0 4/

Transcription

1 04 II 6 7 5

2 0 4/ / / /5 4/ Clebsch-Gordon 4/ / s = / / / Rr i

3 / LS 5/ /0 5/ /7 6/ A 6/ Zeeman 6/0 6/ / WKB 7/ WKB / p L 7/ p L / ii

4 iii

5 0 4/ /8, 4/5, 4/, 5/, 5/3, 5/0, 5/7 6/3, 6/0, 6/7, [6/4 ] 7/, 7/8, 7/5, 7/ 9/ III

6 0.3 i ii Shrödinger eq. iħ ψx, t = Ĥψx, t t ψ x, t = ψx, y, z, t iii = ħ Ĥ Ĥ = m x + V x Ĥφx = Eφx = iet ψx, t = exp φx ħ Shrödinger eq. ψx, t = φx t

7 iv α α β C α = α α 0 α α = 0 α = 0 α pβ + qγ = p α β + q α γ for p, q C α β = β α Â Â α α Â Â α = a α Â α Â β = β Â α Â = Â 3

8 v x φ n x φ n x = x n n ˆx x = x x ˆx = x ˆx ˆx ˆp = ħ ħ x n = x ˆp n ˆp ˆp i x i x [ˆx, ˆp] = iħ Ĥφ nx = E n φ n x Ĥ n = E n n 3 3 ˆp m + V ˆx n = E n n ˆp x m + V ˆx n = x E n n = m = m x ˆp n + x V ˆx n ħ x n + V x x n i x ħ = m = E n x n x + V x φ n x = E n φ n x 3 V x = /mω x φ n x n 4

9 0.4 Â, ˆB [ Â, ˆB] = 0 Â, ˆB n, m, k : { n, m, k = an n, m, k ˆB n, m, k = b m n, m, k  n, m, k k [ˆL, ˆL i ] = 0 ˆL. V = V r [Ĥ, ˆL] = ˆr = ˆx, ŷ, ẑ = ˆx, ˆx, ˆx 3 ˆp = ˆp x, ˆp y, ˆp z = ˆp, ˆp, ˆp 3 [ˆx j, ˆp k ] = iħ δ jk 5

10 04 II 4/ / Â, ˆB [ Â, ˆB] = 0 Â, ˆB  n, m = a n n, m n, m : ˆB n, m = b m n, m  n, m  i  n  n = a n n. ii ˆB n  ˆB n = ˆB n = ˆBa n n = a n ˆB n iii ˆB n  a n  ˆB n n. ˆB n = b n n. iv  n ˆB  - - [-] 6

11 ... L = r p L i = ϵ ijk r j p k ˆL x = ŷˆp z ẑ ˆp y ˆL = ˆr ˆp ˆLi = ϵ ijkˆr j ˆp k = ˆL y = ẑ ˆp x ˆxˆp z ˆL z = ˆxˆp y ŷˆp x Comments + jkl = 3, 3, 3. ϵ jkl = jkl = 3, 3, 3 0 otherwise. ϵ ijk r j p k = 3 j,k= ϵ ijkr j p k 3.. ˆL x = ŷˆp z ẑ ˆp y = ˆp z ŷ ˆp y ẑ = ˆL x ˆL x = iħ y z z y ˆL = ˆr iħ = ˆL y = iħ z x x z ˆL z = iħ x y y x Comments. No. x, y, z r, θ, ϕ. [ˆr j, ˆp k ] = iħδ jk 7

12 .. ˆL = ˆL x + ˆL y + ˆL z ˆL = ħ θ + tan θ θ + sin θ ϕ No...3 4/8 [ˆL j, ˆL k ] = iħϵ jkl ˆLl ˆL x ˆL y [ˆL, ˆL j ] = 0 check [ˆL x, ˆL y ] = [ŷˆp z ẑ ˆp y, ẑ ˆp x ˆxˆp z ] = [ŷˆp z, ẑ ˆp x ] [ŷˆp z, ˆxˆp z ] [ẑ ˆp y, ẑ ˆp x ] + [ẑ ˆp y, ˆxˆp z ] = iħ ŷˆp x iħ ˆp y ˆx = iħˆl z [ˆL, L j ] = ˆL k [ˆL k, ˆL j ] + [ˆL k, ˆL j ]ˆL k = ˆL k iħϵ kjm ˆLm + iħϵ kjm ˆLm ˆL k =

13 [A, B] = 0... L = r p.. L..3 [L i, L j ]..4 4/5 [ˆL j, Ĥ] = 0 [ˆL, Ĥ] = 0. No. 9

14 . [ˆL j, ˆL k ] = iħϵ jkl ˆLl ˆL step 0 step 7 step 0 ˆL = ˆr ˆp.5 Ĵ Ĵ a a =,, 3 [Ĵa, Ĵb] = iħϵ abc Ĵ c step Ĵa = ħĵ a [ĵ a, ĵ b ] = iϵ abc ĵ c ĵ ± = ĵ ± iĵ ĵ = ĵ + ĵ + ĵ 3 [ĵ, ĵ 3 ] = [ĵ, ĵ ± ] = 0 [ĵ 3, ĵ ± ] = ±ĵ ± 3 ĵ = ĵ 3 + ĵ + ĵ + ĵ ĵ + = ĵ 3 ĵ ĵ ĵ + = ĵ 3 ĵ 3 + ĵ + ĵ 4 0

15 step [ĵ, ĵ 3 ] = ĵ ĵ 3 ĵ ĵ 3 λ, µ, k : {ĵ λ, µ, k = λ λ, µ, k ĵ 3 λ, µ, k = µ λ, µ, k k λ, µ λ, µ = δ λλ δ µµ step 3 4 λ, µ λ, µ ĵ λ, µ = λ, µ ĵ3 + + λ, µ ĵ+ĵ ĵ ĵ+ λ = µ + ĵ λ, µ + ĵ + λ, µ µ { λ 0 λ µ λ step 4 λ, µ ĵ ± λ, µ ĵ ĵ± λ, µ ĵ 3 ĵ± λ, µ = ĵ ± ĵ λ, µ = λ ĵ± λ, µ = ĵ ± ĵ 3 ± λ, µ 3 = µ ± ĵ± λ, µ ĵ ± λ, µ ĵ ĵ 3 ĵ ± λ, µ λ, µ ±

16 ĵ 3 µ + µ µ λ, µ + λ, µ λ, µ ĵ + ĵ } } n + n step 5 step 3 ĵ 3 45 { µmax = µ + n + λ µ min = µ n λ λ, µ max ĵ λ, µ max = λ, µ max ĵ + λ, µ max = 0 5 ĵ λ, µ min = 0 6 ĵ 3 ĵ ĵ 0 ĵ + λ, µ max 46 λ = µ max µ max + 7 λ = µ min µ min µ max + µ min µ max µ min + = 0 }{{} >0 µ max = µ min µ max = µ min + n + + n }{{} n 0 { µmax = n = µ min λ = n n +

