Path Integrals and Coherent States
Marcus A.M. de Aguiar Dresden, August 2008 Outline
1. Basic formulations
2. Semiclassical limit
3. Applications Introduction and Motivations
Why Path Integrals?
Why Coherent States?
The propagator in the coherent state representation Motivations for a path integral formulation of QM
−iHTˆ /ℏ The evolution operator U ( T ) = e cannot be computed explicitly, except for very simple problems.
The infinitesimal propagator, U ε ()1 ε = − iH ˆ / ℏ on the other hand, is essentially the Hamiltonian operator, and can be handled more easily.
There is, therefore, a practical motivation to calculate finite time evolutions as a composition of infinitesimal evolutions. DIRAC: is there a Lagrangian formulation of QM?
The infinitesimal propagator:
ε pˆ 2 −i + V( q ˆ ) ℏ qe''|−iε H /ℏ |' q= ∫ qppe ''|' '|2m |' qdp '
ε pˆ 2 ≈∫ qpp''|' '|1 − i + Vqqdp ()|'ˆ ' ℏ 2m
ε p'2 p ' −i + Vq( ') + iqq ( '' − ') 1 ℏ ℏ ≈ e2m dp ' 2πℏ ∫ ε p'2 p ' −i + Vq( ') + iqq ( '' − ') −iε H /ℏ 1 ℏ ℏ qeq'' | | '= e2m dp ' 2πℏ ∫
2 1/2 i mq( ''− q ') − V( q ') ε m ℏ 2 ε 2 = e 2π εiℏ
1/2 i S( q ", q ',ε ) m ℏ = e 2π εiℏ
A The finite propagator
qe''|−iHT/ℏ |' q qe ε | − iHε / ℏ | qqe | − iH / ℏ | q ⋯ = ∫ N NN−1 − 1 N − 2
−iH ε /ℏ ⋯ qe1| | qdq 0N− 121 dq N − dq
i N S( q , q ,)ε dq dq dq ℏ ∑ n n −1 = N−1 N − 2 ⋯ 1 e n=1 ∫ A A A
ℏ = ∫ Dqt[]( ) e iS / with qq0 ≡' qqN = '' FEYNMAM: sum over paths, generalizing the double slit experiment
ℏ ℏ qe''|−iHT/ |' q= ∫ Dqte[ () ] iS /
Quantum mechanics is written in terms of classical paths, making the connection between the two theories less mysterious. Some Properties of the propagator
KqqT( '', ', )≡ qe '' |−iHT /ℏ | q '
Time evolution of wavefunctions Ψ(qT '', ) =∫ KqqT ( '', ', ) Ψ ( q ',0) dq '
ℏ ∗ −iEn T / Spectral decomposition KqqT( '', ', )= ∑ ϕϕ n ( q '') n ( qe ') n
ℏ −iEn T / Trace formula ∫ KqqTdq(, , ) = ∑ e n Coherent states of the Harmonic Oscillator
pˆ2 mω 2 q ˆ 2 Hˆ =+ =ℏω ( aˆ† a ˆ + 1/ 2) 0 2m 2
ω mω 1 m 1 aˆˆ=+ qi p ˆˆˆ a† =− qi p ˆ 2ℏ ω 2mℏω 2 ℏ 2 m ℏ
azˆ 0≡ zz 0 0
ˆ†∗ ˆ 2 ˆ † ˆ ∗ zaza00− − | z 00 | /2 za zDzz0=(,)0 0 0 ≡ e 0 = ee 0
where z 0 is a complex number. Properties:
ℏ zqzˆ =( zzq +∗ ) ≡ 002mω 000
mℏω zpzˆ =− i( zz −≡∗ ) p 002 000
2ℏ 2 m ℏ ω ()∆=qˆ() ∆= p ˆ 2mω 2 Convenient definition:
1 q0 p 0 z0 = + i 2 b c
ℏ b= cm =ℏω bc = ℏ mω More important properties:
2 −1/4 − 1/2 (q− q0 ) ipq 0 〈q| z0 〉 =π b exp −2 + 2b ℏ
2 −1/4 − 1/2 (p− p0 ) iq 0 p 〈p| z0 〉 =π c exp −2 − 2c ℏ
12 1 2 〈zz| 〉 = exp − z − zzz + ∗ 02 2 0 0
n z 2 〈n| z 〉 = 0 e − z0 /2 0 n!
