A Classical Perspective on Non-Diffractive Disorder

A Classical Perspective on Non-Diffractive Disorder The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable Link Terms of Use Klales, Anna. 2016. A Classical Perspective on Non-Diffractive Disorder. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. July 29, 2017 7:28:27 AM EDT http://nrs.harvard.edu/urn-3:hul.instrepos:26718765 This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#laa (Article begins on next page)

r 5 r 5

R = 10MΩ

1 itp2d

6B;m`2 RXR, M `` v Q7 2B;2Mbi i2b rbi? +QMb2+miBp2 2M2`;B2b BM i?2 MQBbBv [mbmib+ TQi2MiB HX h?2 T`272``2/ Q`Bi2Mi ibqmb Q7 i?2 bi i2b + M #2 b22m BM i?bb `` vx k

1.2 1.0 0.8 h (t)i 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 t M δp t = M δp 0. δx t δx 0 ψ G (x, t) =exp{(i/ )[(x x t ) A t (x x t )+p t (x x t )+γ t ]}

fully resolved third recurrence second recurrence first recurrence 0 2 4 6 8 10 12 f A t = 1 2 δp t (δx t ) 1 γ t = φ t + 1 2 i [ln δx t (δx 0 ) 1 ] φ t = p t dx t Et φ t t =0 ψ G (0) ψ G (t)

exp{ nτλ/2} λ M n n ϵ T (ω) = 1 2 π T T exp{iωt} ψ G ψ G (t) dt. λ

λ/ω. ω/λ 1 t

r r 5 r 5

φ =0 r 0 r 5 ψ r,m (ρ, φ) =R r,m (ρ)e imφ ψ r,m ψ r, m ψ r,m = R(r)e imφ ψ r, m = R(r)e imφ. V r,±m = ψ r,m V ψ r,m ψ r,m V ψ r, m. ψ r, m V ψ r,m ψ r, m V ψ r, m ψ + ψ 2 2 ψ + V ψ =0 ψ + cos φ ψ sin φ

ψ + = 1 2 (ψ r,m + ψ r, m ) ψ = 1 2 (ψ r,m ψ r, m ). ψ + cos φ ψ sin φ φ =0

6000 5800 5600 E 5400 5200 5000 2:5 resonant set 3:7 resonant set 0 25 50 75 100 125 150 175 r r m E ω. r 5

H = H 0 + λv H 0 = 2 2µ 2 + r 5 a ψ m V ψ m+a = = e imφ Ve i(m+a)φ dφ e iaφ Vdφ. V a

0 200 400 600 800 1000 1200 0 0 200 400 600 800 1000 1200 200 400 j 600 800 1000 1200 i i ψ i V ψ j

W = ψi 0 V ψ 0 j. ψ 0 i V φ i. W φ 1 = ϵ 1 φ 1 W φ 2 = ϵ 2 φ 2 W φ n = ϵ n φ n. ϵ 1 > ϵ 2 >... > ϵ n ψ = i a i φ i i a i 2 =1.

ψ W ψ = i = i = i = i φ j a i a jw φ i j a i a j φ j W φ i j a i a j φ j ϵ i φ i j a i a jϵ i φ j φ i j i = j ψ W ψ = i a i 2 ϵ i. φ i ψ W ψ ψ φ ϵ φ i V,

itp2d 1:π,

state 3673 state 3678 state 11059 state 11062 state 11070 state 11076 state 20661 state 20671 state 20676 state 30662 state 30670 state 30688 6B;m`2 RXN, ai i2b + H+mH i2/ pb /2;2M2` i2 T2`im`# ibqm i?2q`v QM bk HH bm#b2i Q7 bi i2b rbi?bm `2bQM Mi b2ix k9

state 3673 state 3678 state 11059 state 11062 state 11070 state 11076 state 20661 state 20671 state 20676 state 30662 state 30670 state 30688 6B;m`2 RXRy, ai i2b + H+mH i2/ U HKQbiV 2t +ihv pb /B ;QM HBx ibqm QM p2`v H `;2 bm#b2i Q7 # bbb bi i2bx k8

y 12 10 8 6 4 2 0 0 2 4 6 8 10 12 x 0 2 4 6 8 10 12 x 0 2 4 6 8 10 12 x 1.0 0.5 0.0 0.5 1.0 V

itp2d itp2d

p 0.08 0.00 0.08 / 0 20 15 10 5 0 1.0 0.5 0.0 0.5 1.0 1.5 2.0 x

y p x x x

p n+1 = p n dv n(x) dx x n+1 = x n + p n+1 p x V n (x) x

V n (x) =0, V n (x)v n (x )=v 2 0e x x 2 /ξ 2. t c v 2/3 0

transverse position 200 150 100 50 0 0 500 1000 1500 2000 2500 longitudinal position 8 6 4 2 log( ) transverse position 400 300 200 100 0 0 50 100 150 200 250 longitudinal position 8 6 4 2 log( )

p x p x p x a p x p a x 2 V a x 2 a ( p x 1+ p ) 1 p a x a x. a 1 ( 1, 0]