17 step 6 [Ĵa, Ĵb] = iħϵ abc Ĵ c Ĵa Ĵ Ĵ3 Ĵ j, m = jj + ħ j, m Ĵ 3 j, m = mħ j, m j = n = 0,,, 3, m = j, j +, j, j }{{} j+ j step 7 j, m j, m = δ jj δ mm step 4 Ĵ ± j, m j, m ± ĵ ± j, m = j, m ĵ ĵ ± j, m ĵ = j, m ĵ 3 ĵ 3 ± j, m 4 = jj + mm ± j, m j, m }{{} Ĵ± j, m = jj + mm ± ħ j, m ± phase factor 3

18 .3 4/5 4/. Ĵ [Ĵ, Ĵ] = iħϵĵ] ˆL = ˆr ˆp. ˆL ˆL 3 ˆL l, m = ll + ħ l, m l = 0,, or, 3, ˆL 3 l, m = mħ l, m m = l, l +, l, l l, m r = x, x, x 3 ˆr = ˆx, ˆx, ˆx 3 ˆx i r = x i r for i =,, 3 r 3 r l, m φ l,m r r ˆL l, m = ll + ħ r l, m r ˆL 3 l, m = mħ r l, m 3 r ˆL 3 l, m = r ˆx ˆp y ŷ ˆp x l, m = x ħ i y y ħ r l, m i x = iħ r l, m ϕ r ˆp j α = ħ r α i x j x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ r ˆL l, m = ħ θ + tan θ θ + sin r l, m 5 θ ϕ }{{ } ˆΩθ,ϕ 4 4

19 3 4 5 φ l,m r = r l, m 6 7 ˆΩθ, ϕφ l,m r = ll + φ l,m r 6 ϕ φ l,mr = imφ l,m r 7 φ l,m r = φ l,m r, θ, ϕ Y l,m θ, ϕ Y l,m θ, ϕ by + No [J, J] = iϵħj J, J 3 j, m.3 J = L = r p r l, m Y lm r { l = 0,,, Y l,m θ, ϕ N l,m Pl m cos θe imϕ m = l, l +, l, l N l,m = m l + l m! 4π l + m! Pl m x = l l! x m/ dl+m dx l+m x l m 5

20 ˆL ˆΩθ, ϕy l,m θ, ϕ = ll + Y l,m θ, ϕ ˆL 3 ϕ Y l,mθ, ϕ = imy l,m θ, ϕ d cos θ l=0 m= π Y 0,0 = 4π Y,0 = 0 dϕy l,m θ, ϕy l,m θ, ϕ = δ ll δ mm Y l,m θ, ϕy l,mθ, ϕ = δcos θ cos θ δϕ ϕ 3 3 4π cos θ Y,± = ± 8π sin θe±iϕ r l, m = φ l,m r = φ l,m r, θ, ϕ Y l,m θ, ϕ Comments. Y l,m l, m. Ĵ j, m j, 3 ˆL = ˆr ˆp φr, θ, ϕ + π = φr, θ, ϕ ϕ e imϕ m = ĵ 3 m = ĵ x, y, z. r φ l,m r, θ, ϕ = RrY l,m θ, ϕ.3 Rr 6

21 .4 Clebsch-Gordon 4/ { L = r p L = r p L = L + L ˆL ˆL ˆL = ˆL + ˆL {ˆL l l + ħ ˆL l l + ħ ˆL. Ĵ [ĵ a, ĵ b ] = iϵ abc ĵ c Ĵa = ħĵ a [ĵ a, ĵ b ] = iϵ abc ĵ c [ĵ a, ĵ b ] = 0 ĵ, ĵ, ĵ z, ĵ z 0.4. j, j ; m, m ĵ j, j ; m, m = j j + j, j ; m, m ĵ j, j ; m, m = j j + j, j ; m, m ĵ z j, j ; m, m = m j, j ; m, m ĵ z j, j ; m, m = m j, j ; m, m 7

22 Ĵ a = ĵ a + ĵ a [Ĵa, Ĵb] = iϵ abc Ĵ c [Ĵz, Ĵ ] = 0 Ĵ = Ĵ + Ĵ + Ĵ 3 Ĵz Ĵ Ĵz, Ĵ, ĵ, ĵ Ĵ j, j ; J, M = JJ + j, j ; J, M Ĵ z j, j ; J, M = M j, j ; J, M 3 ĵ j, j ; J, M = j j + j, j ; J, M ĵ j, j ; J, M = j j + j, j ; J, M j, j ; J, M j, j ; m, m 3 j, j ; J, M = j, j ; J, M j j m = j m = j j, j ; m, m j, j ; J, M }{{} Clebsch-Gordan CG j, j ; m, m 8

23 CG j =, j = j, j J, M = m,m =±/ m, m J, M m }{{}, m CG step m, m m = j, m = j, Ĵ z = ĵ z + ĵ z : Ĵ z, =, M = Ĵ = ĴzĴz + + Ĵ Ĵ+ : Ĵ }{{}, = +, J = ĵ + +ĵ, =, step Ĵ = ĵ +ĵ. step 7 jj + mm ±, 0 =, +,, =, 3 step 3 α =,, Ĵ α = 0 Ĵz α = 0 α = 0, 0 4 step 4 4 CG,, = ±,, 0 = ±, 0, 0 = ±,, = 9

24 CG M = m + m j j J j + j Ĵ± = ĵ ± + ĵ ± j, j ; m, m m = j j }{{}, m = j j, j }{{} +j + j + j + j, j ; J, M : M = J J J = j + j j + j + J = j j j j + j + j /. L = r p. j, m.3 r l, m Y l,m θ, ϕ.4 J + J CG 0

25 Clebsch-Gordan II j,j; J, M = j j m= j m= j j,j; m,m j,j; J, M j,j; m,m = + J J ± M + j,j; m,m j,j; J, M ± j ± j m + j,j; m,m j,j; J, M j ± j m + j,j; m,m j,j; J, M Racah j,j; m,m j,j; J, M = δm+m,m [ J +j + j J!j j + J! j + j + J! j + j + J +! ] / [j + m!j m!j + m!j m!j + M!J M!] / k Z k k!j + j J k!j m k!j + m k!j j + m + k!j j m + k! j = any /, j =/ j, ; J, M = m=±/ j, ; M m,m j, ; J, M j, ; M m,m j, ; M m,m j, ; J, M m =/ m = / J = j + J = j j+m+/ j+ j M+/ j+ j M+/ j+ j+m+/ j+ j = any, j = j, ; J, M = j, ; M m,m j, ; J, M j, ; M m,m m=,0, j, ; M m,m j, ; J, M m = m =0 m = J = j + J = j J = j j+mj+m+ j+j+ j+mj M+ jj+ j Mj M+ jj+ j M+j+M+ j+j+ j Mj M+ j+j+ M j Mj+M+ jj+ jj+ j Mj+M jj+ j+mj+m+ jj+ The Review of Particle Physics Particle Data Group, Clebsch-Gordan coefficients 40. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS Note: A square-root sign is to be understood over every coefficient, e.g., for 8/5 read 8/5. Y 0 3 = 4π cos θ Y 3 = 8π Y 0 sin θ eiφ 5 3 = 4π cos θ Y 5 = 8π Y = 4 sin θ cos θ eiφ 5 π sin θ e iφ Y l m = m Y l m jjmm jjjm d l 4π = J j j jjmm jjjm m,0 = l + Y l m e imφ d j m,m = m m d j m,m = d j m, m d 0,0 / =cosθ d /,/ =cosθ d, = +cosθ d / /, / = sin θ d,0 = sin θ d, = cos θ d 3/ 3/,3/ = +cosθ cos θ d 3/ 3/,/ = 3 +cosθ d 3/ 3/, / = 3 cos θ d 3/ 3/, 3/ = cos θ d 3/ /,/ = 3cosθ d 3/ /, / sin θ cos θ sin θ cos θ = 3cosθ + sin θ d, = +cosθ d, = +cosθ 6 d,0 = 4 sin θ d, = cos θ d, = cos θ sin θ sin θ d, = +cosθ cos θ 3 d,0 = sin θ cos θ d, = cos θ 3 cos θ + d 0,0 = cos θ Figure 40.: The sign convention is that of Wigner Group Theory, Academic Press, New York, 959, also used by Condon and Shortley The Theory of Atomic Spectra, CambridgeUniv. Press,NewYork,953,RoseElementary Theory of Angular Momentum, Wiley,NewYork,957, and Cohen Tables of the Clebsch-Gordan Coefficients, NorthAmericanRockwellScienceCenter, ThousandOaks, Calif., 974.

