time evolution -- Density operator -- 12. Quantum mechanics in three dimensions -- The 2-D square well -- Overview of a particle in a central potential -- Hydrogen atom -- 13. Identical particles -- Fermions and bosons -- The Pauli exclusion principle -- The Periodic Table -- 14. Time-independent perturbation theory -- The Stark effect -- 15. Applications of perturbation theory to the hydrogen atom -- The fine structure of hydrogen -- The Zeeman effect -- Hyperfine splitting -- 16. Time-dependent perturbation theory -- "Pictures" in quantum theory -- Two-level systems -- Fermi's' Golden Rule -- The emission and absorption of radiation -- 17. Scattering -- Scattering amplitudes -- The Born approximation.