Planck Bohr

I 30 7 11 1

1 5 1.1 Planck.............................. 5 1. Bohr.................................... 6 1.3..................................... 7 9.1................................... 9.................................... 10.3..................................... 11 3 1 3.1.................................. 1 3................................... 1 3.3..................................... 13 4 1 Schrodinger 14 4.1 Schrodinger........ 14 4. 1 Schrodinger................ 16 4.3................ 17 4.4..................................... 19 5 0 5.1............................. 0 5......... 1 5.3............ 4 5.4............ 6 5.5..................................... 8 6 9 6.1................. 9 6...................................... 31 7 3 7.1....................... 3 7. Schrodinger Heisenberg............... 33 7.3..................................... 34 8 3 Schrodinger 35 8.1 3 Schrodinger................ 35 8...................................... 36

9 37 9.1........................... 37 9......................... 38 9.3..................................... 41 10 43 10.1................................... 43 10...................................... 48 11 50 11.1................... 50 11.................................. 5 11.3 Schrodinger Heisenberg...................... 53 11.4..................... 54 11.5................... 55 11.6..................................... 58 1 59 1.1....................... 59 1........................ 60 1.3...................... 60 1.4........................ 60 1.5.......................... 61 1.6..................................... 6 13 63 13.1................................... 63 13................................. 64 13.3..................................... 65 14 66 14.1 Stern-Gerlach.............................. 66 14.................................. 66 14.3..................................... 69 A 70 B Bloch 70 C 71 D Hermite 7 3

E Legendre 73 E.1 Legendre................................. 73 E. Legendre............................... 74 F Laguerre 75 F.1 Laguerre................................. 75 F. Laguerre............................... 76 G 77 G.1 Bessel.................................. 77 G. Neumann................................ 78 G.3 Hankel................................. 79 H 3 80 H.1................................... 80 H...................................... 81 4

1 1.1 Planck Newton 3 19 Maxwell 19 1900 Planck Planck U(ν, T ) = 8πν n=0 c 3 nhν nhν e kt nhν n=0 e kt = 8πh c 3 ν 3 e hν kt 1, (1) 1 ν c k Boltzmann h h = πħ = 6.66 10 34 J s, () h h π ħ ( ) Planck (1) ν hν E n = nhν, n = 0, 1,, 3, (3) 1905 Einstein (3) ν (= ) E = hν Planck (1) T =.7 K Planck 1 (1) 1 ω E = nħω 5

1: (http://wmap.gsfc.nasa.gov/media/contentmedia/ 990015b.jpg ) 1. Bohr 1885 Balmer Balmer 1890 Rydberg ( 1 ν = Rc m 1 ) n, m = 1,, 3,, n = m + 1, m +,. (4) c R = 1.097 10 7 m 1 Rydberg Balmer m = 0 m = 1, 3, 4 Lyman Paschen Blackett 1911 1913 Bohr Bohr m e v r = e 4πϵ 0 r, (5) : Balmer (http://commons.wikimedia.org/wiki/ File:Emission_spectrum-H.png ) 6

m e e ϵ 0 v r Bohr m e vr = nħ, n = 1,,, (6) r v r = 4πϵ 0ħ m e e n, v = e 4πϵ 0 ħn, (7) E n = 1 m ev e 4πϵ 0 r = m ee 4 1 3π ϵ 0 ħ n, (8) E n E m ν = E n E m h = m ee 4 64π 3 ϵ 0 ħ3 ( 1 m 1 n ), (9) Rydberg (4) Rydberg R = m e e 4 /64π 3 ϵ 0 ħ3 c Frank Hertz 1914 (3) (8) (8) 196 Schrodinger Bohr Schrodinger 1.3 1. Planck ν ( hν kt (Rayleigh-Jeans ) ν ( hν kt (Wien ) 1) 1). Boltzmann T E e E kt E n = nhν (n = 0, 1,, ) 7

3. :390 nm :40 nm :450 nm :500 nm :590 nm :600 nm :700 nm hν 4. m e c = 0.5 MeV 5. ( ) 6. m (r, θ) θ p θ Bohr p θ dθ = nh, n = 1,,, (Bohr-Sommerfeld ) 8

Newton Huygens Huygens Young 1804 19.1 19 Maxwell c Maxwell E(t, x) ( t c )E = 0, E = 0, (10) 3 (10) e πi(k x+νt), (11) ν k λ k = 1/λ (10) ν = ±c k e πi(k x±c k t), (1) ±k c E(t, x) = ( d 3 k A(k)e πi(k x c k t) + A(k) e πi(k x c k t)), (13) A(k) k = 0 { ( 1 )} E = E 0 cos π λ n x νt, n E 0 = 0, (14) ν λ = c/ν n = (n 1, n, n 3 ) n λ E 0 n 3 = = i i = 1 + + 3 n x = n i x i = n 1 x 1 + n x + n 3 x 3 x i x i x i 9

. 0 Planck (3) E = nhν hν n Einstein 3: ν 0 ν 1905 Einstein 1 E p E = hν, p = h n, (15) λ 4 ν λ = c/ν n 1 1 1 m ev = hν hν 0, (16) hν 0 h Einstein ν hν 4 m E p E = p c + m c 4 m = 0 E = p c 10

(15) n E E = E 0 cos { ( 1 )} { 1 π λ n x νt ( ) } = E 0 cos p x Et, (17) ħ 4: 193 Compton X X Compton X λ X λ hc λ + m ec = hc λ + p c + m ec 4, (18) h λ = h h cos θ + p cos ϕ, λ sin θ = p sin ϕ, (19) λ ϕ p θ λ λ = h (1 cos θ), (0) m e c.3 1. L 0 (13). 11

3 19 0 1896 Zeeman (D ) ( ) Lorentz 1897 J. J. Thomson Zeeman Lorentz 5 1911 Rutherford (proton) 1918 Rutherford (neutron) 1935 Chadwick 0 194 de Broglie 3.1 6 1897 J. J. Thomson 1909 Milikan e = 1.60 10 19 C 1 fm = 10 15 m 3. 194 de Broglie (15) V [V] p m e = ev, (1) 5 Lorentz 6 3 1

(15) λ = h p = h me ev = 1.3 10 10 m, () V 100 V 10 10 m 10 10 m 197 Davisson Germer G. P. Thomson 3.3 1. J. J. Thomson E e/m e 5:. 1 fm ( ) E m e c 13