26 .5 5/ ˆ L = ˆ x ˆ p = Ĵ a j j, j z = l, m r x, y, z r, θ, ϕ Ĵ a = ˆ La l = 0,, j, j z = s, s z or + Ĵ a = ˆ Sa s = 0,, r; s, s z ˆ J a = ˆ La + ˆ Sa r, r, ; s, s ˆ J a = ˆ i Lia + ˆ Sia s = 0 π uū d d s = uud, udd s = W, Z 4 N s = 3 ++ uuu s =

27 .6 s = / j = s = { /, / j, m = /, m = /, / = {. {Ŝ3 = ħ Ŝ 3 = ħ Ŝ3 = ħ 0 0 Ŝ ± = Ŝ ± iŝ Ŝ = ħ 0 0 Ŝ = ħ 0 i i 0 Ŝa α = β=, [ ] ħ σ a σ a = βα β 0 0, 0 i i 0, 0 0 Ŝ }{{} a α = [ ] ħ β σ a β=, βα }{{} { [ Ŝ a, Ŝb] = iħϵ abc Ŝ c consistent [σ a, σ b ] = iϵ abc σ c β β Ŝa α = [ ] ħ σ a βα 3

28 . ħ : Ĥφr = m + V r φr = Eφr ˆp : Ĥ φ = m + V ˆr φ = E φ E, φx E, φ r = φr = r φ r = x, x, x 3 ˆx, ˆx, ˆx 3 ˆx j x, x, x 3 = x j x, x, x 3 j =,, 3 : Ĥφ n,l,mx = E n φ n,l,m x : Ĥ n, l, m = E n n, l, m E n = 3.6 ev n n =,, n l m..3 i φr = φr, θ, ϕ = RrY θ, ϕ Rr Y θ, ϕ ii ii i 4

29 . [ˆL j, Ĥ] = 0 [ˆL, Ĥ] = 0 No. [ˆL j, ˆr k ] [ˆL j, ˆp k ] [ˆL j, ˆr ] = [ˆL j, ˆp ] = 0 [ˆL j, Ĥ] = 0,, [ˆL j, ˆL ] = 0. [Ĥ, ˆL ] = [Ĥ, ˆL z ] = [ˆL, ˆL z ] = Ĥ Ĥ, ˆL, ˆL z E, l, m. ˆL ˆL 3 Ĥ E, l, m = E E, l, m 3 ˆL E, l, m = ll + ħ E, l, m 4 ˆL 3 E, l, m = mħ E, l, m 5.3 r E, l, m Y l,m θ, ϕ 5

30 .3 5/ r E, l, m = RrY l,m θ, ϕ r 3 ˆp r m + V ˆr E, l, m = E r E, l, m }{{} RrY l,m θ,ϕ = ħ m + V r r E, l, m }{{} RrY l,m θ,ϕ = r + r r + ˆΩθ, ϕ by r ˆΩθ, ϕy l,m θ, ϕ = ll + Y l,m θ, ϕ [ ħ m r + ] ll + + V r RrY r r r l,m θ, ϕ = ERrY l,m θ, ϕ Shrödinger eq. [ ħ d m dr + r ] d ll + + V r Rr = ERr dr r Comments. V = V r V r = A r 6

31 .4 A e 4πϵ 0 e A = kg m 3 s 37 ħc. V r = mω r = mω x + y + x. Rr = χ lr r [ ħ d V m dr + r + ll + ħ mr Shrödinger eq. V eff r = V r + ] χ l r = Eχ l r ll + ħ mr l V eff r V r ll + ħ mr 5/ 7

32 03 04 = d 3 x φr, θ, ϕ = r dr d cos θ dϕ RrY l,m θ, ϕ = r dr Rr = dr χ l r 3. lim r 0 r V r = 0 V r /r r l 0 r 0 χ l r r s r s ss + ll + = 0 s = l or l + χ l r r l l = 0 χ 0 r r 0 or r χ 0 r r 0 φr /r ħ Ĥφr m + V r r const δr + V r Eφr r r χ l r r l+ r 0 l l = 0 Rr = 0 = 0 8

33 GW. J. V r = V r.. L H.3 Rr r E, l, m = RrY lm θ, ϕ 5/3 9

34 .4 5/3 r V r = A r, A 37 ħc.4. Rr [ ħ d m dr + r d ll + Ar ] dr r E Rr = 0 E < 0 Rr = d 3 x φr, θ, ϕ = r dr d cos θ dϕ RrY l,m θ, ϕ = r dr Rr r = ħ 8mE ρ ρ [ d dρ + d ll + + λ ρ dρ ρ ρ ] Rρ = 0 Rr = 4 Rρ m λ = ħ E A comment Eb λ = where E b ma E ħ mc 3.6 ev 37 E ρ = r where a B = ħ E b ma 37 ħ mc m a B 30

35 Rρ = ρ l e ρ/ G l ρ 3 [ρ d dρ + l + ρ d ] + l λ G l ρ dρ No.4 { case : λ l +, l +, l + 3 case : λ = n n = l +, l +, l + 3 case G l ρ e ρ ρ 3 Rρ e ρ/ Rr r dr = λ = n = l +, l + case G l ρ L l+ n+l ρ 4 ] L m n x = [e dm x dn dx m dx n e x x n L m n n m L l+ n+l n = n = n = 3 l = 0 L = L = x 4 L 3 = 3x + 8x 8 l = L 3 3 = 6 L 3 4 = 4x 96 l = L 5 5 = 0 3

36 34 Rρ = N nl ρ l e ρ/ L l+ n+l ρ N nl Rr r dr = 3 Nnl n l! = na B n[n + l!] 3.4. λ = n= l +, l +, λ = Eb E E b = ma ħ = 3.6 ev E = E b n n = l +, l +, E ρ = r = r a B n E b a B a B = ħ ma = m 3