d2 z dq dp z z≡ z z = 1 ∫ π π ∫ 2 ℏ Phase space view
2 2 22 22 −z − z 0 −−(qq0 )/2 b −− ( pp 0 )/2 c 〈zz| 0 〉 = e = e p b
2 c 〈p| z 0 〉 po
Quantum point in phase space bc =
qo q 2 〈q| z 0 〉 Dynamics under H 0
ℏ −iH0 t / −iω t /2 zt()≡ e z0 ≡ e zt ()
−iω t zt( ) = e z 0
1 qtq( )= cos( ω ω t ) + p sin( t ) 0mω 0
pt( )= − ω mq ω ω 0 sin( tp ) + 0 cos( t ) The Bargmann Representation
∂ψ (q ) q-representation: ψ ψ ψ ψ ψ ψ →=(qq ) ; qˆ → qq ( ); p ˆ →− i ℏ ∂ q
∂ψɶ(p ) p-representation: ψ ψ ψ ψ ψ ψ →=ɶ(pp ) ; pˆ → ppx ɶ ( ); ˆ → i ℏ ∂ p
ˆ† Bargmann-representation: z0 a |z0 )= e 0
∂ψ (z∗ ) ψ ψ ψ ψ ψ ψ →=(zz∗ )( ; aˆ† → zz∗ ( ∗ ); a ˆ → ∂ z∗
analytic function of z * (z does not appear) The Coherent State Propagator
1 1 −||z2 − || z 2 −iHT /ℏ 2f 2 0 − iHT / ℏ KzzT(,,)f0≡ 〈 ze f || z 0 〉 = e( zef || z 0 )
z0
po T
pf z f
qo qf Path Integrals in the Coherent State Representation
N 〈−iH ˆ ε /ℏ 〉 KzzT(,,)f0= z N |∏ e | z 0 j=1
ε ε ε −iHε ˆ/ℏ…… −− iH ˆˆ // ℏℏ iH − iH ˆ / ℏ = 〈zeN | ee e | z 0 〉
N
There are many ways to calculate the infinitesimal propagators First Form: Klauder and Skagerstan (1985)
N 〈−iH ˆ ε /ℏ 〉 KzzT(,,)f0= z N |∏ e | z 0 j=1
ε ε ε iε ˆ ⋯ ii ˆˆ ⋯ i ˆ = 〈zN |1− H 1 −− H 1 H 1 − Hz | 0 〉 ℏ ℏℏ ℏ
2 d2 z d z k+1 z z k z z ∫ k k ∫ k+1 k + 1 π π iε z Hˆ z z1− iHε ˆ /ℏ z = zz 1 − k+1 k k+1 kkk + 1 ℏ zk+1 z k
iε k ≡zk+1 z k1 − H Q ℏ
k ℏ −iε H Q / = zk+1 z k e
k ||z2 | z | 2 iε H z∗ z −k − k +1 − Q = e k+1 k 2 2 ℏ 2 2 k ∗ ||zk | z k | iε H Q z z − −+1 − k+1 k 2 2 ℏ
k 1∗ ∗ 1 ∗ iε H Q =(z −− zz ) ( z − zz ) − 2k+1 kk 2 k + 1 kk + 1 ℏ
i ℏzz∗− ∗ ℏ zz − =kk+1z − kk + 1 z∗ − H k ε ℏ 2i ε ε k 2 i k+1 Q N −1 d2 z N −1 1 1 iε KzzT(,,)=j exp ( zzzzzz∗∗∗ −− ) ( −− ) Hzz (,) ∗ Q f 0 ∫∏ ∑ j+1 jj jj ++ 11 jℏ Qjj + 1 j=1 π j=0 2 2
z Hˆ z The Q-representation of ˆ k−1 k ∗ ˆ H →HQ ( zz , ) = zHz (normal ordering) zk−1 z k