4 1 Schrodinger 4.1 Schrodinger 1 t x E ν p λ de Broglie { ( x )} ψ(t, x) exp πi λ νt E = hν, p = h λ, (3) { i ( ) } exp px Et, (4) ħ ψ(t, x) Eψ = iħ t ψ, pψ = iħ ψ, (5) x Ê = iħ t, ˆp = iħ x, (6) E = p m (5) iħ t ψ = ħ ψ, (7) m x Schrodinger Schrodinger (7) { i ) ψ(t, x) exp px E(p)t ħ( }, E(p) = p m, (8) ψ(t, x) de Broglie Schrodinger Born dx ψ(t, x) = 1, (9) t x x + dx ψ(t, x) dx Born ψ(t, x) 14

(7) ψ (8) ψ 1 ψ(t, x) = 1 πħ dp ψ(p) { i ) exp px E(p)t ħ( }, E(p) = p m. (30) ψ(p) (7) ψ(p) p = p 0 ψ(p) Gauss { ψ(p) = A exp 1 4σ (p p 0) }, A = 1 (π) 1/4, (31) σ1/ (30) p = p 0 + p ψ(t, x) = A πħ = A πħ { dp exp p { i dp exp ħ ( E(p 0 ) + p 0 m p + p ) } t m 4σ + ħ( i p0 + p ) x i ħ ( p0 x E(p 0 )t ) + ħ( i x vg t ) p a p } = A { i ( exp p0 x E(p 0 )t )} { exp (x v gt) } πħ ħ ħ a { dp exp a ( p i x v ) gt }, ħa = A { i ( exp p0 x E(p 0 )t )} { exp (x v gt) } ħa ħ ħ, (3) a v g de dp (p 0) = p 0 m, a 1 σ + it mħ, (33) ψ = A { ħ a exp (x v gt) } ħ σ a, (34) v g v g E(p 0 )/p 0 x = v g t (34) x = v g t x = ħσ a ψ p = p 0 p = σ x p = ħσ a = ħ 15 1 + ( σ t ) ħ mħ, (35)

4. 1 Schrodinger V (x) E = p + V (x), (36) m (5) (36) iħ ψ(t, x) = Ĥψ(t, x), t Ĥ = ħ + V (x), (37) m x V (x) Schrodinger ψ(t, x) ψ dx ψ(t, x) = 1, (38) ρ = ψ x Born t x ρ(t, x) = ψ(t, x) ψ(t, x) Schrodinger ρ ρ t ψ = ψ t + ψ t ψ = iħ ( ψ ψ m x ψ ) x ψ = { iħ x m ( ψ ψ x ψ x ψ )}, (39) ( ) j iħ ( ψ ψ m x ψ ) x ψ, (40) (39) ρ t + j = 0, (41) x ψ ψ x 16

4.3 ψ(t, x) Born t x(t) = dx ψ (t, x)ˆxψ(t, x), ˆx = x, (4) ψ(t, x) p(t) = dx ψ (t, x)ˆpψ(t, x), ˆp = iħ x. (43) Fourier p(t) = 1 πħ = = dx dp dp dp ψ (t, p )p ψ(t, p)e i ħ (p p )x dp ψ (t, p )p ψ(t, p)δ(p p ) dp ψ (t, p)p ψ(t, p), (44) x p ˆx ψ(t, p) = iħ p ψ(t, p), ˆp ψ(t, p) = p ψ(t, p), (45) [ˆx, ˆp] = iħ, (46) x p ( x) (x x ) = x x, ( p) (p p ) = p p, (47) ẑ = t(ˆx x ) + i(ˆp p ) z z = t ( x) ħt + ( p) 0 ħ x p, (48) 17

Fourier ψ(t, x) Fourier ψ(t, x) = A dp ψ(t, p)e i ħ px, (49) A p Dirac δ(x x ) = (49) ψ(t, p) = 1 πħa dp πħ e i ħ p(x x ), (50) dx ψ(t, x)e i ħ px, (51) A dx ψ (t, x)ψ(t, x) = dp ψ (t, p) ψ(t, p), (5) A = 1 πħ Gauss Gauss P (x) = 1 πσ e (x x 0 ) σ (53) Gauss x x x = (x x 0 ) (= x x 0 ) (x x 0 ) = dx P (x) = 1, (54) dx xp (x) = x 0, (55) dx (x x 0 ) P (x) = 1 πσ dx x e x σ = σ, (56) x x 0 x (x x 0 ) = σ 18

4.4 1. dp e ap = π a.. 3. t = 0 dp p e ap = 1 π a a. ψ(x) = Ae x 4σ, A x = x 4. Fourier ψ(p) = 1 πħ p = p 5. 6. dx ψ(x)e i ħ px, { i ( ) } { i ( ) } ψ(t, x) = A exp px E(p)t + B exp px E(p)t, ħ ħ ( ) 7. ψ(t, x) = 1 πħ dp ψ(p) { i ( ) } exp px E(p)t, ħ ( ) 8. Schrodinger d dt x = 1 m p 9. Schrodinger d dv dt p = dx (Ehrenfest ) 19

5 5.1 E ψ(t, x) Ĥψ = Eψ ψ(t, x) = e i ħ Et ϕ(x), (57) Schrodinger (37) ϕ(x) d ϕ dx = m (E V )ϕ, (58) ħ E (40) x j(x) = iħ ( ϕ dϕ m dx dϕ ) dx ϕ, (59) V = V 0 ( ) E V 0 > 0 (58) ϕ(x) = Ae i ħ px + A e i ħ px, p = m(e V 0 ), (60) A A (59) j(x) = p m A p m A, p = m(e V 0 ), (61) A x p/m A x p/m V 0 E E V 0 < 0 ϕ(x) = Be ρx + B e ρx, ρ = m(v0 E), (6) ħ B B (59) j(x) = iħρ m ( B B B B ) m(v0 E), ρ =, (63) ħ x ± ϕ 0 B = B = 0 0

5. [ V 0 < E < 0 ] 6 E 6: Schrodinger (58) Ae ρx + A e ρx, x 0, ϕ(x) = Be ipx + B e ipx, 0 x a, (64) Ce ρx + C e ρx, a x, A, A, B, B, C, C ρ p me m(e + V0 ) ρ =, p =, (65) ħ ħ Born ψ (= ϕ ) x x = ± A = C = 0 x = 0 x = a A = B + B, (66) Be ipa + B e ipa = C e ρa, (67) x = 0 x = a = ρa = ip(b B ), (68) ip(be ipa B e ipa ) = ρc e ρa, (69) 1

(66) (68) A B A = 1 ( 1 i ρ ), p B A = 1 ( 1 + i ρ ), (70) p (67) (69) A C A ρ = cos pa + ρ p sin pa ρ p sin pa = C A e ρa, (71) cos pa = C A ρ p ρ p ρ p e ρa, (7) ±1 cos pa =, (73) sin pa mv 0 p ħ p E 5 5 10 15 0 5-5 7: (73) mv 0a ħ = (8π) [0 < E ] 6 E x = x = Schrodinger (58) Ae ikx + A e ikx, x 0, ϕ(x) = Be ipx + B e ipx, 0 x a, (74) Ce ikx + C e ikx, a x,