37 N nl 5/7 = R nl r r nab 3 dr = R nl ρ ρ dρ nab 3 = N nl ρ l+ e [ ρ L l+ n+l ρ] dρ k=m j=m = = L m k x yk k! = ym xy exp y m+ y k=m y k z j k! j! y m z m y m+ z m+ x m+ e x L m k xl m j x y m z m m +! y m+ z m+ = m +! ym z m y z yz m+ y k z k k m x m+ e x [L m k! k x] [ y x m+ exp y + z [ y y + z z + ] z + x dx ] m+ k +! k! = m +! + k m! m +! k m! m +! }{{}}{{} yz k m from yz m yz k m from yz m k + mk! = k m! ρ l+ e ρ [ L l+ n+l ρ] dρ = n[n + l!] 3 n l! N nl 33

38 .4.3 Schrödinger eq Ĥ φ = E φ H = p m + V r, V r = A r E < 0 E = E n = E b n E b = ma ħ = 3.6 ev Ĥ φ n = E n φ n φ n = l,m c l,m n, l, m Ĥ n, l, m = E n n, l, m n =, ˆL n, l, m = ll + ħ n, l, m l = 0,, n ˆL 3 n, l, m = mħ n, l, m E b m = l, l +, l r φ n = c }{{} l,m r n, l, m }{{} l,m =φ nr =φ n,l,m r,θ,ϕ φ n,l,m r, θ, ϕ = R n,l r Y l,m θ, ϕ na B R nl x = N nl e x/ x l L l+ n+l x 34

39 .4.4 3, l, m s, l, m s l, m = 0, 0,,,, 0,,, 0, 0 s s = / ŝ z ±/ r r, θ, ϕ n, l, m r; s z or n, l, m; s z s z = ± n, l, m = 0, 0, 0 0, 0, 0; ± Schrödinger eq. Dirac eq. 3 LS 35

40 m r p m r p r H = p m + p m + V r r = p CM M + p m + V r Schrödinger eq. M = m + m m = m m m + m r = r CM m /M m /M p m /M m /M = p CM Ĥφ total r, r CM = E total φ total r, r CM φ total r, r CM = φ CM r CM φr ˆp CM φ M CMr CM = E CM φ CM r CM φ CM e ik x ˆp + V r φr = Eφr m E total = E CM + E r r p p 36

41 I S 3 3. n B n B H = µ B where µ = IS n 3. 5/3 e L = mrv n I = e πr/v S = πr µ = IS n = e m L ˆ µ L = e ˆ m L l 0 Ĥ B = ˆ µ L B = e ˆ L m B ˆ µ S g = }{{} + Dirac eq. 5/3 α π + }{{} QED e ˆ S m ˆ µ S = g e ˆ S m = }{{} vs 0 37

42 5/0.4. L 3 3x = 6 V = V r H = µ B µ L = e m L µ S = g e m S g =.00 5/0 3 6 or III 38

43 3.3 LS 5/0 Ĥ LS = ξrˆ L ˆ S LS ξr = mc dv r Dirac eq. r dr Ĥ = ˆp m + V r + ξrˆ L ˆ S.4 A ˆL, Ŝ, ˆL 3, Ŝ3 ˆL l, s; m, s z = ll + ħ l, s; m, s z Ŝ = ss + ħ l, s; m, s z : ˆL 3 = mħ Ŝ 3 = s z ħ n, l, m ĤLS ˆ L ˆ S [ Ĥ, ˆL 3 ] 0 [Ĥ, Ŝ3] 0 l, s; m, s z Ĥ B ˆL, Ŝ, Ĵ, Ĵ3 Ĵa = ˆL a + Ŝa ˆL l, s; J, M = ll + ħ l, s; J, M Ŝ = ss + ħ l, s; m, s z : Ĵ = JJ + ħ Ĵ 3 = Mħ ˆL, Ŝ, Ĵ, Ĵ3 Ĥ l, s; J, M r Ĥ A B.4 CG 39

44 3.4 Ĥ = Ĥ0 +ĤLS +ĤB = ˆp m + V r +ξrˆ L ˆ S e B m ˆ L + g ˆ S Ĥ = g e ˆ S m B / ψt = αt + βt Schrödinger eq. iħ ψt = Ĥ ψt t αt = g e B βt m ħ σ αt βt iħ t αt βt T = π ω [ = exp i g e B 4m σ = [ cos ωt = π 4m ge B 0 0 ] α0 t β0 + i sin ωt B σ B ψt Ŝ ψt T/ ] α0 β0 where ω = g e B 4m S B 40

45 5/0 σ i 0 = for i =,, σ i σ j + σ j σ i = δ ij 0 v σ = v i σ i = v + v + v3 i 0 v n for n = even v σ n = 0 v n v σ for n = odd v e i v σ = i v σn n! n=0 = i n 0 n! v n + 0 n=even n=odd 0 v σ = cos v + i sin v 0 v 0 0 = v i n σ v n v n! v 0 0 4

46 Ĥ ψ n = E n ψ n Ĥ = Ĥ0 + Ĥ Ĥ 0 n = E n 0 n E n 0 n Ĥ0 = ˆp + m mω x Ĥ = Ax Ĥ = λx 4 Ĥ 0 ψ n n E n E 0 n Ĥ 0 E 0 n n Ĥ 0 Ĥ = Ĥ0 + Ĥ E n ψ n ψ n E n Ĥ Ĥ = λx 4 Ĥ = Ax 4

47 4.. 5/0 5/7 Ĥ0 + λĥ ψ n λ = E n λ ψ n λ Ĥ0 n = E n 0 n { m n = δmn n n n = Ĥ 0 n E 0 m E 0 n for m n n λ E n λ = ψ n λ = p=0 λ p E p n = E 0 n + λe n + λ E n + 3 λ p n p = n + λ n + λ n + 4 p=0 λ 0 ψ n λ n E n λ E 0 n 4 ψ n λ + cλ ψ n λ ψ n λ ψ n λ ψ n λ = * n ψ n λ = ** 34 Ĥ0 + λĥ λ p n p = λ q E n q λ p n p 43

48 λ p Ĥ 0 n p + Ĥ n p = p k=0 E n p k n k Ĥ0 + λĥ ψ n λ = E n λ ψ n λ E n λ = ψ n λ = = Ĥ0 n p + Ĥ n p = p=0 λ p E p n λ p n p p=0 p k=0 E n p k n k E n p n p E p n p = n Ĥ n p k= E p k n n n k 5 n p = c p n n + m n m n p m 6 where, for m n, m n p = E n 0 E m 0 [ p m Ĥ n p k= E p k n m n k ] 7 c p n 44

49 proof m Ĥ0 n p }{{} + m Ĥ n p = E m 0 m n p p k=0 E p k n m n k p + k= = m n =δ mn = E n p m n 0 }{{} E p k n m n k + E 0 n m n p m = n p n Ĥ n p = E n p + E n p k n n k = 5 k= m n p E m 0 m n p + m Ĥ n p = E n p k m n k +E n 0 m n p = 7 proof k= 45

50 567 n Ĥ m = H nm p = E n = H nn 8 n = c n n + H mn m m n E n 0 E m 0 p = E n = n Ĥ n E n n n = c n n Ĥ n }{{} =H nn=e n H mn = m n n = c n n + m n = E n 0 E m 0 + m n H mn E n 0 E m 0 n Ĥ m E n c n }{{} =H nm=h mn 9 E n 0 E m 0 m Ĥ E n n m comments. n p c p n E n p. E n 0 E m 0 0 E n 0 E m 0 =

