( ) s n (n = 0, 1,...) n n = δ nn n n = I n=0 ψ = n C n n (1) C n = n ψ α = e 1 2 α 2 n=0 α, β α n n! n (2) β α = e 1 2 α 2 1

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1 ( ) 03032s n (n = 0,,...) n n = δ nn n n = I n=0 ψ = n C n n () C n = n ψ α = e 2 α 2 n=0 α, β α n n (2) β α = e 2 α 2 2 β 2 n=0 =0 = e 2 α 2 β n α 2 β 2 n=0 = e 2 α 2 2 β 2 +β α β n α! n (3) = e 2 (β α βα ) e 2 β α 2 0 β α 2 * α α d2 α π = (4) ψ ψ = β ψ = d 2 α α α ψ (5) π d 2 α β = π α α β d 2 α = π α e 2 α 2 2 β 2 +α β (6) *

2 ˆF â â ˆF = ˆF (â, â ) ˆF ˆF = ˆF n n = F n n (7) F n = ˆF n ˆF n ˆF = π 2 d 2 β d 2 α β β ˆF α α (8) β ˆF α = n F n β n α = e 2 ( β 2 + α 2) F (β, α) (9) F (β, α) = n F n (β ) (α) n! (0) ˆF = π 2 d 2 β d 2 α e 2 ( β 2 + α 2) F (β, α) β α () ˆF λ ˆF = λ λ λ λ (2) F n = λ λ λ λ n (3) F n λ λ λ λ λ n λ = T r ˆF (4) F n F (β, α) β α ˆF ˆF (9),(0) α ˆF α e α α = n α α [ n+ ( α ˆF ] α e α α! α α n α α n! ˆF n (5) α =0,α=0 = ˆF n (6) ˆF 2

3 2 ˆx ˆp ˆx ˆp Quadrature *2 ˆX ˆX 2 ˆX = â + â 2 ˆX 2 = â â 2i (7) ˆX = α 2 (â + â ) α = 2 (α + α ) = Re(α) (8) ˆX 2 = α 2 (â â ) α = 2 (α α ) = I(α) (9) α ˆX ˆX 2 ( ˆX ) 2 = 4 = ( ˆX 2 ) 2 (20) α = α e iθ α (X, X 2 ) ( ) *3 α ˆX θ α θ θ α = 0 ( ) θ = 2π n n n 2π ( ) α αe iωt αe iωt ˆX αe iωt = α cos ωt αe iωt ˆX 2 αe iωt = α sin ωt (2) error circle 2 ˆ E x (z, t) = ε o (â + â ) sin(kz) = 2ε o sin(kz) ˆX (22) *2 ˆxˆp Introductry Quantu Optics 2-3 *3 ˆX = 2 = ˆX 2 Introductry Quantu Optics 3-3

4 ( 2 αe iωt ˆ E x αe iωt = 2ε o α sin(kz) cos(ωt) (23) ˆX 2 error circle 3 ψ ψ 2... ˆρ = i p i ψ i ψ i (24) p i i T r(ˆρ) = i p i = (25) ô ô = T r(ôˆρ) = i p i ψ o ô ψ i (26) 4

5 7 ˆρ = ρ n n (27) ρ n = ˆρ n ˆρ p n = ρ nn n 8 ˆρ = α ˆρ α α α d2 α d 2 α π 2 (28) ˆρ ˆρ = P (α) α α d 2 α (29) P (α) Glauber-Sudarshan P α ˆρ P (α) P (α) T rˆρ = T r P (α) α α d 2 α = P (α) n α α n d 2 α n = P (α) α n n α d 2 α n = P (α)d 2 α = P (α) P (α) P (α) *4 ˆρ P (α) u u 29 u ˆρ u = P (α) u α α u d 2 α = P (α)e 2 u 2 2 α 2 u α e 2 α 2 2 u 2 +α u d 2 α = e u 2 P (α)e α 2 e α u αu d 2 α (30) (3) { g(u) = f(α)e α u αu d 2 α f(α) = π g(u)e u α uα d 2 u 2 (32) g(u) = e u 2 u ˆρ u (33) *4 P (α) 0 P (α) δ(a α) 5