1 0 δs = δp 2 + δx 2 = δp ( p 1+( x p) 2 = 2 δx 1+ x) δp δx p x p δs > x 0 q 1+2 p x +2 p S drift = q x 1+ p x 2 q 2 q 2 < p x < 0

S kick = 1+ p x + 2 V n 2 q dx 2 p 2 q x 1+ p x 2 q 2 V n q dx 2 q. αp = p βx = x p x = α β p 0 < x < 2 α β α = β =1

r(t) = log M(x 0,x t ) d(x 0 ) d(x 0 ) x 0 M(x 0,x t ) x 0 x t

p 0.0004 0.0002 0.0000 0.0002 0.0004 0.0 0.5 1.0 1.5 x 4.5 3.0 1.5 0.0 1.5 3.0 4.5 6.0 log( ) v 0 = 10 6 ξ =0.1 x =0 x =1 λ c λ c v 0 λ c v 2/3 o p(x) =0

40 30 20 v 0 =2 20 v 0 =2 18 v 0 =2 16 v 0 =2 14 v 0 =2 12 r 10 0 10 0.0 0.2 0.4 0.6 0.8 1.0 t/t c v 0 t c v 0 =2 10 v 0 =2 12 v 0 =2 14 v 0 =2 16 v 0 =2 18 v 0 =2 20 0 2 4 6 8 10 c λ c v 0

p n+1 = p n dv n(x) dx x n+1 = x n + p n+1 p x

V n (x)v n (x )=v 2 0e x x 2 /ξ 2 x v 0 ξ 2 c 0 v 0 ξ 2 m = mv m 2 V 1+m V m p x V 2 V x 2 m c V c 0 V m c = c 0 c 2 0 +4c 0 2 c 0 m c = c 1/2 0. V m

V,m<<1 m =(mv m 2 V )(1 m + V ) m<<m c m 2 << V m = V mm c m 2 V m = m 2 V m>>m c m 2 >> V m = m 2 m c m n (n n 0 ) 1 n 0

L n+1 L n = 2 = 2 (( V 1 2 dx dp) 2 + 1 4 dx2 dx 2 + dp 2 ( = 2 ( V dx ) 2 1 2 dp dx + 1 4 1+ dp 2 dx V 1 2 dp dx 1+ dp 2 dx ) 2 + 1 4 1 2 1 2 ) 1 2 r = r n+1 r n =ln(l n+1 /dx 0 ) ln(l n /dx 0 ) = ln(l n+1 /L n ) r = ln ( 2 = 1 2 ln V 1 2 dp dx ( 2 1+ dp 2 dx V 1 2 dp dx 1+ dp 2 dx ) 2 + 1 2 1 2 ) 2 + 1 2

dp dx = m ( r = 1 ( 2 V 2 ln 1 2 m) ) 2 + 1 2 1+m 2 ln(x) x 1 ( r 1 ( 2 V 1 2 m) ) 2 + 1 2 2 1+m 2 1 ( 1 ( 2 V 1 2 m) 2 + 1 2 1 ) m2 2 1+m 2 ( ( 2 V 2 1 2 V mv 1 2 V + 1 4 + 1 2 m mv + 1 2 m + m2 + 1 4 1 2 1 2 m2) ) 1 2 1+m 2 ( V 2 1 2 V mv 1 2 V + 1 4 + 1 2 m mv + 1 2 m + m2 + 1 4 1 2 ) 1 2 m2 1+m 2 ( ) V 2 V + m 2mV + 1 2 m2 1+m 2 (1 + m 2 ) 1 1 m 2 r V 2 V + m 2mV + 1 2 m2 r m V

m m 1 t t c t c r ln(t t c ) r = m +ṁ m<<m crit α = (x dx 1,p dp) β = (x, p) γ = ((x + dx 2,p+ dp)) α = (x + p dx 1 dp V 1,p dp V 1) β = (x + p V,p V ) γ = (x + p + dp + dx 2,p+ dp V 2)