51 Stark Ĥ 0 = ˆp m A A r 37 ħc Ĥ = eez z Ĥ0 Ĥ 0 n, l, m = E 0 n,l,m E 0 n,l,m = A a B n n, l, m, a B = ħ ma 37 ħ mc Ĥ0 n, l, m =, 0, 0 E,0,0 Ĥ Ĥ = λ A a B z λ = ee a B A λ 89 E,0,0 =, 0, 0 Ĥ, 0, 0 n, l, E m Ĥ, 0, 0,0,0 = E 0,0,0 E 0 n,l,m n,l,m,0,0 47

52 n, l, m Ĥ, 0, 0 = = d 3 x d 3 x d 3 x n, l, m x x Ĥ x x, 0, 0 d 3 x φ n,l,m x λ A a B zδ 3 x x φ,0,0 x = λ A a B.4.3 φ n,l,m r, θ, ϕ = R nl ry l,m θ, ϕ R nl r = R nl r na B n, l, m Ĥ, 0, 0 =λ A a B 0 π r dr d cos θ dϕ φ n,l,m x r cos θ φ,0,0 x 0, Rnl x = N nl e x/ x l L l+ n+l x d cos θdϕ Yl,mθ, ϕ cos θ Y 0,0 θ, ϕ } {{} = 3 δ l,δ m,0 r dr Rnlr r R }{{},0 r a B l= }{{} = I n = = 6n 7/ n n 5/ n + n 5/ E 0,0,0 = A a B E,0,0 = 0 E,0,0 = = n,, 0 Ĥ, 0, 0 n= n= E n,,0 E n,l,m λ A a B 3 I n A a B = λ A a B 3 n n= I n /n } {{ }

53 Stark z V Ĥ = Ĥ0 + λĥ E,0,0 λ = E 0,0,0 + λ E,0,0 + ψ,0,0 λ =, 0, 0 k e /λ k=0 5/7 d cos θdϕ Yl,mθ, ϕ cos θ Y 0,0 θ, ϕ = = d cos θdϕ Yl,mθ, ϕ 3 4π 3 Y,0θ, ϕ d cos θdϕ Yl,mθ, ϕy,0 θ, ϕ } {{ } =δ l, δ m,0 4π d cos θdϕ = [ r = na ] B x 0 0 = n4 a 3 B 6 = n4 a 3 B 6 r drrn,r r R,0 r a B r dr R r n, na B 0 0 = n4 a 3 B 6 N n,n,0 r a B R,0 r a B x dx R n, x x R,0 nx x dx [ N n, e x/ xl 3 n+x ] x [N,0 e nx/ L nx ] 0 dx x 4 exp n + x L 3 n+x 49

54 .4 3 n! N n, = na B n[n +!] = 3 n n +!a 3/ n + nn B 3 N,0 = a B = a 3/ B L m k x yk k! = ym y y k k! k=3 k=m dx x 4 exp n + x m+ exp L 3kx = y3 xy y y 4 = y3 y 4 y n+ dx x 4 exp n + x L 3 n +! n+x [ ] n 3 4! 5 = n + n n +! 5 n 3! n + 4! = 6 n n n n + n+ 0 = n4 a 3 B 6 = 6 n 7/ r drrn,r r R,0 r a B n n +!a 3/ B n n 5/ n + n+5/ = n + nn 0 n + [ n + dx x 4 exp + y ] x y + y 5 4! 4! 5 n + 5 y3 y a 3/ B n! I n y n n + y [ n n + 5 ] n n +! 4! n +! 6 n n n n + n+ 50

55 4..3 5/7 6/3 Stark n = Ĥ 0 = ˆp m A r λĥ = eez 4.., l, m Ĥ 0, l, m = E 0, l, m Ĥ 0 λ = 0, 0, 0,, 0,, ± E 0 = A 8a B Ĥ = Ĥ0 + λĥ E 0, l, m 3 5/7 5

56 6/3 4..A 6/3 step Ĥ ψ a = E a ψ a { n } n m = δ nm n n = n n Ĥ m Ĥ ψ a = E a ψ a A n H ψ a = n E a ψ a m m n H m m ψ }{{} a = E }{{} a n ψ a H nm u a,m H nm u a,m = E a u a,n H H H u a, u a, = E a u a, u a, B AB H nm E a Ĥ E a u a, u a = u a, = ψ a ψ a ψ a = m m m ψ a = m u a,m m 5

57 H nm E a Hnm = n Ĥ m = m }{{} Ĥ n = H mn =Ĥ ψ a ψ a ψ b = δ ab u a u b = n ψ a n n ψ b = ψ a ψ b = δ ab u a U = u u u U U = u u u = = H nm u a,m = E a u a,n m H u u = u u }{{}}{{} U U U HU = E E E E 53

58 step Ĥ0 + λĥ ψ n λ = E n λ ψ n λ n : Ĥ 0 n = E n 0 n H nm = n Ĥ0 + λĥ m = E n 0 δ nm + λ n Ĥ m }{{} H nm E 0 E 0 + λ H H H H ψ n λ ψ n λ } {{ } u n λ E n λ = E n 0 + λe n + λ E n + u n λ = u 0 n + λ u n + λ u n + k ψn λ = k n 0 + λ k n + λ k n + = E n λ ψ n λ ψ n λ } {{ } u n λ λ 0 E 0 E 0 u 0 n = E 0 n u 0 n u 0 n = n 0 λ E 0 E 0 u n + H H H H u 0 n = E n 0 u n +E n u 0 n 54

59 n E n 0 u n n + H nn = E n 0 u n n + E n E n = H nn m n E m 0 u n m + H mn = E n 0 u n + 0 m 4.. step 3 u n m = H mn E n 0 E m 0 = m n Ĥ 0 n, α = E 0 n n, α α =,, N n Ĥ 0 n Ĥ0 m = E 0 E 0 E 0 n E 0 n E 0 n N n N n n, α Ĥ0 n, β = E n 0 δ αβ E 0 E n 0 Nn Nn + λ H u = E u λ 0 E E n 0 E n,α λ = E 0 u n,α λ = u 0 n + n,α + k= k= λ k E k n,α λ k u k n,α 55 α =, N n

60 λ 0 E n 0 Nn Nn u 0 n,α = E 0 n u 0 n,α u 0 n,α 0 u 0 n,α = χ α } N n 0 λ E n 0 Nn N n u n,α+ H u 0 n,α = E 0 n u n,α+e n u 0 n,α [H ] Nn N }{{ n } χ α = E n χ α n, α Ĥ n, β α, β =, N n χ α [H ] Nn N n u 0 n,α E n [H ] Nn N n [H ] Nn N n λ Ĥ 0 λ = 0 Ĥ = Ĥ0 + λĥ E n 0 n, α E n, 3 E n,nn ψ n, λ ψ n,nn λ 56

61 λ 0 ψ n,α λ λ 0 n, α 0 = β = β n, β n, α 0 n, β χ α β n, β n, α u n,α λ λ 0 u 0 n,α = 0 χ α 0 57