A, A, B, B, C, C k p k = me, p = ħ m(e + V0 ), (75) ħ x = x = C = 0 x = 0, a A + A = B + B, (76) Be ipa + B e ipa = Ce ika, (77) x = 0, a (76) (78) k(a A ) = p(b B ), (78) p(be ipa B e ipa ) = kce ika, (79) B = A B = A (77) ( 1 + k p ( 1 k p ) + A ) + A ( 1 k ), p ( 1 + k ), (80) p ( C = A cos pa + i k )e p sin pa ika + A ( cos pa i k ) p sin pa e ika, (81) B, B, C (79) A (i sin pa + k ) p cos pa + A ( i sin pa k ) p cos pa = k p A ( cos pa + i k p sin pa ) + k p A ( cos pa i k p sin pa ), (8) A A = (81) C A = ( ) i 1 k p sin pa k p cos pa i( ), (83) 1 + k p sin pa k p e ika k p cos pa i( 1 + k p ) sin pa, (84) 3

( ) 1 k p sin pa j r j i = j t j i = 4k p 4k p cos pa + ( 1 + k p ) sin pa, (85) 4k p cos pa + ( 1 + k p ) sin pa, (86) x = pa = nπ (n = 1,, ) p λ p = π/λ a = nλ/ 5.3 [0 < E < V 0 ] 8 0 < E < V 0 x = 8: Schrodinger (58) Ae ikx + A e ikx, x 0, ϕ(x) = Be ρx + B e ρx, 0 x a, Ce ikx + C e ikx, a x, (87) A, A, B, B, C, C ρ k ρ = m(v0 E), k = ħ me, (88) ħ 4

x = (40) E m ( A A ), x 0, (V j(x) = i 0 E) m (B B B B ), 0 x a, E m ( C C ), a x, x 0 x j i = E/m A x j r = E/m A a x x j t = E/m C x = C = 0 0 x a x = 0 x = a (89) A + A = B + B, (90) Be ρa + B e ρa = Ce ika, (91) x = 0 x = a (90) (9) ik(a A ) = ρ(b B ), (9) ρ(be ρa B e ρa ) = ikce ika, (93) B = A B = A (91) ( 1 + i k ρ ( 1 i k ρ ) + A ) + A ( 1 i k ), ρ ( 1 + i k ), (94) ρ ( C = A cosh ρa + i k )e ρ sinh ρa ika + A ( cosh ρa i k ) ρ sinh ρa e ika, (95) B, B, C (93) ( A sinh ρa + i k ) ρ cosh ρa + A ( sinh ρa i k ) ρ cosh ρa = i k ρ A ( cosh ρa + i k ρ sinh ρa ) + i k ρ A ( cosh ρa i k ρ sinh ρa ), (96) A A = ( k ρ + 1 ) sinh ρa ( k 1 ) sinh ρa + i k (97) ρ ρ cosh ρa, 5

(95) C A = j r j i = i k ρ e ika ( k 1 ) sinh ρa + i k (98) ρ ρ cosh ρa, ( k + 1 ) ρ sinh ρa j t j i = ( k ρ 1 ) sinh ρa + 4k ρ cosh ρa, (99) 4k ρ ( k ρ 1 ) sinh ρa + 4k ρ cosh ρa, (100) 0 α ( α ) 8 V 0 < E 5. V 0 V 0 5.4 1 a V (x) V (x) = σ 0 δ(x na), (101) n= ( 9 ) (58) V (x) = V (x a) ϕ(x) = ϕ(x a) ϕ ϕ(x) = e iθ ϕ(x a), (10) θ Bloch 7 Bloch a x a Ae ipx + A e ipx, a x 0, ϕ(x) = e iθ( Ae ip(x a) + A e ip(x a)), 0 x a, (103) 7 ϕ(x) ϕ(x a) 1 ϕ(x) ϕ(x a) ϕ Bloch (10) 6

9: 1 x = 0 A + A = Ae iθ e ipa + A e iθ e ipa, (104) x = 0 ϵ x ϵ ϵ 0 dϕ dx (0 +) dϕ dx (0 ) = mσ 0 ħ ϕ(0), (105) (103) Ae iθ e ipa A e iθ e ipa A + A = i mσ 0 ħ p (A + A ), (106) (104) (106) A A A = 1 + eiθ e ipa 1 e iθ e ipa A = 1 eiθ e ipa mσ i 0 ħ p 1 e iθ e ipa + i mσ A, (107) 0 ħ p e iθ + 1 = e iθ( cos pa + mσ 0 ħ p sin pa ), cos θ = cos pa + mσ 0a ħ sin pa pa, (108) 1 cos θ 1 pa 1 cos pa + mσ 0a sin pa ħ 1, (109) pa 10 pa ( E ) 7

1.0 0.5-10 -5-0.5 5 10-1.0 10: pa mσ 0a ħ = 5 5.5 1. 1 ( Ĥϕ 1 = Eϕ 1 Ĥϕ = Eϕ ϕ 1 = cϕ c ). V (x) = V ( x) 3. 11(a) 4. 11(b) V (x) = σ 0 δ(x) x = 0 5. 11(c) 0 < E < V 0 6. 7. V 0 < E 8. (a) (b) (c) 11: 8

6 6.1 V (x) = 1 mω x, (110) m ω Shrodinger (37) (58) d ϕ ( me dx + ħ m ω ħ x ) ϕ = 0, (111) (111) (111) y = ( mω ħ ) 1 x, (11) d ϕ dy + ( E ħω y) ϕ = 0, (113) y ± 0 y ± ϕ(y) e 1 y ϕ(y) ϕ(y) = H(y)e 1 y, (114) (113) H(y) d H dy dh ( E ) y dy + ħω 1 H = 0, (115) 0 H(y) e 1 y H(y) Hermite Hermite ϕ(x) = 1 n πn! ( E n = ħω n + 1 ), (116) ( mω ħ ) 1 4 H n (y)e 1 y, y = ( mω ħ ) 1 x, (117) 9

0.6 0.4 0. -4-4 -0. -0.4-0.6 1: n = 0, 1,, 3 d H n dy Hermite y dh n dy + nh n = 0, (118) Hermite n = 0, 1,, H 0, H 1, H, H n (y) = i=0 c iy i (i + )(i + 1)c i+ = (n i)c i n c i c n 0 dn H n (y) = ( 1) n e y dy n (e y ), (119) H 0 = 1, H 1 = y, H = 4y, (10) H n dyh m (y)h n (y)e y = δ mn n πn!, (11) 30

