Time evolution of a quantum system

Quantum Mechanics (Winter, 2012)

January 30, 2012

Professor Susskind opens the lecture by presenting the four fundamental principles of quantum mechanics that he touched on briefly in the last lecture.  He then discusses the evolution in time of a quantum system, and describes how the classical concept of reversibility relates to the quantum mechanical principle of conservation of information, which is actually the conservation of distinctions or distinguishability of states.  The evolution in time of a quantum system is represented by unitary operators which preserve distinctions and overlap.

Professor Susskind then derives the time-dependent Schrödinger equation, and describes how to calculate the expected value of an observable, and how it changes with time.  This discussion introduces the commutator operator.  Professor Susskind closes the lecture by showing the connection between the quantum mechanical commutator and the Poisson bracket formulation of classical physics, thus showing how the time evolution of the expected value of an observable is closely related to classical equations of motion.

  • Four fundamental principles of quantum mechanics
  • Unitarity and unitary evolution of a system
  • Reversibility, conservation of information, preservations of distinctions, and conservation of overlap of states
  • Derivation of the time-dependent Schrödinger equation
  • Time evolution of expectation value and equivalence to classical equations of motion
  • Parallel between quantum mechanical commutator and classical Poisson bracket