Since azˆ= zz za ˆ † = zz∗
we calculate H Q by simply writing H in normal order :
ˆ †n m ∗n m H=∑ Haanmˆ ˆ → H Q = ∑ Hzz nm n, m n, m pˆ2 q ˆ 2 Example: ( ħ=m=1) Hˆ = + + q ˆ 4 2 2
1qˆ† 1 q ˆ aˆ=+ ipb ˆˆ a =− ipb ˆ 2b 2 b
1q∗ 1 q z=+ ipb z =− ipb 2b 2 b
p2 x 2 1 3 H=+++ xbx43 22 + ( bb 22 +− ) + b 4 Q 22 4 4 Second form: Klauder and Skagerstan (1985)
N −1 〈−iH ˆ ε /ℏ 〉 KzzT(,,)f0= z N |∏ e | z 0 j=1 ε ε ε iε ˆ …… ii ˆˆ i ˆ = 〈zN |1− H 1 −− H 1 H 1 − Hz | 0 〉 ℏ ℏℏ ℏ
d2 z Hˆ = Hzzz(∗ , ) k z ∫ Pkkkπ k d2 z 1 ε−iHε ˆ /ℏ = 1 − iHk / ℏ zk z ∫ P kπ k
2 k ℏ d z = e−iε H P / zk z ∫ kπ k
(1 ε−iHε ˆ /1ℏ)( − iH ˆ / ℏ )
2 kℏ k +1 ℏ d z = e− εiHε P/ − iH P / zzzk z ∫ kkk+1 + 1 π k N −1 d2 z N −1 1 1 iε KzzT(,,)=j exp ( zzzzzz∗∗∗ −− ) ( −− ) Hzz (,) ∗ P f 0 ∫∏ ∑ j+1 jj jj + 1 + 1 jℏ Pjj j=1 π j=0 2 2
The P-representation of H (anti-normal ordering)
Example:
pˆ2 q ˆ 2 p2 x 2 1 3 Hˆ = + + q ˆ 4 H=++− xbx43 22 − ( bb 22 +− ) + b 4 2 2 P 22 4 4 If we take the limit N → ∞ we get
ℏ i i ∗ɺ ɺ ∗ ∫ (zzzz− ) − HQ dt ∫ Dzt[ ( )] e ℏ 2
Kz(f , z0 ,) T = ℏ i i ∗ɺ ɺ ∗ ∫ (zzzz− ) − HP dt ∫ Dzt[ ( )] e ℏ 2 It is remarkable that two apparently different path integrals, involving actions computed with different Hamiltonians, represent the same quantum object.
In fact the difference lies not only in the Hamiltonian, but also in the way it enters in the action:
* HP( z k ,z k) in one case
* HQ(z k+1 ,z k) in the other
We will show for a simple case that the two forms are indeed identical. This emphasizes that we have to be extra careful to take the continuum limit. FURTHER QUESTIONS:
1 – If coherent states are the most classical states, why these path integrals are not governed by the classical Hamiltonian?
2 – What is the transformation that takes the Hamiltonian operator into the classical Hamiltonian function?