6. V (x) = 1 mω x E n = ħω(n + 1 ) Shrodinger ϕ(x) = C n H n (y)e 1 y C n y = (mω/ħ) 1/ x Hermite H n n dyh m(y)h n (y)e y = δ mn n πn! 1. H n (y) Hermite d H n dy ( ) y dh n dy + nh n = 0. H n (y) = a=0 c ay a c a+ = (a n) (a+)(a+1) c a n = 0, 1,, c a H(y) e y 3. C n 4. Hermite H 0 (y) E 0 ( x) = ˆx ˆx 5. H 1 (y) ( x) 6. H (y) ( x) 7. H 3 (y) ( x) 8. H n (y) = ( 1) n e y d n dy n (e y ) ( ) yh n+1 = (n + 1)H n + H n+ 9. H n (y) Hermite 10. m < n dyym H n (y)e y = 0 11. dyyn H n (y)e y = πn! 1. Hermite 31

7 7.1 1 x m V (x) mẍ = dv dx, (1) (1) Lagrangian L(x, ẋ) L(x, ẋ) = 1 mẋ V (x), (13) action S[x] Lagrangian S[x] = t t 1 dt L(x, ẋ), (14) action S[x] x (1) x x(t) x(t)+δx(t) δx(t 1 ) = δx(t ) = 0 ( 13 ) t ( L L ) t { L 0 = δs[x] = dt δx + t 1 x ẋ δẋ = dt t 1 x d ( L )} δx, (15) dt ẋ δx L x d ( L ) = 0, (16) dt ẋ Euler-Lagrange Lagrangian (13) (1) 13: 3

(x, ẋ) Euler-Lagrange (x, p) Hamiltonian x p Hamiltonian H(x, p) Hamiltonian δh = δpẋ + pδẋ L x p L ẋ, (17) H(x, p) pẋ L(x, ẋ), (18) δx L ẋ L δẋ = δpẋ δx = δpẋ ṗδx, (19) x H x p (16) (17) ẋ = H p, ṗ = H x, (130) x p O(x, p) Ȯ = O x ẋ + O p ṗ = O H x p H O x p {O, H} P.B., (131) Poisson 7. Schrodinger Heisenberg Schrodinger ψ(t, x) Ô Schrodinger (37) ψ(t, x) = e i ħ Ĥt ψ(0, x), (13) Ô O = dx ψ(t, x) Ôψ(t, x) = ÔH dx ψ(0, x) e i ħ Ĥt Ôe i ħ Ĥt ψ(0, x), (133) Ô H e i ħ Ĥt Ôe i ħ Ĥt, (134) d = i [ÔH, Ĥ], (135) dtôh ħ Heisenberg ψ H (x) ψ(0, x), (136) 33

ÔH O = dx ψ H (x) Ô H (t)ψ H (x), (137) Heisenberg (131) (135) Heisenberg {O 1, O } P.B. i ħ [Ô1, Ô], (138) {x, x} P.B. = 0 i [ˆx, ˆx] = 0, ħ {x, p} P.B. = 1 i [ˆx, ˆp] = 1, (139) ħ {p, p} P.B. = 0 i [ˆp, ˆp] = 0, ħ ˆx ˆp Heisenberg Born ˆx ˆp Heisenberg 7.3 1. {x n, p } P.B. [ˆx n, ˆp ]. Heisenberg 3. [ˆx, ˆp] = iħ1 ˆx ˆp 34

8 3 Schrodinger 8.1 3 Schrodinger 3 Schrodinger t 3 x ν λ { ( 1 )} ψ(t, x) exp πi λ n x νt, (140) (15) Eψ = iħ ψ, pψ = iħ ψ, (141) t ˆp = iħ, (14) m V (x) E = p m + V (x), (143) (141) (143) iħ ψ(t, x) = Ĥψ(t, x), t Ĥ = ħ m + V (x), (144) 3 Schrodinger ψ(t, x) 1 ψ d 3 x ψ(t, x) = 1, (145) ρ = ψ(t, x) x Born ρ ρ t ψ = ψ t + ψ t ψ = iħ { ψ ψ ψ ψ } m { iħ ( = ψ ψ ψ ψ )}, (146) m 35

(146) j iħ ( ψ ψ ψ ψ ), (147) m ρ + j = 0, (148) t ψ ψ x 8. 1. Schrodinger d dt x = 1 m p. Schrodinger d dt p = V (Ehrenfest ) 3. 36

9 9.1 m 1 1 x 1 m x H = 1 m 1 ẋ 1 + 1 m ẋ + V ( x 1 x ) = 1 M Ẋ + 1 µ ẋ + V ( x ), (149) M X µ x M = m 1 + m, X = m 1x 1 + m x m 1 + m, µ = m 1m m 1 + m, x = x 1 x, (150) (center of mass) Ĥ = ħ M X ħ µ x + V ( x ), (151) X X gradient x x gradient E total Ψ(X)ψ(x) Schrodinger ψ ħ M XΨ Ψ ħ µ xψ + V ( x )Ψψ = E total Ψψ 1 ħ Ψ M XΨ 1 ħ ψ µ xψ + V ( x ) E total = 0 ħ M XΨ = E cm Ψ, ħ µ xψ + V ( x )ψ = (E total E cm )ψ, (15) Ψ(X) Schrodinger ψ(x) V ( x ) Schrodinger P ( i Ψ(X) = C exp ), ħ P X P E cm = M, (153) 1 4.1 Gauss 37

ħ { r + µ r r + 1 ( r θ + cos θ sin θ θ + 1 )} sin θ ϕ ψ + V (r)ψ = Eψ, (154) E = E total E cm (x, y, z) (r, θ, ϕ) x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, (155) r = sin θ cos ϕ x + sin θ sin ϕ y + cos θ z, θ = r cos θ cos ϕ x + r cos θ sin ϕ y r sin θ z, (156) ϕ = r sin θ sin ϕ x + r sin θ cos ϕ y, x = sin θ cos ϕ r + 1 r cos θ cos ϕ θ sin ϕ r sin θ ϕ, y = sin θ sin ϕ r + 1 r cos θ sin ϕ θ + cos ϕ r sin θ ϕ, (157) z = cos θ r 1 r sin θ θ, = r + r r + 1 ( r θ + cos θ sin θ θ + 1 ) sin θ ϕ. (158) 9. ŷˆp z ẑ ˆp y ˆL = ˆx ˆp = ẑ ˆp x ˆxˆp z, (159) ˆxˆp y ŷˆp x 38

