1. We will begin with the notion of symmetry, conserved charges and generators in particular, for the space and time translation symmetry, establishing the commutator algebra as well as the Schrodinger equation for states carrying the quantum numbers of the conserved charges.
2. Next we will generalize to rotation symmetry, and construct irreducible representations of the group. For SO(3) we will introduce the spherical harmonics, and solve the Schrodinger equation with rotation invariant potentials on this basis.
3. We will introduce the covering group of SO(3), SU(2), and construct its irreducible representation. We will then consider how to recast tensor product of representations into irreps.
4. Equipped with irreps, we will demonstrate how the harmonic oscillator and the Coloumb potential exhibits hidden symmetry, from the fact that the degeneracy pattern is larger than the irreps.
5. We will then move on to discrete symmetries, including lattice as well as time reversal symmetry, and its implications on the energy spectrum.
6. Using the propagator, we will introduce the path integral formalism. We will demonstrate how Lorentz invariance can be reestablished by introducing the world line metric, and thus recasting quantum mechanics as a one-dimensional quantum gravity theory.
7. The path integral formalism will allow us to naturally introduce the coupling the electromagentic fields, and derive the corresponding Hamiltonian. Some phenomenon, such as AB effects and magnetic monopoles will be discussed.
8. We will introduce the semi-classical approximation to the propagator utilizing the path integral formalism, i.e. the WKB approximation.