6 f(α) = P (α)e α 2 32 P (α) = e α 2 π 2 e u 2 u ˆρ u e u α uα d 2 u (34) β ˆρ = β β u ˆρ u = u β β u (35) = e β 2 e u 2 e u β+uβ P (α) = e α 2 e β 2 π 2 e u (α β) u(α β ) d 2 u δ 2 (α β) = δ[re(α) Re(β)] δ[i(α) I(β)] = π 2 e u (α β) u(α β ) d 2 u (36) P (α) = δ 2 (α β) (37) n n (ˆρ = n n ) u ˆρ u = u n n u = e u 2 ( u u) n P (α) = e α 2 (38) π 2 ( u u) n e u α αu d 2 u (39) P (α) = e α 2 = e α 2 2n α n α n π 2 ( u u) n e u α αu d 2 u 2n α n α n δ(2) (α) F (α, α ) F (α, α 2n [ ) α n α n δ(2) (α)d 2 2n F (α, α ] ) α = α n α n 6 α=0,α =0 (40) (4)

7 â â Ĝ(N) (â, â ) â â Ĝ (N) (â, â ) = C n (â ) n â (42) Ĝ(N) (â, â [Ĝ(N) ] ) = T r (â, â )ˆρ = T r P (α) C n (â ) n â α α d 2 α = P (α) C n α (â ) n â α d 2 α = P (α) C n α n α d 2 α = P (α)g (N) (α, α )d 2 α (43) P â α, â α â â O(â, â ) : O(â, â ) : O (N) (â, â ) (44) ˆn = ââ ˆn = â â = P (α) α 2 d 2 α (45) ˆn 2 = â ââ â : ˆn 2 := (â ) 2 â 2 (46) : ˆn 2 : = (â ) 2 â 2 = P (α) α 4 d 2 α (47) ˆρ P ˆB P ˆB = B p (α, α ) α α d 2 α (48) ˆB ˆB = T r( ˆB ˆρ) = n B p (α, α ) α α ˆρ n d 2 α n = B p (α, α ) α ˆρ α d 2 α (49) 7

8 Q ˆB = Î Q(α) = α ˆρ α π (50) Q(α)d 2 α = (5) P Q ˆB Q B Q (α, α ) α B α = e α 2 n B n B n = n B ˆB ˆρ P (α ) n (α) (52) (!) 2 ˆB = T r( ˆB ˆρ) = T r ˆBP (α) α α d 2 α = n ˆBP (α) α α n d 2 α n = P (α) α ˆB α d 2 α = P (α)b Q (α, α )d 2 α (53) ˆρ P ˆB Q (49) ˆB P ˆρ Q Q P *5 Weigner ˆρ W (q, p) 2π h q + 2 x ˆρ q 2 x e ipx h dx (54) q ± 2x ˆρ = ψ ψ W (q, p) 2π h ψ (q 2 x)ψ(q + ipx x)e h dx (55) 2 q + 2 x ψ = ψ(q + 2 ) W (q, p)dp = 2π h = ψ (q 2 x)ψ(q + 2 x) ψ (q 2 x)ψ(q + 2 x)δ(x)dx = ψ (q 2 x)ψ(q + 2 x) = ψ(q) 2 e ipx h dpdx (56) *5 8

