Uncertainty, unitary evolution, and the Schrödinger equation

Quantum Mechanics (Winter, 2012)

February 6, 2012

Professor Susskind begins the lecture by introducing the Heisenberg uncertainty principle and explains how it relates to commutators. He proves that two simultaneously measurable operators must commute. If they don't then the observables corresponding to the two operators cannot be measured simultaneously.

He then reviews the time evolution of a system and the Schrödinger equation. Unitary operators represent the time evolution of a system, and the quantum mechanical Hamiltonian generates the time evolution. Professor Susskind reviews the derivation of the time-dependent Schrodinger equation, the computation of expectation values of observables, and the parallels between the quantum mechanical commutator and the classical Poisson bracket.

Professor Susskind then demonstrates how to solve the Schrödinger equation for a general quantum mechanical system. This solution is the origin of the connection between the energy of a system and oscillations of the wave function. This is the Heisenberg matrix formulation of quantum mechanics.

The lecture concludes by solving a practical example of a single spin in a constant magnetic field.

  • Pure states
  • Heisenberg uncertainty principle
  • Commutator
  • Time evolution of a system
  • Quantum mechanical Hamiltonian
  • Time-dependent Schrödinger equation
  • Solving the Schrödinger equation
  • Expectation values of observables
  • Heisenberg's matrix formulation of Quantum Mechanics
  • Spin in a magnetic field