Wigner PhaseSpace Approach to Quantum Mechanics HaiWoong Lee

Wigner Phase-Space Approach to Quantum Mechanics Hai-Woong Lee Department of Physics KAIST

Mechanice • Classical (Newtonian) Mechanics • Relativistic Mechanics • Quantum Mechanics

Modern Physics • Theory of Relativity High speed If , then relativistic mechanics classical mechanics Time Dilation

Modern Physics • Quantum Mechanics Microscopic world If , then quantum mechanics classical mechanics ?

Phase space From Wikipedia, the free encyclopedia In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.

Isaac Newton (1643 -1727) British William Rowan Hamilton (1805 -1865) Irish

Phase Space=(q, p) space • Hamilton’s equations Initial Condition

Free Particle Phase-space trajectory

Harmonic Oscillator

Initial Condition: Free Particle • Classical Treatment • Quantum Treatment Uncertainty principle Probability What about initial momentum?

Initial Condition: Harmonic Oscillator in its Ground State • Classical Treatment • Quantum Treatment

Wigner Distribution Function Wigner, Phys. Rev. 40, 749 (1932) Wigner in Perspectives in Quantum Theory (MIT, 1971) Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949) Phase-space distribution function Comments (1) (2) (3) Is bilinear in

Eugene Wigner (1902 -1995): Hungarian Nobel prize in 1963

Wigner quasi-probability distribution From Wikipedia, the free encyclopedia The Wigner quasi-probability distribution (also called the Wigner function or the Wigner-Ville distribution) is a special type of quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to supplant the wavefunction that appears in Schrödinger's equation with a probability distribution in phase space.

Gaussian Wave Packet centered at and

Harmonic Oscillator in the Ground State

Harmonic Oscillator in the first excited state The Wigner distribution function can take on negative values!! Quasiprobability function

Morse Oscillator in the Ground State Morse, Phys. Rev. 34, 57 (1929) Morse potential Not much different from the Wigner distribution function of the harmonic oscillator in the ground state

Philip Morse (1903 -1985): American

Dynamics Schroedinger equation Equation of motion for Moyal bracket

Jose Enrique Moyal (1910 -1998): Australian

Dynamics limit If , then quantum dynamics = classical dynamics (Free particle) (Harmonic oscillator)

Classical vs. Quantum Treatment • Classical Treatment (1) Initial condition dynamics (2) Initial condition dynamics

Classical vs. Quantum Treatment • Quantum Treatment (1) Initial condition dynamics (2) Initial condition dynamics

Free Particle • Classical Phase-Space Approach Initial Condition p Dynamics q

Free Particle • Wigner Phase-Space Approach Initial Condition Dynamics

Spreading of a Free Wave Packet

Harmonic Oscillator • Classical Phase-Space Approach Initial Condition Dynamics

Harmonic Oscillator • Wigner Phase-Space Approach Initial Condition Dynamics

Nonlinear Oscillator • Duffing Oscillator Classical phase-space approach Wigner phase-space approach

1 d He-H 2 Collision H 2 He

(Quasi)classical Method Initial condition ( ) Dynamics : Transition probability from state 0 to state m

Wigner Phase-Space Method Lee and Scully, J. Chem. Phys. 73, 2238 (1980) Initial condition ( Dynamics: Classical Transition Probability: )

Transition Probability ( QM Wigner QC 0 0 0. 060 0. 046 0 0 1 0. 218 0. 202 0. 375 0 2 0. 366 0. 351 0. 200 0 3 0. 267 0. 294 0. 250 0 4 0. 089 0. 106 0. 175 )

References E. Wigner, Phys. Rev. 40, 749 (1932) E. P. Wigner in Perspectives in Quantum Theory, edited by W. Yourgrau and A. van der Merwe (MIT, Cambridge, (1971) J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949) M. Hillery, R. F. O’Connel, M. O. Scully and E. P. Wigner, Phys. Rep. 106, 121 (1984) H. W. Lee, Phys. Rep. 259, 147 (1995)