62 4..3 Stark n = Ĥ0 = E 0 E 0 E 0 E 0 E 0, 0, 0, l, m 4 4, l, m Ĥ, l, m λĥ = eez = eer cos θ 4.., l, m Ĥ, l, m δ l,l±δ m,m [H ] 4 4 =, 0, 0,, 0,,,,, 0, 0,, 0,,,, 0 H H λh 0 = λh 0 = 3eEa B E 0 Ĥ 0, l, m 3 Ĥ E 0 + λh 0 E 0 E 0 λh 0 +Oλ, 0, 0 +,, 0,, &,,, 0, 0 +,, 0 +Oλ 58

63 , l, m λĥ, l, m = ee, l, m ẑ, l, m = ee d 3 xd 3 x, l, m x x ẑ x x, l, m = ee d 3 xd 3 x φ,l,m x r cos θδ3 x x φ,l,m x = ee r drr,l rrr,lr d cos θdϕyl,m θ, ϕ cos θy l,mθ, ϕ d cos θdϕ ϕ δ m,m Y l,m θ, ϕ l cos θ θ l, l, or, l, l =, 0 or 0, m, m = 0, 0 d cos θdϕy0,0θ, ϕ cos θy,0 θ, ϕ = d cos θdϕy,0θ, ϕ cos θy 0,0 θ, ϕ = d cos θdϕ 3 4π cos θ 4π cos θ = 3 r drr,rrr,0 r = r drr,0rrr, r = r dr R r,0 r a R r, B a B = a 4 B x dx R,0x x R, x = a 4 BN,0 N, x dx e x/ L x x e x/ xl 3 3x = a 4 B 96 dx e x x 4 x 4 6 3a 3 B, l, m λĥ, l, m = = 3 3a B { 3eEaB for l, l, m, m =, 0, 0, 0, 0,, 0, 0 0 otherwise 59

64 4..4 Zeeman 6/0 6/7 3.4 Ĥ = Ĥ0 +ĤLS +ĤB = ˆp m + V r +ξrˆ L ˆ S e m B ˆ L + g ˆ S ξ = const. z g =.003 H B = e m B zˆl z + Ŝz Ĥ case. Ĥ0 + ĤLS ĤB Zeeman case. Ĥ0 + ĤB ĤLS Paschen-Back case 3. Ĥ l = s = / Ĥ 0, ˆL, Ŝ, ˆL z, }{{ Ŝz n, l, s, l } z, s z Ĥ0, ˆL, Ŝ, Ĵ, }{{ Ĵz n, l, s, J, M } 60

65 Ĥ0, Ĥ LS, Ĥ B Ĥ0, ˆL, Ŝ n, l, s,, Ĥ n, l, s,, δ n nδ l lδs s = n, l, s =, 0, / n, l, s =, 0, / n, l, s =,, / n, l, s n, l, s Ĥ ξ = ξr Ĥ0 ĤLS n, ĤLS n, n n n 0 = n [Ĥ, Ĥ0] n = n ĤĤ0 Ĥ0Ĥ n = n n n Ĥ n ˆL, Ŝ 6

66 case. step B 0 Ĥ0 + ĤLS Ĥ0 + ĤLS J, M = Ĥ LS = ξ ˆ L ˆ S = ξĵ ˆL Ŝ n, l, s, }{{}}{{} J, }{{} M Ĵ Ĵ z E n + ξħ }{{} E LS [JJ + ll + ss + ] J, M s = / l J = l ± Ĥ0 ĤLS + J = l +, M = E n + l E LS J = l +, M Ĥ0 ĤLS + J = l, M l + = E n + E LS J = l, M Ĥ 0 Ĥ 0 + ĤLS J = l + / E n + l E LS J + = l + E n 3 E n + l+ E LS J + = l J = l / J, M Ĥ0 + ĤLS J, M l E n 4l+ 4l+ + E LS l+ l+ J = l + l+ E LS l l J = l 6

67 step ĤB Ĥ B = eb z m ˆL z + Ŝz = eb z m Ĵz + Ŝz Ĵ z J, M Ĵz J, M = Mħδ MM Ŝ z CG J = l ± l ± M + /, M = ± l + l z = M, s z = l M + / + l + l z = M +, s z = [ l ± M + / J, M Ŝz J, M = + l M + / l + ħ + ] l + ħ δ MM = ± l + Mħδ MM ĤB J, M ĤB J, M = ± l + JJ + ll + + ss + g j = + JJ + s 0 g j M eħb z } m {{} E B δ MM [ ] Lande g-factor 63

68 Ĥ 0 + ĤLS J = l + / l + E n + l E LS 3 +ĤB + l + E l+ B + l E l+ B l E n + l+ E LS J = l / 3 l+ l E B l+ l 3 E B s = 0 J = l l + s = / J = l ± l + l + s s = J = s + l + } }{{} l s 64

69 case. step ĤLS 0 Ĥ0 + ĤB n, l, s, l }{{} z, s }{{} z }{{} ˆL 3 Ŝ z Ĥ0+ Ĥ }{{} B l z, s z = E n + eħb l z +s z l z, s z m e m B }{{} zˆl z + Ŝz E B s = / l z + s z { m +, l z, s z = l z +s z = m, E = E n +me B m, l z, s z Ĥ0 + ĤB l z, s z E n 4l+ 4l+ l + l l +E B = l, / = l, / l = l, / = l, / m m = m +, / = m, / l + l + l l 65

70 step ĤLS ξ ˆ L ˆ S = ξ ˆL z Ŝ z + ˆL + Ŝ + ˆL Ŝ + ˆL z, ˆL +, ˆL l z ± [ĤLS ] m, ξ ˆ L ˆ S m +, = 0 l z, s z ξ ˆ L ˆ S mlz, s z = ξħ }{{} E LS l z s z m +, / m, / = m + m E LS l, / E n + l + E B + le LS l, / E n + le B + l E LS l, / E n + l E B + l E LS l, / 3 En + l E B le LS m +, / E n + me B + m E LS m, / 3 En + me B m + E LS 6/0 66

71 6/ case case case 3. l =, s = / l =, s = / E B A E n + E LS + 3 E B B case B 3 E B C E B A E n E LS + 3 E B B 3 E B C E n + E B + E LS A E n + E B + 0 B case B E n + 0 E LS BC l = l E n E B + 0 C E n E B + E LS A ĤB = E B /ħˆl z +Ŝz ĤLS = E LS /ħ ˆLz Ŝ z + ˆL + Ŝ + ˆL Ŝ + l z, s z Ĥ l z, s z, /, / 0, / 0, /, /, / E LS + E B E n E LS E LS E LS +E B E B LS E LS E LS E LS E B 67

72 r B = E B /E LS E n + E LS ± 4r B A E n + E LS r B ± rb + r B E n + E LS r B ± rb r B B C r B r B case r B r B case ABC 6/7 68

73 4. 6/7 iħ d ψt = Ĥ ψt dt 4. ħ = Ĥ/ħ Ĥ 4.. Ĥ = Ĥ0 Ĥ0 n = E n n E n, n i d dt ψt = Ĥ0 ψt ψt = e iĥ0t ψ0 iĥ0t k = ψ0 t d k! dt ψt = iĥ0 ψt k=0 [ n ψt = n n ψt n }{{} n ψt = m n e iĥ0t m m ψ0 }{{} e iemt m = e ie nt n ψ0 e ie nt ] 69

