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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