[ˆL x, ˆL y ] = iħˆl z, [ˆL y, ˆL z ] = iħˆl x, [ˆL z, ˆL x ] = iħˆl y, (160) (157) sin ϕ θ ˆL = iħ cos ϕ θ cos θ cos ϕ sin θ ϕ cos θ sin ϕ sin θ ϕ ϕ, (161) ˆL = ħ ( θ + cos θ sin θ θ + 1 ) sin θ ϕ, (16) ˆL ħ ( θ + cos θ sin θ θ + 1 ) sin θ ϕ Y (θ, ϕ) = ħ λ Y (θ, ϕ), (163) (163) θ ϕ Y (θ, ϕ) = Θ(θ)Φ(ϕ) d dϕ Φ(ϕ) = λ ϕφ(ϕ), (164) ( d dθ + cos θ d sin θ dθ + λ ) ϕ sin Θ(θ) = λθ(θ), θ (165) (164) (165) (164) ϕ π λ ϕ = m, Φ(ϕ) = 1 π e imϕ, m Z, (166) π 0 dϕ Φ(ϕ) = 1 (165) z = cos θ (1 z ) d Θ dθ ( z dz dz + λ m ) 1 z Θ = 0, (167) Legendre λ = l(l + 1), l = 0, 1,,, m = l, l + 1,, l 1, l, (168) Θ(θ) Legendre Pl m Θ(θ) = ( 1) m+ m (l + 1)(l m )! P m (l + m )! l (cos θ), (169) 39

π 0 dθ sin θ Θ(θ) = 1 l m ˆL Y m l (θ, ϕ) = ħ l(l + 1)Y m l (θ, ϕ), Y m l (θ, ϕ) = ( 1) m+ m ˆLz Y l m (θ, ϕ) = ħmy l m (θ, ϕ), (l + 1)(l m )! P m 4π(l + m )! l (cos θ)e imϕ, (170) l = 0, 1,,, m = l, l + 1,, l 1, l, Y m l (θ, ϕ) l m Y 0 0 = 1 3, Y1 0 ±1 3 = cos θ, Y 1 = 4π 4π 8π sin θe±iϕ, (171) 5 15 15 Y 0 = 16π (3 cos θ 1), Y ±1 = 8π sin θ cos θe±iϕ, Y ± = 3π sin θe ±iϕ, Legendre (1 z ) d Pl m dz z dp l m ( dz + l(l + 1) m ) 1 z Pl m = 0, (17) Legendre l = 0, 1,, m = 0, 1,, l 1 z 1 m = 0 Legendre d l P l (z) = 1 l l! dz l (z 1) l, (173) P 0 (z) = 1, P 1 (z) = z, P (z) = 1 (1 3z ), (174) Legendre Legendre P m l 1 1 Pl m (z) = (1 z ) m d m P l dz m, (175) Legendre dzpl m (z)pl m (z) = δ (l + m)! ll (l + 1)(l m)!, (176) 40

14 Y l m (θ, ϕ) l m 14: Y l m 9.3 1. 0 ϕ π π 0 Φ m (ϕ) = 1 π e imϕ, dϕ Φ m(ϕ)φ m (ϕ) = δ mm,. Legendre d dz (1 z ) d dz P l(z) = l(l + 1)P l (z), l = 0, 1,, l 0, 1,, P l (z) = n=0 c nz n z = 1 ( ) 41

3. (Legendre ) d l P l (z) = 1 l l! dz l (z 1) l, l = 0, 1,, Legendre ( ) 4. k < l 1 1 dzzk P l (z) = 0 5. 1 1 dzzl P l (z) = l+1 (l!) (l+1)! 6. Legendre 1 1 dzp l (z)p l (z) = δ ll l + 1, 7. (Legendre ) Legendre Pl m (z) = (1 z ) m d m P l dz m, d dz (1 z ) d dz P l m (z) = l(l + 1)P l (z) + m 1 z P l m (z), l = 0, 1,,, m = 0, 1,, l 8. Legendre 9. 1 1 dzpl m (z)p l m (z) = δ ll Y lm (θ, ϕ) = ( 1) m+ m (l + m)! (l + 1)(l m)!, (l + 1)(l m )! (l + m )! P m l (z)φ m (ϕ), z = cos θ l = 0, 1,, m = l, l + 1,, l 1, l l = 1 r(θ, ϕ) = Y lm (θ, ϕ) (ϕ ) 10. l = r(θ, ϕ) = Y lm (θ, ϕ) (ϕ ) 11. ˆL ± ˆL x ± iˆl y = ±ħ e ±iϕ( θ ± i cos θ ) sin θ ϕ, ˆL Y 1 1 ˆL Y 1 0 ˆL Y 1 1 1. ˆL+ Y 1 1 ˆL + Y 1 0 ˆL + Y 1 1 4

10 10.1 Ze V (r) = α r, α Ze 4πϵ 0, (177) Schrodinger ħ ( r + µ r r ˆL ) ħ r ψ + V (r)ψ = Eψ, (178) E = E total E cm E < 0 ψ(r, θ, ϕ) ψ(r, θ, ϕ) = R(r)Y l m (θ, ϕ) R(r) ( d dr + r d ) { µ R(r) + (E dr ħ + α ) l(l + 1) } r r R(r) = 0, (179) r d R + µe R = 0 R exp( µe dr ħ ħ r) r 0 d R + dr dr r dr l(l+1) R = 0 r R r l ρ = 8µE r, (E < 0), (180) ħ (179) ( d dρ + d ) { R(ρ) + 1 ρ dρ 4 + α µ ħ E ρ d R dρ + ρdr dρ + { 1 4 ρ + α ħ 1 l(l + 1) } ρ ρ R(ρ) = 0, µ E ρ l(l + 1) } R = 0, (181) R(ρ) = F (ρ)ρ l e ρ, (18) F (ρ) ρ d F dρ + ( l + ρ ) df ( α µ ) dρ + ħ E l 1 F = 0, (183) 43

Laguerre α µ = n, n = 1,, 3,, l = 0, 1,, n 1, (184) ħ E E n = µα ħ 1 n = Z e 4 µ 3π ϵ 0 ħ 1 n, (185) Z = 1 µ = m e Bohr (8) (183) Laguerre Laguerre ( ) 3 (n l 1)! R nl (r) = na 0 n(n + l)! ρl e ρ L l+1 n l 1 (ρ), (186) 8 n 0 drr R(r) = 1 a 0 ħ µα, (187) Z = 1 µ = m e Bohr a 0 ρ = r na 0 ( 1 ) 3 ( R 10 = e r a 1 ) 3 ( 0, R 0 = r )e r a 0, a 0 a 0 a 0 ( 1 ) 3 r R 1 = e r a 0, (188) a 0 3a0 n = 1,, 3,, l = 0, 1,, n 1, (189) m = l, l + 1,, l 1, l, 3 Legendre Pl m L l+1 n l 1 ( r ) le ψ(r, θ, ϕ) = R nl (r)y m l (θ, ϕ) = C nlm na 0 ( ) 3 (n l 1)! m+ m C nlm = ( 1) na 0 n(n + l)! r na 0 L l+1 n l 1 ( r ) na 0 Laguerre P m (l + 1)(l m )!, 4π(l + m )! l (cos θ)e imϕ, (190) 8 mathematica L k n+k = ( 1) k (n + k)!l k n 44

