Wigner Phase-Space Approach to Quantum

MechanicsHai-Woong Lee

Department of PhysicsKAIST

Mechanice

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

Modern Physics

• Theory of Relativity

v c

High speed

If , then relativistic mechanics classical mechanics

Time Dilation 002

21

tt t

v

c

v c 1 0t t

Modern Physics

• Quantum Mechanics

0

Microscopic world

If , then quantum mechanics classical mechanics

( )0

V qmq

q

2 2

2[ ( )] ( , )2

i V q q tt m q

0 ?

Phase spaceFrom 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

2

( )2

pH V q

m

dq H p

dt p m

( )dp H V q

dt q q

Initial Condition

0( 0)q t q 0( 0)p t p

Free Particle2

2

pH

m

dq p

dt m

0dp

dt

00( )p

q t q tm

0( )p t p

q

p

Phase-space trajectory

Harmonic Oscillator2

2 21

2 2

pH m q

m

dq p

dt m

2dpm q

dt

q

p

Initial Condition: Free Particle

• Classical Treatment

• Quantum Treatment

0( 0)q t q 0( 0)p t p

202

0

( )[ ]

/4( )2 1/ 4

1( , 0)

[2 ( ) ]

q q

ip qqq t e eq

Uncertainty principle Probability

202

( )[ ]

2 2( )2 1/ 2

1| ( , 0) |

[2 ( ) ]

q q

qq t eq

What about initial momentum?/1

( , 0) ( , 0)2

ipqp t dq q t e

202

( )[ ]

2 2( )2 1/ 2

1| ( , 0) |

[2 ( ) ]

p p

pp t ep

2

q p

2| ( ) |p2| ( ) |q

q p0p0q

qp

Initial Condition: Harmonic Oscillator in its Ground State

• Classical Treatment

• Quantum Treatment

22 200

1 1

2 2 2

pm q

m

21

4 20( , 0) ( ) ( )

m qmq t q e

212 2

0| ( ) | ( )m qm

q e

21

2 20

1| ( ) | ( )

p

mp em

/

0 0

1( ) ( )

2ipqp dq q e

Wigner Distribution Function

2 /1( , , ) ( , ) ( , )ipxW q p t dxe q x t q x t

( , ) ( , , )q t W q p t

( , , )W q p t

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) 2( , , ) | ( , ) |dpW q p t q t 2( , , ) | ( , ) |dqW q p t p t

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 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.

Gaussian Wave Packet

202

0

( )[ ]

/4( )2 1/ 4

1( )

[2 ( ) ]

q q

ip qqq e eq

0q q 0p pcentered at and

2 20 02 2

( ) ( )

2( ) 2( )1( , )

q q p p

q pW q p e e

2q p

2

q

q

2

p

p

0q

0p

Harmonic Oscillator

21

4 20 ( ) ( )

m qmq e

in the Ground State

2 2/ /1( , ) m q p mW q p e e

p

m

/

q

m

Harmonic Oscillator

2 22 2

/ /1 2 2( , ) ( 1)m q p m m q p

W q p e em

in the first excited state

21

4 21

2( ) ( )

m qm mq qe

The Wigner distribution function can take on

negative values!! Quasiprobability function

Morse Oscillator

0( ) 2( ) [ 1]q qV q D e

12 2

0 ( ) ( 1) (2 )(2 1)

di d dyq e e dyd

0( )q qy e

in the Ground StateMorse, Phys. Rev. 34, 57 (1929)

Morse potential

2mDd

0 0

2 1(2 1)( ) ( )

2

2(2 )( , ) (2 )

(2 1)

dd q q q q

i p

dW q p e K de

d

Not much different from the Wigner distribution function of the harmonic oscillator in the ground state

Philip Morse (1903-1985): American

Dynamics( , ) ( , , )q t W q p t

( , , )W q p t

2 3 3 4 5 5

3 3 5 5

( , , ) ( / 2 ) ( / 2 )

3! 5!

W q p t p W V W i V W i V W

t m q q p q p q p

Schroedinger equation Equation of motion for

1 2 1 21 1 2 2 ,

1 2 2 1

2sin[ ( )] ( , ) ( , , )

2 q q q p p pH q p W q p t

q p q p

[ , ]MH FMoyal bracket

Jose Enrique Moyal (1910-1998): Australian

Dynamics0

( , , )W q p t p W V W

t m q q p

2 3 3 4 5 5

3 3 5 5

( , , ) ( / 2 ) ( / 2 )

3! 5!