p (x + dx 2,p+ dp) (x dx 1,p dp) (x, p) x p (x + p dx 1 dp V1 0,p dp V 0 1 ) x (x + p + dp + dx 2,p+ dp V 0 2 ) (x + p V 0,p V 0 )

m 2 = dp dx 2 m = dp dx 1 c = dp dx 2 dp dx 1 dx 2 = m 2 m dx 2. c = m 2 m dx 2 c = ( ) ( ) dp V 2 +V dp+v dp+dx 2 V 2 +V 1 V dx 1 +dp+v 1 V dp + dx 2 V 2 + V = 1 dx 2 m 2 V 2 m 2 +1 V 2 m V 1+m V m 2 +1 V 2 c c = 1 dx 2 = 1 dx 2 ( ( ) ( ) m2 V 2 m 2 m V +1 V 2 1+m V m 2 +1 V 2 (m 2 m) m 2 m V 2 + V (m 2 +1 V 2 )2 (1 + m V ) m 2 + m ) c c 1 ( (m2 m V 2 + V )( 2m 2 +1+2V 2 )(1 m + V ) dx 2 m 2 + m )

dx 2 (c c) V V 2 +4m 2 V 2 2m 2 2 mv 2 + mm 2 2V 2 2 + V 2 V V m 2 2mV + m 2 + V 2 t t =6

6.5 3200 6.0 5.5 2400 2d periodic potential Intensity 5.0 y y 4.5 1600 4.0 3.5 800 3.0 2.5 0 0 50 100 150 200 250 time time 300 350 6B;m`2 kxrk, S2`BQ/B+ #` M+?2/ ~Qr Q7 K MB7QH/ Q7 +H bbb+ H i` D2+iQ`B2b 8e

6B;m`2 kxrj, S2`BQ/B+ #` M+?2/ ~Qr Q7 [m MimK K2+? MB+ H TH M2 r p2 7Q` p `B2iv Q7 `2T2 i H2M;i?bX 8d

6B;m`2 kxr9, +QKT `BbQM Q7 i?2 /2MbBiv Q7 +H bbb+ H i` D2+iQ`B2b UH27i T M2HV rbi? r p2t +F2i H mm+?2/ BM i?2 b K2 TQi2MiB H M/ KmHiBTHB2/ #v e ie0 t/! iq ;2i i?2 `2bmHiBM; [m MimK bi i2 rbi? 2M2`;v E0 U`B;?i T M2HVX h?2 v2hhqr ``Qr BM/B+ i2b #` M+? 2pB/2Mi BM i?2 +H bbb+ H /2MbBiv i? i Bb #b2mi BM i?2 +Q``2bTQM/BM; [m MimK bi i2x "Qi? T M2Hb b?qrm 7mHH irq@/bk2mbbqm H 2pQHmiBQM 83

e ie0t/ E 0 E 0. 0.

V I R R = 10MΩ V

R T = V I.

n m = πn/v F

Ω n 0 n (r) n n(r) =n 0 + n (r)+n (r). L = 1 2 m(r)ṙ2 L d L =0 q i dt q i [ ] 1 2 m(r)ṙq2 t + r (m(r)ṙq) = 0 1 2 m(r)ṙ2 = m(r) r + ṙ (ṙ m(r))

m(r) = π v F n(r) r = [ ṙ2 1 ) (ˆṙ m(r) 2 m(r) m(r) ˆṙ]. r B = eṙb z /m(r) r = [ ṙ2 1 ) (ˆṙ m(r) 2 m(r) m(r) ˆṙ] + eṙb z /m(r).

] [ 2 + ω2 c 2 n2 (r) ψ =0. ψ = e iωs(r)/c S(r) S S = n 2 (r). n(r) dr ds = S [ d n(r) dr ] ds ds = n(r). d/ds = ˆk dr/ds = ˆk ] [ˆk n(r) ˆk + n(r) [ˆk ˆk] = n(r). ˆk = n(r)v/c F 2n(r) 2 [v n(r)] v + n(r)[v v] =c 2 F n(r)

2 1 y 0 1 2 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x d dt = t + v v dv dt =(v )v d 2 r dt 2 = c2 F ( ) n(r) n(r) 3 2 ( v n(r) n(r) ) v. m(r) n(r).