74 4.. Ĥ = Ĥ0 + Ĥ t Ĥ Ĥ 0 n i d ψt = Ĥ ψt 3 dt Ĥ = const ψt = e iĥt ψ0 e iĥ t e iĥ0t ψ0 = iĥ t + e iĥ0t ψ0 Ĥ ψt H 0 = } e iĥ0t {{} ψ0 t factorize ψt = e iĥ0t ψt I 4 ψt }{{} I e +iĥ0t ψt 4 3 i d dt e iĥ0t ψt I = Ĥ0 + Ĥ t e iĥ0t ψt I 70

75 i d dt ψt I = ĤIt ψt I 5 where ĤIt e +iĥ0t Ĥ te iĥ0t Ĥ t ĤIt t 5 ψt I 4 ψt [ψt I n ψt I = n n ψt }{{} I n 5 i d dt n ψt I = m n ĤIt m m ψt I 5 n ψt I 4 ψt = n ψt n n }{{} = n e iĥ0t ψt I ] = e ient n ψt I 5 5 i d ft = htft dt i ft = e 7 t 0 ht dt f0

76 5 i ψt I??? =??? e t No. 0 dt Ĥ I t ψo I 6 CHECK t 0 dt Ĥ I t = ˆϕt 6 = e i ˆ ϕt ψ0 I = i d dt 6 = = n=0 n= n! n! i d dt n k=0 n=0 i ˆ n ϕt ψ0 I n! i ˆ ϕt n ψ0 I k d ϕt dt ˆϕt }{{} Ĥ I t i ˆ = = i ˆ ϕt n k ψ0 I [ [ĤIt, ˆϕt] ] 0 ĤIt ĤIt n! n i ˆ n ϕt ψ0 I n= }{{} = ĤIt 6 i ˆϕt e 7

77 i ψt I = T e t 0 dt Ĥ I t ψ0 I 7 T: T {ât ˆbt t > t T ât ˆbt = ˆbt ât t > t CHECK [ 7 = T n! i d dt 7 = n= n=0 [ ˆϕt = n! T t 0 ] iϕt ˆ n ψ0 I [ n k=0 ] iϕt ˆ k ĤI t iϕt ˆ n k ψ0 I = = Ĥ I t dt t > t ĤIt [ = ĤIt n! T n i ˆ ] n ϕt ψ0 I n= }{{} i ˆϕt Te ] = ĤIt 7 [7 5 [ t exp i n ψt I = }{{} T t [ ] HI t m 0 [ ] ] dt HI t m ψ0 I nm ] [ HI t = kl k ĤIt l ] 73

78 7 ψt I = ψ0 I + n! T i n= = ψ0 I + n= = ψ0 I i t 0 t 0 tn dt n dt n t 0 0 dt Ĥ I t n ψ0 I t3 0 t dt dt Ĥ I t n ĤIt ψ0 I dt Ĥ I t ψ0 I Ĥ 0 i f f i P i f t = f e iĥt i = f ψt }{{} ψ0 }{{} ψt 3 = f e} iĥ0t {{} e ie f t t 8 = f i i dt f Ĥ }{{} I t i + OH 0 }{{} I =0 e +iĥ0t Ĥ te iĥ0t t = dt f Ĥ t i e ie f E i t + OH 0 ψt I = f ψt I Ĥ t = const for [0, t] sin E t/ f = Ĥ i E f Ĥ i E P i f πħ E 3 4 t 6/7

79 04 II 6/ / Wigner-Eckart

80 4.3 WKB 7/ 4.3. WKB Schrödinger eq. x ħ d m dx + V x φx = Eφx φx = e i ħ Sx Sx = S 0 x + ħs x + ħ S x m S x iħ m S x + V x E e i ħ Sx = 0 S x m E V x iħs x = 0 Schrödinger eq.. Sx = S 0 x + ħs x +ħ S }{{} x WKB [ ] Comments Sx Sx + const S 0 x, S x, const. term. ħ = ħ 3. S 0x E > V x px 76

81 ħ 0 : S 0 = m E V x ħ : S 0S is 0 = 0 WKB ħ : S 0S + S is = 0 S 0x = S x = i { ±px ± me V x for E > V x ±iκx ±i mv x E for E < V x S 0 S 0 = i ln S 0 3 is x = ln S 0 + const e is x = S 0 x const φx = e i ħ S 0x+ħS x+ħ S x S 0 x e i ħ S 0x+ħS x 77

82 A: E > V x B: E < V x φx c e i x dx ħ px + c e +i x dx ħ px px p cl E = p cl m + V x p clx = ± me V x S 0x = ±px φx c e x dx ħ κx + c e x dx ħ κx κx WKB Schrödinger eq. C: E V x 3 px, κx 0 WKB 78

83 WKB S = S 0 + ħs + S 0 ħ S 3 ħ 4 m E V x 3/ V x 4 x = x 0 V x = V x 0 +V x }{{} 0 x x 0 + =E ħ 4 x x 0 3m V x 0 /3 E < V x x 0 Schrödinger eq. 79

84 4.3. V 0 Ae ikx/ħ Be ikx/ħ E = k m < V 0 κ = mv 0 E = T = j C = C j A A = a j = iħ m φ φ φφ Ce ikx/ħ V 0 κa + 4EV 0 E sinh ħ [ T exp ħ b a mv x Edx ] 80

85 φx = φ,in x + φ,out x = C i x px dx,in eħ px + C,out e i x px dx φx ħ px T = j 3 j,in = = C 3 C,in κx e ± ħ x κx dx iħ m φ φ φφ 3 iħ m φ φ φφ,in = [ exp ħ b a mv x Edx ] φx = φ 3 x = C i x px dx 3 eħ px x a, x b 8

86 とても 大雑把な説明 a < φx < b での波動関数を φx φ+ x + φ x x x κx dx κx dx C+ + ℏ C ℏ a a = e e + κx κx とすると 波動関数の変化は φx b C+ em/ℏ + C e M/ℏ, φx a C+ + C 一方 x b で進行波 φ ei/ℏ つなげるためには φ+ b φ b pdx where M = b dx mv x E a に 片方だけだと φ e i/ℏ pdx も出てくる 後でもうちょいちゃんと見る よって C+ em/ℏ C e M/ℏ 以上より C e M/ℏ φx b e M/ℏ M/ℏ φx a C e + C j3 j,in φx b φx a M ℏを仮定 e M/ℏ a < x < b の間に波動関数が e M/ℏ だけ減衰する というのが効い ている 7/ ココまで 8

87 /8 x a V a > 0 WKB V V x E + V ax a Schrödinger eq. 0 = ħ d m dx + V x E φx ħ d m dx + V ax a φx λ mv a ħ, y λ /3 x a d dy y φ = 0 83

88 φy = C A πai y + C B πbi y π Airy.5 B i y A i y y E Airy [ cos y /4 3 y3/ π ] y 4 [ cos y /4 3 y3/ + π ] y 4 a πai x πbi x x V x y y y /4 exp exp y/4 [ 3 y3/ ] [+ 3 y3/ ] y ± [ y C y φy /4 A cos 3 y3/ π ] [ + C B cos 4 3 y3/ + π ] 4 CA y + [ y /4 exp 3 ] y3/ + C B exp [+ 3 ] y3/ 3 84