3 – Is it possible to construct a path integral formulation where the classical H appear? The Weyl Symbol of the Hamiltonian operator is defined as
ℏ x x Hqp( , ) = eipx / q − Hqˆ + dx W ∫ 2 2 and has a number of important properties. In particular:
ˆ (a) HKpVq=+( ˆ) ( ˆ ) → HqpW ( , ) =+ KpVq( ) ( )
Hˆ=Ψ H ˆ Ψ= HqpWqpdqdp(,) (,) (b) ∫ w Ψ
ℏ where Wqp(,)∗ ( qx /2)( qx /2) edxipx / Ψ =Ψ∫ + Ψ− is the Wigner function. Third form:
N 〈−iH ˆ ε /ℏ 〉 KzzT(,,)f0= z N |∏ e | z 0 j=1 ε ε ε iε ˆ ⋯ ii ˆˆ ⋯ i ˆ = 〈zN |1− H 1 −− H 1 H 1 − Hz | 0 〉 ℏ ℏℏ ℏ
write ∗ ∗ 2 2 zwk− z w k d z d w Hˆ= H( w∗ , wDzze )(,) ˆ ∗ k ∫ W k k π
Weyl Hamiltonian
† ∗ Dzzˆ ( ,∗ ) = e zaˆ− za ˆ displacement operator 2 2 2 ∗∗ ∗∗ d z d w zHzˆ = Hwwe(∗ , ) −+−|z | /2 zzzzkk−1 zze zwzw kk − k kk−1 ∫ Wkk kk − 1 π
ˆ ℏ ∗ℏ 2 ∗∗ ∗∗ zez−iH ε / e−iHwwWkk( , )/ε e −+− ||/2 z zzzz kk−1 zze zwzw kk − k k −1 = ∫ k k − 1 d2 z d2 w × k π ˆ ℏ ∗ℏ 2 ∗ ∗ 2 2 ∗ zeze−iH ε / −iHwwWkk(,)/ε e −+− ||/2 z zzzz kk−1 e −−+ ||/2| z kk z − 1 |/2 zz kk − 1 k k −1 = ∫ 2 2 ∗ ∗ d z d w ezwk− z w k × k π
Integrals over z and z * are quadratic and can be performed exactly:
2 ˆ ℏ ∗ℏ 2 ∗∗ 2 2 ∗ d w ze−iH ε / z= 2 e −iHwwWkk( , )/ε −+−−−+ 2||2 w k zwzwz kkkkk 2 ||/2| z k−1 |/2 zz kk − 1 k k k −1 ∫ π N 2 d w j 12 1 2 KzzT( , , ) exp z z 2 CzCzzz∗ 2 ∗ ∗ W f 0 =∫∏ −−++−+0 fψ N Nf N0 f 0 j=1 π /2 2 2
N N k iε 2 ∗j + 1 ψ N=−−+∑Hw Wk2 4 ∑∑ ww kkj+1 + 1 − ( − 1) k=1 h k=1 j = 1
N k +1 CN=∑( − 1) w N+1 − k The Weyl -representation of H k =1 (symmetric ordering) Coelho-Aguiar J. Phys. A (2006) Example:
2 2 2 2 p x pˆ q ˆ 4 4 Hˆ = + + q ˆ HW = + + x 2 2 2 2 In the limit of the continuum we may formally write
12 1 2 −z − z + zz∗ KzzTe(,,)20 2 f f 0 Dww [,]exp∗ 2 Cz ∗ 2 Cz ∗ W f 0 =∫ ψ +f − 0
The special case z f = z 0 = z becomes simply
KzzT Dww∗ Cz ∗ Cz ∗ W (,,)=∫ [, ]expψ + 2 − 2 Application: The harmonic oscillator in the three representations
ℏ 2 2 2 ∂ mω 2 † Setting Hˆ =− − xaa =ℏω () + 1/ 2 2m∂ x 2 2
we obtain
∗∗ℏ ∗∗ ℏ ∗∗ ℏ ωHzzQ (,) ω =+ω ( zz 1/2) HzzW (,) = zz Hzz P (,) =− ( zz 1/2)
All integrals in K Q , K P and K W are quadratic and can be performed exactly. We find: ∗ N ∗ 2 2 KzzTQ(,,)exp f 0 =−{ ωε iTω /2(1 +− i ) zzz0f − | 0 |/2| − z f |/2 }
N ∗ −N 1− iωε ∗ 2 2 KzzTW(,,)(1 f 0 =+ iωε /2)exp zzz0f −− ||/2| 0 z f |/2 1+ iωε
∗ ∗ − N z0 z f 2 2 KzzTP(,,)(1 f 0 =+ iωωε )exp iT /2 +N −− ||/2| z0 z f |/2 (1+ iωε )
whereas the exact result is
∗ −iω T ∗ 2 2 KzzT(,,)expf 0 =−{ iTω /2 + ezzz0f − ||/2| 0 − z f |/2 } Except for the normalization and the overall multiplication phase, the * non-trivial piece of the propagator is the term f=exp(-iωT) multiplying z i zf . Taking ωT=2 π and N=100 we find f=1 for the exact propagator and