1 1 y x n 3

2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) 0.0-1.0-2.0 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm)

2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) 0.0-1.0-2.0 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm)

n 3 n 0 R = n 3 n 0 N N n tot = N i e l i/l l i i σ(ρ) = qa 2π (ρ 2 + a 2 ) 3/2 q a ρ 2ρ =2a 2 2/3 1 σ = q 2πa 2

10 11 2

y(µm) 4 3 2 1 0 0 1 2 3 x(µm) 16 12 8 4 0 4 8 12 16 B =0.12T n = 10 12 2

y(µm) y(µm) 1.0 0.0-1.0 1.0 0.0-1.0 1.0 B = 0.00 T B = 0.06 T B = 0.07 T B = 0.08 T B = 0.09 T B = 0.10 T B = 0.11 T B = 0.12 T B = 0.13 T 0.08 0.06 0.04 0.02 0.00 T/T 0.02 0.04 y(µm) 0.0-1.0 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.06 0.08 10 1 2 2

1.0 B = 0.00 T B = 0.06 T B = 0.07 T y(µm) 0.0-1.0 0.016 0.012 y(µm) 1.0 0.0-1.0 1.0 B = 0.08 T B = 0.09 T B = 0.10 T B = 0.11 T B = 0.12 T B = 0.13 T 0.008 0.004 0.000 T/T 0.004 0.008 y(µm) 0.0-1.0 0.012 0.016 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm)

2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) 0.0-1.0-2.0 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) x =1µ y =0

B =0.10T,

2.0 B = 0.05 T B = 0.06 T B = 0.07 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.08 T B = 0.09 T B = 0.10 T 1.0 y(µm) 0.0-1.0-2.0 2.0 B = 0.11 T B = 0.12 T B = 0.13 T 1.0 y(µm) 0.0-1.0-2.0 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm) 0.0 1.0 2.0 x(µm)

Ψ β (q, t) = dq G (q, q,t)ψ β (q, 0), G (q, q,t)= 1 2πi 2 S(q, q ) q q 1/2 [ ] exp is (q, q,t). S(q, q ) q 0 q t

( ) ( ) S S S (q, q ) = S(q t,q 0 )+ (q q t )+ (q q 0 ) q t q 0 q 0 q t + 1 ( 2 ) S 2 qt 2 (q q t ) 2 + 1 ( 2 ) S q 0 2 q0 2 (q q 0 ) 2 q t ( 2 ) S + (q q 0 )(q q t ) q 0 q t + 1 ( 3 ) S 6 q0 3 (q q 0 ) 3 + 1 ( 3 ) S q t 6 qt 3 (q q t ) 3 q 0 + 1 ( 3 ) S 2 q0 2 q (q q t )(q q 0 ) 2 + 1 ( 3 ) S t 2 q 0 qt 2 (q q t ) 2 (q q 0 ). Ψ β (q, t) = 1 N 2πi ( 3 ) S + q 0 qt 2 dq 2 ( S 3 S + q 0 q t q0 2 q t 1/2 (q q t ) exp { i [ Ξq 3 +Υq 2 +Ωq +Λ ]} ) (q q 0 )

Ξ = 1 ( 3 ) S 6 q0 3, q [ t ( Υ = 1 2 ) ( S 3 ) ( S 3 ) ( S 3 ) ] S 2 q0 2 + q t q0 2 q q t q0 3 q 0 q t q0 2 q q t + A, t ( ) ( S 2 ) ( S 2 ) ( S 3 ) S Ω = + (q q t ) q 0 q t q 0 q t q0 2 q 0 + q t q0 2 q (q 0 q t qq 0 ) t + 1 ( 3 ) ( S 2 q0 3 q0 2 3 )( S 1 + q t q 0 qt 2 2 q2 t + 1 ) 2 q2 qq t 2Aq β + ξ, ( ) S Λ = q + 1 ( 2 ) S q t q 0 2 qt 2 q 2 + 1 ( 3 ) ( ) S S q 0 6 qt 3 q 3 q 0 q 0 q 0 q t ( 2 ) S qq 0 1 ( 3 ) S q 0 q t 2 q 0 qt 2 q 2 q 0 + 1 ( 2 ) S 2 q0 2 q0 2 q t + 1 ( 3 ) S 2 q0 2 q qq0 2 t 1 ( 3 ) ( ) ( S S 6 q0 3 q0 3 2 ) S q t q t q t q 0 qt 2 qq t 1 ( 3 ) S q 0 2 qt 3 q 2 q t q 0 ( 2 ) ( S 3 ) S + q 0 q t + q 0 q t q 0 qt 2 qq 0 q t 1 ( 3 ) S 2 q0 2 q q0q 2 t t ( 2 ) S + 1 2 qt 2 qt 2 q 0 + 1 ( 3 ) S 2 qt 3 qqt 2 1 ( 3 S 2 q 0 qt 2 +Aq 2 β + S + γ ξq β. ) q 0 q 2 t 1 6 ( 3 ) S q 3 t q 0 q 3 t