9 q q W (q, p)dq = φ 2 (57) φ ψ(q) 57 W (q, p) 4 x x ˆρ(x) ρ(x) 0 (58) ρ(x) dx = (59) x n x n = dx x n ρ(x) (60) x n ˆρ(x) C(k) = e ikx = dxρ(x) = (ik) n x n (6) ρ(x) = dke ikx C(k) (62) 2π x n C(k) ˆρ(x) x n = d n C(k) i n dk n (63) k=0 n=0 C W (λ) = T r[ˆρe λâ λ â] = T r[ˆρ ˆD(λ)] (Wigner) C N (λ) = T r[ˆρe λâ e λ â] (norally ordered) C A (λ) = T r[ˆρe λâ e λ â] (antinorally ordered) (64) * 6 C W (λ) = C N (λ)e 2 λ 2 = C A (λ)e 2 λ 2 (65) *6 eâ+ ˆB = eâe ˆBe 2 [Â, ˆB] = e ˆBeÂe 2 [Â, ˆB] 9

10 (â ) â n = T r[ˆρ(â ) â n (+n) ] = λ ( λ ) n C N (λ) λ=0 (â â ) n = T r[ˆρ( â â ) n (+n) ] = λ n ( λ ) C A(λ) λ=0 {(â ) â n } W = T r[ˆρ{(â ) â n (+n) } W ] = λ ( λ ) n C W (λ) λ=0 (66) Cahill Glauber s C(λ, s) = T r[ˆρe λâ λ â+s λ 2 /2 ] (67) C(λ, 0) = C W (λ), C(λ, ) = C N (λ), C(λ, ) = C A (λ) (68) (antinorally) C A (λ) = T r[ˆρe λ âeλâ ] = T r[e λâ ˆρe λ â] = d 2 α α e λâ ˆρe λ â α π = d 2 αq(α)e λα λ α (69) Q Q(α) = π 2 C A (λ)e λ α λα d 2 λ (70) (29) P ˆρ C N (λ) = T r[ˆρe λâ e λ â] = P (α) α e λâ e λ â α d 2 α = P (α)e λα λ α d 2 α (7) P P (α) = π 2 C N (λ)e λ α λα d 2 λ (72) Wigner Wigner W (α) π 2 C W (λ)e λ α λα d 2 λ = π 2 C N (λ)e λ 2 2 e λ α λα d 2 λ (73) 0

11 P ˆρ T h ˆρ T h = + n n=0 ( ) n n n n (74) + n n n = exp( hω k B T ) (75) Q Q(α) = α ˆρ T h α /π = ˆρ π e α 2 T h n (α ) α n (!) /2 ( ) n n (α α) n = e α 2 π( + n) + n ) = ( π( + n) exp α 2 + n (76) 69 C A (λ) = π( + n) ) d 2 αexp ( α 2 e λα λ α + n (77) α = (q + ip)/ 2,λ = (x + iy)/ 2,d 2 α = dqdp/2 C A (x, y) = 2π( + n) [ exp (q2 + p 2 ] ) e [i(yq xp)] dqdp (78) 2( + n) π e as2 e ±βs ds = a e β 2 4a (79) C A (λ) = exp[ ( + n) λ 2 ] (80) 65C N (λ) = C A (λ)e λ 2 72 P (α) = α 2 = α 2 π n e n e n λ 2 e λ α λα d 2 λ Q Wigner (ˆρ = β β ) Q (8) Q(α) = π α β 2 = π e α β 2 (82)

12 (ˆρ = n n ) Q(α) = π α n 2 = π exp( α 2 ) α 2n (83) 3 Q β 73Wigner W (α) = 2 (84) π e 2 α β 2 W (α) = 2 π ( )n L n (4 α 2 )e 2 α 2 (85) L n (z) = 2πi e zt/( t) dt ( t)tn+ (Laguerre Polynoial) (86) ( 4) Wigner Q Wigner 3 (β = 0) (n = 4) Q α 4 (β = 0) (n = 3) Wigner α 2

13 5 ( (4) (2) α α d 2 α = e α 2 n α n α! n d 2 α (87) α = re iθ,d 2 α = rdrdθ α α d 2 α = n n 2π dre r2 r n++! 0 0 dθe i(n )θ } {{ } 2πδ n (88) r 2 = y,2rdr = dy α α d 2 α = π n n n dye y y n 0 } {{ } (89) α α d 2 α = π n = 0 n n = π (90) (4) 3

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh