Wigner Phase-Space Approach to Quantum Mechanics

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

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

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

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

5. 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.

6. William Rowan Hamilton (1805-1865) Irish IsaacNewton (1643-1727) British

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

8. Free Particle Phase-space trajectory

9. Harmonic Oscillator

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

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

13. 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) Is bilinear in (3)

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

15. 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 quasi-probability 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.

16. Gaussian Wave Packet centered at and

18. Harmonic Oscillator in the Ground State

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

21. 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

22. Philip Morse (1903-1985): American

24. Dynamics Schroedinger equation Equation of motion for Moyal bracket

25. Jose Enrique Moyal (1910-1998): Australian

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

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

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

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

30. Free Particle • Wigner Phase-Space Approach Initial Condition Dynamics

31. Spreading of a Free Wave Packet

32. Harmonic Oscillator • Classical Phase-Space Approach Initial Condition Dynamics

33. Harmonic Oscillator • Wigner Phase-Space Approach Initial Condition Dynamics

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

36. 1d He-H Collision 2 H He 2

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

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

39. Transition Probability ( )

40. 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)

41. 문제(주제: Wigner distribution function) Wave function 는 그 절대값의 제곱이 고려하고 있는 계를 시각 t에 지점 q에서 발견할 확률이라는 물리적 의미를 갖는다. 비슷하게 Wigner distribution function 는 그 계를 시각 t에 phase-space point 에서 발견할 확률로 해석할 수 있다고 생각할 수 있다. 그러나 불행하게도 그렇게 해석할 수가 없다. 그 근본적인 이유는 Heisenberg uncertainty principle에 의해서 한 phase-space point에서 발견할 확률의 개념이 허용되지 않기 때문이다. 따라서 Wigner distribution function이 음의 값을 갖지 못할 이유가 없으며, probability function이 아니고 "quasiprobability" function이라고 부르는 이유도 여기에 있다. 그런데 어떤 계를 와 의 사이 및 와 의 사이에서 발견할 수 있는 확률, 즉 와 에서 발견할 확률의 개념은 인 이상 Heisenberg uncertainty principle에 위배되지 않는다. 따라서 양자물리에서도 허용되는 개념이다. (1) 이러한 확률을 나타내는 Lee(or your name) distribution function 를 정의해 보시오. (Note: 확률함수 는 의 normalization을 만족시킴) (2) 가 nonnegative임을 증명하시오. (3) 의 time evolution을 기술하는 equation of motion을 구하시오. (4) 간단한 계들(예: free wave packet, harmonic oscillator)이 를 사용해서 어떻게 기술되는지를 설명하시오.