W q p t p W V W i V W i V W

t m q q p q p q p

limit

3 5

3 50

V V

q q

0V 2 21

2V m q

dq p

dt m dp V

dt q

If , then quantum dynamics = classical dynamics

(Free particle)

(Harmonic oscillator)

Classical vs. Quantum Treatment

• Classical Treatment

0( 0)q t q (1) Initial condition

0( 0)p t p

dynamics ( )0

V qmq

q

( )q t

(2) Initial condition

dynamics

( )p t

0( 0)q t q 0( 0)p t p

( )q t ( )p tdq p

dt m dp V

dt q

Classical vs. Quantum Treatment

• Quantum Treatment ( , 0)q t (1) Initial condition

dynamics

(2) Initial condition

dynamics

2 2

2[ ( )] ( , )2

i V q q tt m q

( , , 0)W q p t

2 3 3 4 5 5

3 3 5 5

( , , ) ( / 2 ) ( / 2 )

3! 5!

W q p t p W V W i V W i V W

t m q q p q p q p

( , , )W q p t

( , )q t

Free Particle

• Classical Phase-Space Approach

0q 0p

dq p

dt m 0

dp

dt

Initial Condition

Dynamics

00( )p

q t q tm

0( )p t p

q

p

Free Particle

• Wigner Phase-Space Approach

2 20 02 2

( ) ( )

2( ) 2( )1( , )

q q p p

q pW q p e e

Initial Condition

Dynamics2

q p

( , , )W q p t p W

t m q

220

02 2

( ) ( )

2( ) 2( )1( , , ) ( , , 0)

ptq q p pm

q pptW q p t W q p t e e

m

Spreading of a Free Wave Packet

Harmonic Oscillator

• Classical Phase-Space Approach

0qInitial Condition

0p

Dynamicsdq p

dt m 2dp

m qdt

( ) sin( )q t A t ( ) cos( )p t Am t

0 sinq A 0 cosp Am

22 200

1 1

2 2 2

pm q

m

Harmonic Oscillator

• Wigner Phase-Space Approach

2 20 02 2

( ) ( )

2( ) 2( )1( , )

q q p p

q pW q p e e

Initial Condition

Dynamics

2q p

2 2/ /1( , ) m q p mW q p e e

2( , , )W q p t p W Wm q

t m q p

dq p

dt m 2dp

m qdt

p

m

/

q

m

Nonlinear Oscillator

• Duffing Oscillator 2 4( )V q q q

Classical phase-space approach

dq p

dt m 32 4

dpq q

dt

Wigner phase-space approach

2 33

3

( , , ) ( / 2 )(2 4 ) (24 )

3!

W q p t p W W i Wq q q

t m q p p

1d He-H Collision2

HeH2

2 2( ) 2 221

( , , , )2 2 2

qQP p

H Q q P p Ae m qM m

(Quasi)classical Method

0( 0)Q t Q 0 1Qe 0 0( 0) 2P t P ME

Initial condition

0( 0)q t q

0( 0)p t p

22 200

1 1

2 2 2

pm q

m

22 21

( 1)2 2f

f

pm m q m

m

( )

Dynamics dQ H P

dt P M

dq H p

dt p m

( )2

qQdP H

Aedt Q

( ) 22

2

qQdp H

Ae m qdt q

: Transition probability from state 0 to state m0 mP

Wigner Phase-Space Method

0( 0)Q t Q 0 1Qe 0 0( 0) 2P t P ME

Initial condition

0( 0)q t q

0( 0)p t p

2 20 0/ /

0 0

1( , ) m q p mW q p e e

( )

Dynamics: Classical

0 mPTransition Probability:

Lee and Scully, J. Chem. Phys. 73, 2238 (1980)

Transition Probability ( )0

19

2E

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

ReferencesE. 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)

문제 ( 주제 : Wigner distribution function)

Wave function       는 그 절대값의 제곱이 고려하고 있는 계를 시각 t 에 지점 q 에서 발견할 확률이라는 물리적 의미를 갖는다 . 비슷하게 Wigner distribution function         는 그 계를 시각 t 에 phase-space point         에서 발견할 확률로 해석할 수 있다고 생각할 수 있다 . 그러나 불행하게도 그렇게 해석할 수가 없다 . 그 근본적인 이유는 Heisenberg uncertainty principle 에 의해서 한 phase-space point 에서 발견할 확률의 개념이 허용되지 않기 때문이다 . 따라서 Wigner distribution function 이 음 의 값 을 갖 지 못 할 이 유 가 없 으 며 , probability function 이 아 니 고 "quasiprobability" function 이라고 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 principle 에 위배되지 않는다 . 따라서 양자물리에서도 허용되는 개념이다 . (1) 이러한 확률을 나타내는 Lee(or your name) distribution function         를 정의해 보시오 . (Note: 확률함수 는 의 normalization 을 만족시킴 ) (2)         가 nonnegative 임을 증명하시오 . (3)         의 time evolution 을 기술하는 equation of motion 을 구하시오 . (4) 간단한 계들 ( 예 : free wave packet, harmonic oscillator) 이 를 사용해서 어떻게 기술되는지를 설명하시오 . 

( , )q t( , , )W q p t

( , )q p

q q p ppq( , )q q q ( , )p p p / 2q p

( , , )L q p t( , , ) 1dq dpL q p t

( , , )L q p t

( , , )L q p t( , , )L q p t

( , , )L q p t