89 3 WKB px = me V x mv aa x = ħλ /3 y κx = mv x E mv ax a = ħλ /3 y Lx ħ Mx ħ x a x a px dx = 3 y3/ κx dx = 3 y3/ 3 ħ / λ /6 [ x < a C A cos Lx + π ] [ + C B cos Lx π ] px 4 4 φx a = ħ/ λ /6 CA + ic B e ilx + ic A + C B e +ilx px + i + i ħ / λ /6 CA x > a κx e Mx + C B e +Mx WKB C = ħ / λ /6 C A C + = ħ / λ /6 C B ic + C + e +ilx + + i px } {{ } C,in C + ic + + i } {{ } C,out e ilx px x < a φx a x > a C e Mx + C + e +Mx 4 κx κx 85

90 x b V b < 0 C e Mx + C + e + Mx x < b φx b x > b κx κx C + + i C e +i Lx i + C + + C e i Lx 5 + i px + i px } {{ } C 3 }{{} C 3,out = 0! Lx x ħ b px dx Mx ħ x b κx dx 45 a < x < b C e Mx = C e Mx C + e +Mx = C + e + Mx C = C [ + = e C Mx Mx = exp C + ħ C 3,out = 0 b a ] κxdx e A i C + + C = 0 T = C 3 C,in = C + + i C ic + C + 4 = C + 4e A C + + e A C + = e A + e A 4 e A

91 x b V b < 0 Schrödinger eq. 0 V x E + V bx b ħ d d m dx + V bx b φx λ mv b ħ, z λ /3 x b dz + z φ = 0 φz = C A πai z + C B πbi z z ± CA z [ z φz /4 exp 3 ] z3/ [ z + C z /4 A cos 3 z3/ π ] 4 + C B exp [+ 3 ] z3/ ] [ + C B cos 3 z3/ + π 4 px = me V x mv bx b = ħλ /3 z κx = mv x E mv bb x = ħλ /3 z Lx ħ Mx ħ x b x b px dx = 3 z3/ κx dx = 3 z3/ ħ / λ /6 CA x < b κx e+ Mx + C B e Mx φx b ħ / λ /6 i x > b C A + C B e i Lx + C A + i C B e px + i + i C + = ħ / λ /6 CA / C = ħ / λ /6 CB i Lx

92 5 5. p L 7/8 5.. p a = real a check Û a = e i a ˆ p/ħ = n=0 Û a x = x + a i a ˆ p n n! ħ Û a = e ˆX [ ˆx j, ˆX ] = ia k ħ [ˆx j, ˆp k ] }{{} =iħδ jk ˆX i a ˆ p ħ = a j [ˆx j, ˆX ] = ˆX [ ˆx j, ˆX ] [ ˆX] + ˆx j, ˆX = aj ˆX [ˆx j, ˆX ] n = n ˆX n a j ] [ˆx j, Û a = Û a a j ˆx j Û a x = Û aˆx j + a j x = x j + a j Û a x Û a x x + a Û a x = f a x + a Û a Û a = x x = δ x x Û aû b = Û a + b f a = expi k 0 a with k 0 = const ˆ p ˆ p+ a f a = 88

93 5.. L n = θ = real Ûθ n = e iθ n ˆ L/ħ = k=0 n θ Rθ n Ûθ n x = Rθ n x n = 0, 0, Z check iθ n ˆ k L k! ħ Ûθ n = eŷ Ŷ iθ n ˆ L ħ ˆx ˆx ˆx 3 [ˆx j, Ŷ ] = iθ ħ [ˆx j, ˆL3 ] = }{{} =ˆx ˆp ˆx ˆp, Ŷ = θ θ 0 θˆx j = θˆx j = 0 j = 3 ˆx ˆx ˆx 3 or ˆX, Ŷ e Ŷ ˆXeŶ = ˆX + [ ˆX, Ŷ ] + [[ ˆX, Ŷ ], Ŷ ] +! ˆx e Ŷ ˆx ˆx 3 eŷ = n! θ θ 0 n ˆx ˆx ˆx 3 = cos θ sin θ sin θ cos θ } {{ } Rθ n ˆx ˆx ˆx 3 ˆx j Ûθ n = Ûθ n [Rθ nˆx] j 89

94 ˆx j Ûθ n x = Ûθ n [Rθ nˆx] j x = [Rθ nx] j Ûθ n x Ûθ n x Rθ n x Û Û = phase factor 7/8 7/5 5. 7/5 Â Â = Â Û Û Û = ÛÛ = Â e iâ = n=0 n! iân 90

95 5.3 [Ĥ, Â] = 0 e iĥt/ħ Âe iĥt/ħ = Â 5.4 e iαâĥe iαâ = Ĥ 5.5 Ĥ = ˆ p m [Ĥ, ˆp k] = 0 e iĥt/ħˆp k e iĥt/ħ = ˆp k 5.4 e i a ˆ p Ĥe i a ˆ p = Ĥ : p Ĥ = ˆ p m + V }{{} r [Ĥ, ˆL k ] = 0 e iĥt/ħ ˆLk e iĥt/ħ = ˆL k 5.4 e i θ ˆ L Ĥe i θ ˆ L = Ĥ : L 9

96 5.4 A  [Ĥ, Â] = 0 Schrödinger Schrödinger eq. αt, βt d d d dt αt  βt = dt αt  βt + αt  dt βt }{{}}{{} iħ αt Ĥ iħĥ βt = i ] [Ĥ, ħ αt  βt = 0 [ ]  time-independent  ψt  ψt time-indepedent. Heisenberg  H t = e iĥt/ħ Âe iĥt/ħ = ÂH0 =  time-independent Ĥ = ˆ p m [Ĥ, ˆ p] = 0 = p Ĥ = ˆ p m + V r [Ĥ, ˆ L] = 0 = L 9

97 対称性 5.5. ユニタリー演算子と対称性 5.5 U : U U = U U = ˆ 例 U a = ei a p a 実数 任意の α β に対して ユニタリー変換 { α α = U α β β = U β α β = α U U β = α β 系に対称性がある ユニタリー変換 U があって [H, U ] = 0 U H U = H このとき d ψt = H ψt なら dt d iℏ U ψt = U H ψt = H U ψt dt iℏ U ψt も ψt と同じ時間発展 あるいは β e ih t/ℏ α = β U e ih t/ℏ U α = β e ih t/ℏ α α β の振幅と α β の振幅が同じ 93

98 H Û H = p /m Ûa = e ia ˆp H = p /m + V x Û x = x Ĥ Û ÛÛ Û = Û 94

99 5.5. Û : {Û Û = [Ĥ, Û] = 0 Û N Ûϵ = + i N ϵ a ˆTa + Oϵ a= ˆT a Û Û = iϵ a ˆT a + iϵ a ˆTa = + Oϵ ˆT a = ˆT a [Ĥ, Û] = 0 [Ĥ, ˆT a ] = 0 ˆT a Ûϵ Ûϵ Ûϵ Ûϵ = Ûfϵ, ϵ [ ˆT a, ˆT b ] = i N fab c ˆT c c= Ûa = e ia ˆp/ħ Û θ = e i θ ˆ L/ħ Ûϵ = iϵˆp/ħ + Û ϵ = i ϵ ˆ L/ħ + ˆp ˆ L f c ab = ϵ abc SU 95

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