C nlm n (185) l = 0, 1,, 3, 4, 5 s, p, d, f, g, h, n = 1 1s n = s p (3 ) n = 3 3s 3p (3 ) 3d (5 ) (185) n 1 (l + 1) = n, (191) l=0 Bohr Schrodinger Bohr Laguerre ρ d L k h dρ + (k + 1 ρ)dlk h dρ + hlk h = 0, (19) Laguerre h = 0, 1,, k = 0 Laguerre L h (ρ) L h (ρ) L h (ρ) = 1 dh eρ n! dρ h (e ρ ρ h ), (193) L 0 (ρ) = 1, L 1 (ρ) = 1 ρ, L (ρ) = 1 (ρ 4ρ + ), (194) Laguerre Laguerre L k h (ρ) = ( 1)k dk L h+k dρ k, (195) L k h (ρ) Laguerre 0 0 dρ e ρ ρ k L k g(ρ)l k h (ρ) = δ gh dρ e ρ ρ k+1 L k h (ρ)lk h (h + k)!, (196) h! (h + k)! (ρ) = (h + k + 1), (197) h! 45

15 16 17 ψ(r, θ, ϕ) Bohr 1 1869 Z Z 1. +. 1 1 (Pauli ) 1s + 1s 1 H He 1s s p 3s 3p 4s 3d 4p, (198) 3d 4s 15: n = 1, 46

16: n = 3 17: n = 4 47

10. 1. Laguerre ρ d L h dρ + (1 ρ)dl h dρ + hl h = 0, h = 0, 1,, h 0, 1,, L h (ρ) = a=0 c aρ a e ρ. (Laguerre ) L h (ρ) = eρ d h h! dρ h (e ρ ρ h ) = h ( 1) a hc a a! ρa, Laguerre ( ) a=0 3. g < h 0 dρ e ρ ρ g L h (ρ) = 0 4. 0 dρ e ρ ρ h L h (ρ) = ( 1) h h! 5. Laguerre 0 dρ e ρ L g (ρ)l h (ρ) = δ gh, 6. Laguerre ρ d L k h dρ + (k + 1 ρ)dlk h dρ + hlk h = 0, h = 0, 1,, h 0, 1,, L k h (ρ) = a=0 c aρ a e ρ 7. (Laguerre ) L k h (ρ) = eρ ρ k h! d h dρ h (e ρ ρ h+k ) = h a=0 ( 1) a h+kc a+k a! ρ a = ( 1) k dk L h+k dρ k, Laguerre ( ) 8. g < h 0 dρ e ρ ρ k ρ g L k h (ρ) = 0 9. 0 dρ e ρ ρ k ρ h L k h (ρ) = ( 1)h (h + k)! 10. Laguerre 0 dρ e ρ ρ k L k g(ρ)l k h (ρ) = δ gh (h + k)!, h! 48

11. 0 dρ e ρ ρ k+1 ρ h L k h (ρ) = ( 1)h (h + k + 1)!(h + 1) 1. 0 dρ e ρ ρ k+1 L k (h + k)! h (ρ)lk h (ρ) = (h + k + 1), h! 13. ( ) 3 (n l 1)! R nl (r) = na 0 n(n + l)! ρl e ρ L l+1 n l 1 (ρ), ρ = r na 0 n = 0, 1,, l = 0, 1,, n 1 n = 14. n = 3 15. n = 1, l = 0 r 49

11 1 3 11.1 Schrodinger 9 Dirac x ˆx x = x x, (199) x x x x = δ(x x ), (00) ψ ψ = dx x ψ(t, x), (01) ψ = x x ψ = dx ψ (t, x) x, (0) dx x x ψ(t, x ) = ψ(t, x), (03) x ψ x = dx ψ (t, x ) x x = ψ (t, x), (04) 9 50

dx x x ψ = dx x ψ(t, x) = ψ, (05) dx x x = 1 (06) ˆp p ˆp p = p p, p p = δ(p p ), ψ = dp p ψ(t, p), ψ = ψ(t, x) = (49) dp x p p ψ = dp p p = 1, dp ψ (t, p) p, (07) dp x p ψ(t, p), (08) x p = 1 πħ e i ħ px, (09) ϕ ψ ϕ ψ = = dx dx ϕ (t, x ) x x ψ(t, x) dx ϕ (t, x)ψ(t, x) = ψ ϕ, (10) ϕ ( λ 1 ψ 1 + λ ψ ) = λ 1 ϕ ψ 1 + λ ϕ ψ, ( λ 1 ϕ 1 + λ ϕ ) ψ = λ 1 ϕ 1 ψ + λ ϕ ψ, (11) ψ ψ = ψ = 0 dx ψ (t, x)ψ(t, x) 0, (1) 51

11. Ô(ˆx, ˆp) ˆx ˆp Ô(ˆx, ˆp) ψ = = dx x O(x, iħ x )ψ(t, x) (13) dp p O(iħ p, p) ψ(t, p), (14) ψ Ô(ˆx, ˆp) = = dx O (x, iħ x )ψ (t, x) x (15) dp O (iħ p, p) ψ (t, p) p, (16) Ô Ô Schrodinger iħ ψ = Ĥ ψ, t Ĥ = ˆp + V (ˆx), (17) m x ψ Ô O ψ = ψ Ô ψ = dx ψ (t, x)o(x, iħ x )ψ(t, x), (18) O ψ = dx O (x, iħ x )ψ (t, x)ψ(t, x) = ψ Ô ψ, (19) Ô Ô Ô = Ô ˆx ˆp Ĥ ˆp = iħ x ψ p ψ = = dx (iħ x ψ (t, x))ψ(t, x) dx ψ (t, x)( iħ x ψ(t, x)) = ψ p ψ, (0) ˆp Ô1 Ô (Ô1Ô) = Ô Ô 1, (1) 5

