Physics / advanced
Symmetry, spin, and geometric quantum phases
Use symmetry representations and geometric phases to organize quantum spectra, selection rules, discrete transformations, and adiabatic transport.
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Module 01 advanced
Symmetry, spin, and geometric quantum phases
Use symmetry representations and geometric phases to organize quantum spectra, selection rules, discrete transformations, and adiabatic transport.
- 01
Unitary and antiunitary symmetry operators
Identify probability-preserving transformations without confusing antilinearity with a minus sign.
30 min - 02
Translation generators and momentum conservation
Derive the active translation sign and the precise conditions under which momentum is conserved.
30 min - 03
Rotation generators and SU(2) representations
Build angular-momentum multiplets and track the distinction between a spatial rotation and a spinor representative.
30 min - 04
Adding angular momentum with Clebsch–Gordan coefficients
Resolve coupled spin states into product amplitudes and use total spin to compute exchange energies.
30 min - 05
The Wigner–Eckart theorem and selection rules
Separate angular geometry from reduced dynamics and distinguish an allowed transition from a guaranteed transition.
30 min - 06
Parity and electric-dipole transitions
Use inversion symmetry to prove matrix-element zeros and combine parity with angular selection rules.
30 min - 07
Time reversal and Kramers degeneracy
Prove the paired-state result with antilinearity and distinguish it from rotational degeneracy.
30 min - 08
Berry connection and gauge dependence
Compute a geometric connection with an explicit sign convention and separate local gauge freedom from loop holonomy.
30 min - 09
Measuring an adiabatic geometric phase
Separate dynamical and geometric phases while checking the spectral gap and the interferometric reference.
30 min - 10
Aharonov–Bohm phase and holonomy
Calculate a magnetic flux phase with explicit charge and orientation conventions and identify the topology of the accessible region.
30 min