Review of quantum mechanics and introduction to symmetry

Advanced Quantum Mechanics (Fall, 2013)

September 23, 2013

The course begins with a brief review of quantum mechanics and the material presented in the core Theoretical Minimum course on the subject. The concepts covered include vector spaces and states of a system, operators and observables, eigenfunctions and eigenvalues, position and momentum operators, time evolution of a quantum system, unitary operators, the Hamiltonian, and the time-dependent and independent Schrodinger equations.

After the review, Professor Susskind introduces the concept of symmetry.  Symmetry transformation operators commute with the Hamiltonian.  Continuous symmetry transformations are composed from the identity operator and a generator function.  These generator functions are Hermitian operators that represent conserved quantities.

The lecture closes with the example of translational symmetry.  The generator function for translational symmetry is the momentum operator divided by ħ.

  • Vector space
  • Observables
  • Hermitian operators
  • Eigenvectors and eigenvalues
  • Position and momentum operators
  • Time evolution
  • Unitarity and unitary operators
  • The Hamiltonian
  • Time-dependent and independent Schrödinger equations
  • Symmetry
  • Conserved quantities
  • Generator functions