fQ : 1.22+ 0.01 i
fW : 0.999998+ 0.002 i
fP : 0.82+ 0.007 i
which seems to indicate a better convergence (as N → ∞) of the Weyl form. Relation with the Weyl Propagator
The Weyl symbol of the evolution operator
UqpT(,,)=∫ qs − /2 e−iHTˆ /ℏ qs + /2 e ips / ℏ ds
which is also a phase space representation of the evolution operator, can be related to the coherent state propagator as
2 2 ˆ ℏ ℏ d z d z UqpT(,,)= qszze − /2−iHT/ zzqs + /2 e ips / 0 f ds ∫ 0 0 f f π Inserting the Weyl propagator, the integrals over z 0 and z f become quadratic and can be performed exactly. These integrals result in a delta function involving the variable s, which is also performed. The final result is
∗ ∗ ∗ 2 UqpT(,,)=∫ Dww [,]expψ +−+ 2 Cz 2 Cz 2|| C
where 1 q pb z= + i 2 b ℏ
N * ∗ 2i and 22CzN− Cz N =∑ ( QpPq kk − ) is independent of b. k=1 ℏ
1 Qk P k b Above we defined the integration variables wk = + i 2 b ℏ Therefore, the path integral for Weyl symbol of the evolution operator is the ‘unsmoothed’ version of the path integral for diagonal coherent state propagator.
Conversely, the path integral for diagonal coherent state propagator is identical to that for the Weyl symbol of the evolution operator, but where each path is smoothed by the Gaussian factor |C| 2. Some bibliography:
The Lagrangian in Quantum Mechanics Dirac, P. A. M. 1933 Phys. Z. Sowjet. 3, 1. Reprinted in Schwinger, J. (ed) 1958 Quantum electrodynamics. New York: Dover. Schwinger (1958), pp. 312-320.
Space-Time Approach to Non-Relativistic Quantum Mechanics Feynman, R. P. I948 Rev. mod. Phys. 20, 367-387.
Techniques and Applications of Path Integration L.S. Schulman (1981) Dover
Handbook of Feynman Path Integrals (Springer Tracts in Modern Physics) (Berlin: Springer) C. Grosche and F. Steiner 1998
Coherent States, Applications in Physics and Mathematical Physics J.R. Klauder and B.S. Skagerstam 1985 (Singapore: World Scientific)
On a Hilbert Space of Analytic Functions and an Associated Integral Transform V. Bargmann, Comm. on Pure and Appl. Math. 14, 187 (1961).
Semiclassical approximations in phase-space with coherent states M. Baranger, M.A.M. de Aguiar, F. Keck H.J. Korsch and B. Schellaas J. Phys. A: Math. Gen. 34 7227-7286 (2001).
Coherent State Path Integrals in the Weyl Representation L.C. dos Santos and M.A.M. de Aguiar J. Phys. A39 13465 (2006).
The Weyl representation in classical and quantum mechanics A. M. Ozorio de Almeida Phys. Rep. 295 (1998) 265.