φ Ψ 1 β (q, t) =N 2πi i dq A(q, t)exp{iφ} : φ = Ξq 3 +Υq 2 +Ωq +Λ ( = Ξ q 3 + Υ Ξ q 2 + Ω Ξ q + Λ ) Υ [ ( = Ξ q + 1 ) 3 Υ 1 3 Ξ 3 [( = Ξ q + 1 ) ( 3 Υ Ω + 3 Ξ Ξ 1 3 1 Υ 3 Ξ ( Ω Ξ 1 3 ( Υ Ξ ( ) 2 Υ q 1 Ξ 27 ) 2 ) ( Υ Ξ 1 27 ( ) 3 Υ + Ω Ξ Ξ q + Λ Ξ ) ) 2 ( q + 1 ) Υ 3 Ξ ( ) 3 Υ + Λ ]. Ξ Ξ ] t = ( q + 1 ) Υ 3 Ξ = φ =Ξt 3 +Ξ ( Ω Ξ 1 3 ( ) ) ( Υ 2 t 13 Ξ Υ Ω Ξ 1 3 ( ) ) Υ 2 1 Υ 3 Ξ 27 Ξ 2 +Λ. Ψ β (q, t) = N 1 2πi i exp { i [ Ξq 3 +Υq 2 +Ωq +Λ ]} = N 1 2πi exp { i [ i + 1 Υ 3 Ξ [ dq 2 ( S 3 ) ( S + q 0 q t q0 2 q (q 3 ) ] 1/2 S q 0 )+ t q 0 qt 2 (q q t ) + 1 Υ 3 Ξ Ξt 3 +Ξ ( [ 2 ( S 3 S dt + q 0 q t q0 2 q t Ω Ξ 1 3 ( Υ Ξ ) 2 ) t 13 Υ ( Ω Ξ 1 3 ) ( (q 3 S q 0 )+ q 0 qt 2 ( Υ Ξ ) 2 ) ) ] 1/2 (q q t ) 1 Υ 3 27 Ξ 2 +Λ ]}.

q q 0 Ψ β (q, t) = N 1 + 1 Υ 3 Ξ 2πi [ 2 S + q 0 q t { [ exp i + 1 Υ 3 Ξ t ( 3 S Ξt 3 +Ξ = N 1 + 1 Υ 3 Ξ 2πi [ 2 S + q 0 q t { [ exp i q 0 q 2 t + 1 Υ 3 Ξ ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) ) ] 1 2 n ( 3 S (q q t ) n=0 ( Ω Ξ 1 3 t ( 3 S Ξt 3 +Ξ = N 1 + 1 Υ 3 Ξ 2πi [ 2 S + q 0 q t n n m { [ m=0 exp i q 0 q 2 t + 1 Υ 3 Ξ n=0 ( Υ Ξ ) n (q q 0 ) n q0 2 q t ) ) ( 2 t 13 Υ Ω Ξ 1 3 ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) ) ] 1 2 n ( 3 S (q q t ) ( Ω Ξ 1 3 t ( 3 S n=0 ( Υ Ξ ) n ( t 1 Υ 3 q0 2 q t ) ) ( 2 t 13 Υ Ω Ξ 1 3 ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) ) ] 1 2 n ( 3 S (q q t ) q0 2 q t q 0 qt 2 ( 1 ) Υ n m 3 Ξ q 0 t m Ξt 3 +Ξ ( Ω Ξ 1 3 ) n ( ) ) ( Υ 2 t 13 Ξ Υ Ω Ξ 1 3 ( ) ) ]} Υ 2 1 Υ 3 Ξ 27 Ξ 2 +Λ ( Υ Ξ Ξ q 0 ) ) 2 ) n 1 Υ 3 27 Ξ 2 +Λ ]} ( ) ) ]} Υ 2 1 Υ 3 Ξ 27 Ξ 2 +Λ