ψ (Ô1Ô) = = = ( ψ Ô dx (O 1 O ψ) x dx O 1(O ψ) x )Ô 1, () Ô1 Ô [Ô1, Ô] = 0, (3) m Ô1 n Ô 1 m j = m m j, j = 1,, n (4) Ô 1 (Ô m j ) = Ô Ô 1 m j = m ( Ô m j ), (5) Ô m j Ô1 m Ô m j m j Ô m j = n m i C i j, (6) i=1 C C i j n n P 1 CP = D D P ( Ô m j P j ) i = m k P k j(p 1 CP ) j ( i = d i m j P j i), (7) j j,k j Ô Ô1 Ô Ô1 Ô 11.3 Schrodinger Heisenberg Schrodinger (17) ψ(t) = e i ħ Ĥt ψ(0), (8) Ô O = ψ(t) Ô ψ(t) = ψ(0) e i ħ Ĥt Ôe i ħ Ĥt ψ(0), (9) 53

Ô H (t) = e i ħ Ĥt Ôe i ħ Ĥt, ψ H = ψ(0), (30) O = H ψ ÔH(t) ψ H, (31) Schrodinger Heisenberg Heisenberg d = i [ÔH, Ĥ], (3) dtôh ħ d dt O = {O, H} P.B. O H x p H O x p, (33) 11.4 ˆx ˆp mω 1 â = ħ ˆx + i mωħ ˆp, â = mω 1 ħ ˆx i ˆp, (34) mωħ ( Ĥ = ħω â â + 1 ), (35) â â ˆN = â â ˆN n [â, â ] = 1, (36) [ ˆN, â] = â, [ ˆN, â ] = â, (37) ˆN n = n n, (38) 54

n ( Ĥ n = ħω n + 1 ) n, (39) ħω ( n + 1 ) (37) ˆNâ n = (n 1)â n, ˆNâ n = (n + 1)â n, (40) â ˆN 1 â ˆN 1 â â â ħω â 0 (38) 0 â 0 = 0, n = 1 n! (â ) n 0, n = 0, 1,,, (41) 0 0 = 1 ϕ n (x) = x n â = 1 ( y + d ), â = 1 ( y d ) = 1 e 1 d y dy dy dy e 1 y, (4) y = mω ħ x x â 0 = 0 ϕ 0 (x) 1 ( y + d ) ϕ 0 (x) = 0, (43) dy ϕ 0 (x) = C 0 e 1 y, C 0 = ( mω πħ ) 1 4, (44) C 0 dxϕ 0 (x) = 1 ϕ n (x) = 1 n! x (â ) n 0 n ϕ n (x) ϕ n (x) = 1 ( 1 e 1 d y n! dy e 1 y) n ϕ0 (x) = 1 n n! C 0( 1) n e 1 y dn dy n e y, (45) (117) 11.5 11.5 (160) ˆL ˆL z [ˆL, ˆL z ] = [ˆL x, ˆL z ] + [ˆL y, ˆL z ] = ˆL x [ˆL x, ˆL z ] + [ˆL x, ˆL z ]ˆL x + ˆL y [ˆL y, ˆL z ] + [ˆL y, ˆL z ]ˆL y = 0, (46) 55

[ˆL, ˆL x ] = [ˆL, ˆL y ] = 0 ˆL ˆL z l, m ˆL l, m = ħ l(l + 1) l, m, ˆLz l, m = mħ l, m (47) ˆL ± = ˆL x ± iˆl y, (48) ˆL = 1 (ˆL + ˆL + ˆL ˆL+ ) + ˆL z, [ˆL z, ˆL ± ] = ±ħˆl ±, [ˆL +, ˆL ] = ħˆl z, (49) ˆL z ˆL± l, m = (ˆL ± ˆLz ± ħˆl ± ) l, m = ħ(m ± 1)ˆL ± l, m ˆL + ˆL z ħ ˆL ˆL z ħ ˆL + l, m = c m+1 l, m + 1, ˆL l, m = d m l, m 1, (50) c m d m m ˆL + l, m H = 0 ( ) (50) (49) 1 d m H = l, m H (ˆL + ˆL + ˆL ˆL+ ) l, m H = ħ l(l + 1) ħ m H, (51) (50) (49) 3 (51) (5) d m H = l, m H [ˆL +, ˆL ] l, m H = ħ m H, (5) m H = l, d m H = lħ, (53) ˆL l, m L = 0 (50) 1 (49) 1 c (m L +1) = l, m L (ˆL + ˆL + ˆL ˆL+ ) l, m L = ħ l(l + 1) ħ m L, (54) (50) 1 (49) 3 (54) (55) c (m L +1) = l, m L [ˆL +, ˆL ] l, m L = ħ m L, (55) m L = l, c m L = lħ, (56) 56

m H m L = l l = 0, 1, 1, 3,,, (57) l = 0, 1,, m = l, l + 1,, l 1, l Y l m l = 1, 3, Yl m (x) = x l, m ˆL + = ħ e iϕ( θ + i cos θ ) sin θ ϕ, ˆL = ħ e iϕ( θ i cos θ ) sin θ ϕ, ˆLz = iħ ϕ, (58) x ˆL z l, l = lħ x l, l x ˆL l, l = 0 Y l l (θ, ϕ) iħ ϕ Y l l (θ, ϕ) = lħy l l (θ, ϕ), ) ħ e iϕ( θ i cos θ sin θ ϕ Y l l Y l l (θ, ϕ) = (θ, ϕ) = ħ e iϕ( θ l cos θ sin θ (l + 1)! 4π ) Y l l (θ, ϕ) = 0, (59) 1 l l! sinl θ e ilϕ, (60) dθdϕ sin θ Y l l = 1 l, m = 1 c m c m 1 c l+1 (ˆL + ) l+m l, l = 0 m Yl m Y m l (θ, ϕ) = = = (l + 1)(l m)! 4π(l + m)! (l + 1)(l m)! 4π(l + m)! (l + 1)(l m)! 4π(l + m)! = ( 1) m 1 l l! 1 l l! (l + 1)(l m)! 4π(l + m)! (θ, ϕ) m 1 a= l m 1 a= l 1 ( 1 l l! sinm θ sin θ (l m)! ( ) ˆL+ l+m l, l, (61) (l + m)!(l)! ħ ( d dθ acos θ ) sin l θ e imϕ sin θ ( sin a θ d ) dθ sin a θ sin l θ e imϕ 1 l l! (1 z ) m d ) l+m sin l θ e imϕ dθ d l+m dz l+m (z 1) l e imϕ, (6) a z = cos θ (170) m < 0 (170) 57

c m d m c m d m (50) (49) 1 c m+1 + d m = l, m (ˆL ˆL+ + ˆL + ˆL )l, m = ħ { l(l + 1) m }, (63) (50) (49) 3 c m+1 + d m = l, m [ˆL +, ˆL ] l, m = ħ m, (64) c m = d m = ħ (l + m)(l m + 1), (65) m l l + 1 m 0 11.6 1.. 3. 4. V (r) r 5. n â = 1 (y d dy ) â = 1 (y + d dy ) y = mω ħ x x â 0 = 0 ϕ 0 (x) = x 0 6. 1 = â 0 1 ϕ 1 (x) = x 1 7. n = 1 n! (â ) n 0 n ϕ n (x) = x n ( â = 1 (y d dy ) = 1 e 1 y d dy e 1 y ) 8. l, m x ˆL 0, 0 = 0 x ˆL z 0, 0 = 0 Y0 0 (x) = x 0, 0 9. x ˆL 1, 1 = 0 x ˆL z 1, 1 = ħ x 1, 1 Y 1 1 (x) = x 1, 1 10. x ˆL l, l = 0 x ˆL z l, l = lħ x l, l Y l l (x) = x l, l 58