{ [ ( = N 1 1 exp i 2πi 3 Υ Ω Ξ 1 3 ( 1) n (2n)! (1 2n)(n!) 2 (4 n ) n=0 n n m=0 m { [ 1 6 ( 1)m (iξ) m 3 1 ( ( 1 ) Υ n m 3 Ξ q 0 ( ) ) Υ 2 Ξ [ 2 ( S 3 S + q 0 q t q 0 qt 2 + 1 Υ 3 27 Ξ 2 Λ ) (q q t ) ]} ] 1 2 n ( 3 S q 2 0 q t ( ) ( m +1 m 2(iΞ) 2/3 Γ 1F 2 3 3 + 1 3 ; 1 3, 2 3 ; Z 3 ) 27(iΞ) + Z 2 3 ( ) ( m +2 m (iξ)γ 1F 2 3 3 + 2 3 ; 2 3, 4 3 ; Z 3 ) 27(iΞ) ( m ) ( m + ZΓ 3 +1 1F 2 3 + 1; 4 3, 5 3 ; Z 3 ) )] 27(iΞ) + 1 ( ) ( 6 ( (iξ)) m m +1 m 3 [2( (iξ)) 1 2/3 Γ 1F 2 3 3 + 1 3 ; 1 3, 2 3 ; Z 3 ) 27(iΞ) ( ( m ) ( m + Z ZΓ 3 +1 1F 2 3 + 1; 4 3, 5 3 ; Z 3 ) 27(iΞ) 2 3 ( ) ( m +2 m (iξ)γ 1F 2 3 3 + 2 3 ; 2 3, 4 3 ; Z 3 ) )]}. 27(iΞ) ) n Z = Ξ ( Ω Ξ 1 3 ( ) ) Υ 2, Ξ

x n e αx3 e px dx = 0 x n e αx3 e px dx + 0 ( x) n e α( x)3 e p( x) dx [ = L x x n e αx3] (p)+l x [( x) n e α( x)3] ( p) = 1 ( ) ( 6 ( 1)n α n n +1 n 3 {2α 1 2/3 Γ 1F 2 3 3 + 1 3 ; 1 3, 2 3 ; p 3 ) 27α [ ( ) ( +p 2 3 n +2 n αγ 1F 2 3 3 + 2 3 ; 2 3, 4 3 ; p 3 ) 27α ( n ) ( n +pγ 3 +1 1F 2 3 + 1; 4 3, 5 3 ; p 3 ) ]} 27α + 1 ( ) ( 6 ( α) n n +1 n 3 {2( α) 1 2/3 Γ 1F 2 3 3 + 1 3 ; 1 3, 2 3 ; p 3 ) 27α [ ( n ) ( n +p pγ 3 +1 1F 2 3 + 1; 4 3, 5 3 ; p 3 ) 27α 2 3 ( ) ( n +2 n αγ 1F 2 3 3 + 2 3 ; 2 3, 4 3 ; p 3 ) ]}. 27α M(t) = p t p 0 q0 q t p 0 q0 p t q 0 p0 q t q 0 p0.

Ṁ(t) =K(t) M(t) K(t) = 2 H q p 2 H p 2 2 H q 2 2 H p q. H = p2 2m + V (q) K(t) = 0 2 V (q) q 2 1 m 0. 2 S = 1, q 0 q t M 21 2 S = M 22, q 0 q 0 M 21 2 S = M 11, q t q t M 21 M(0) = I 2 2 Γ(t) = K(t) (p, q) ijk M(t) kl M(t) jm + K(t) ij Γ(t) jlm,

Γ(t) 2N 2N 2N M (p 0,q 0 ) K(t) p = 3 H q p 2 3 H p 3 3 H q 2 p 3 H q p 2, K(t) q = 3 H q 2 p 3 H p 2 q 3 H q 3 3 H q 2 p. H = p2 2m + V (q) K(t) p =0 I 2 2, K(t) q = 0 3 V (q) q 3 0 0. 3 S q0 3 3 S q0 2 q t 3 S q 0 qt 2 = = = = 3 S q t q t q t = 1 M 21 1 (M 21 ) 2 1 M 21 Γ 211 ( Γ 222 M 22 ) Γ 221 M ( 22 M 21 (M 21 ) 2 Γ 212 M ) 22 Γ 211, M 21 ) Γ 211, M 21 ) M ( 11 (M 21 ) 2 Γ 212 M ) 22 Γ 211, M 21 ( Γ 212 M 22 ( Γ 112 M 22 M 21 Γ 111 (M 21 ) 3, ( 1 Γ 111 M ) 11 Γ 211. M 21 M 2 21 Γ(0) = 0 I 2 2 2