1 1.1 Noether Noether ( ) 3 ψ(t, x) Schrodinger ψ (t, x) = ψ(t, x) + iθδψ(t, x), (66) θ ψ (t, x) Schrodinger δψ(t, x) Shorodinger iħ ħ δψ(t, x) = t m δψ(t, x) + V (x)δψ(t, x), (67) j t (t, x) ψ δψ, (68) t j t = ( t ψ )δψ + ψ t δψ = iħ { ( ψ )δψ + ψ δψ } m = iħ m {ψ δψ ( ψ )δψ } = j (69) j = iħ { ψ δψ ( ψ )δψ }, (70) m Q = d 3 x j t (t, x), (71) dq dt = d 3 x t j t (t, x) = d 3 x j = 0, (7) 59

1. ψ(t, x) Schrodinger ψ (t, x) = e iθ ψ(t, x) ψ(t, x) + iθψ(t, x), (73) δψ(t, x) = ψ(t, x) Q = d 3 x ψ(t, x), (74) 1.3 t ψ(t, x) t = t + ϵ ψ (t, x) ψ (t, x) = ψ(t, x) ψ (t, x) ψ(t, x) ϵ t ψ(t, x), (75) ϵ ħϵ δψ(t, x) = iħ t ψ(t, x), (76) E = d 3 x ψ(t, x) ( ) iħ t ψ(t, x), (77) 1.4 x i (i = 1,, 3) ψ(t, x) x i = x i a i ψ (t, x ) a i ψ (t, x ) = ψ(t, x) ψ (t, x) ψ(t, x) + a i i ψ(t, x), (78) a i ħa i a i δ i ψ δ i ψ(t, x) = iħ i ψ(t, x), (79) P i = d 3 x ψ(t, x) ( ) iħ i ψ(t, x), (80) 60

1.5 R R ( ) ( ) cos θ sin θ R = = e ix 0 i, X = θ, (81) sin θ cos θ i 0 3 3 3 R = e ix R 1 3 = R T R = 1 3 i(x T + X) + X T = X X X X = θ i X i = θ 1 X 1 + θ X + θ 3 X 3, (8) 0 0 0 0 0 i 0 i 0 X 1 = 0 0 i, X = 0 0 0, X 3 = i 0 0, (83) 0 i 0 i 0 0 0 0 0 X i Baker-Campbell-Hausdorff e ix e iy = e i(x+y ) 1 [X,Y ]+, (84) X i [X i, X j ] = iϵ ijk X k, (85) (Lie ) ϵ ijk 10 (160) (X i ) jk = iϵ ijk θ i x i = R i jx j x i ϵ i jkθ k x j, (86) x i (i = 1,, 3) ψ(t, x) x i ψ (t, x ) ψ (t, x ) = ψ(t, x) ψ (t, x ) ψ(t, x) + ϵ i jkθ k x j i ψ(t, x), (87) θ k ħθ k θ k δ k ψ(t, x) δ k ψ(t, x) = iħϵ kij x i j ψ(t, x), (88) L i = d 3 x ψ(t, x) ( iħϵ ijk x j k) ψ(t, x), (89) 10 ϵ 13 = ϵ 31 = ϵ 31 = ϵ 13 = ϵ 31 = ϵ 13 = 1 0 61

1.6 1. δψ. δψ 3. δψ 4. Baker-Campbell-Hausdorff e X e Y = e X+Y +[X,Y ]+ [X, Y ] 6

13 Faraday Faraday Zeeman Zeeman 1 3 13.1 ψ(t, x) ψ (t, x) = e iθ ψ(t, x) e iθ iħ t ψ + ħ m ψ V (x)ψ = e iθ( iħ t ψ + ħ ) m ψ V (x)ψ = 0, (90) ψ (t, x) ψ (t, x) = e iθ(t,x) ψ(t, x), (91) ψ (t, x) Shrodinger 11 x µ = (ct, x i ), µ = (c 1 t, i ), µ = 0, 1,, 3, (9) c (91) µ ψ (x) = e iθ(x) µ ψ(x) + i( µ θ)e iθ(x) ψ(x), (93) D µ = µ i e ħ A µ(x), (94) A µ (x) SI [V s/m] (93) A µ(x) = A µ (x) + ħ e µθ(x), (95) 11 1 63

D µ ψ (x) = e iθ(x) D µ ψ(x), (96) Schrodinger iħcd 0 ψ(x) = ħ m D id i ψ(x) + V (x)ψ(x), (97) U(1) Ĥ = ħ ( i i e )( m ħ A i i i e ħ Ai) + V (x) eca 0, (98) A µ (x) A µ (x) ϕ(x) A i (x) A µ (x) = (c 1 ϕ(x), A i (x)), (99) F µν = µ A ν ν A µ, (300) E i = cf i0, B i = 1 ϵijk F jk, (301) 13. µ m e z B A µ (x) = (0, yb, xb ), 0, (30) (98) Ĥ = ħ {( x + i eby m e ħ = ħ m e { i eb ħ ) ( + y i ebx ) } + ħ z + V (x) ( ) e B (x + y ) } x y y x 4ħ + V (x) = ħ m e + V (x) B eˆl z m e + e B (x + y ) 8m e, (303) 64

B Ĥ = ħ m e + V (x) B ˆµ z, ˆµ z = eħ m e ˆLz ħ, (304) ˆµ z ˆµ z ˆµ Ĥ = ħ m e + V (x) B ˆµ, ˆµ = eħ m e ˆL ħ, (305) µ B = eħ/m e 1 (304) (190) E = E n eħb m e m, m = l, l + 1,, l 1, l, (306) n l (l + 1) Faraday Zeeman D 13.3 1. m q L = m ẋ V (t, x), V (t, x) = qa µ dx µ dt = q dx ϕ(t, x) + qa(t, x) c dt, : da i A i t + dxj dt ja i dt =. ˆp = iħ 3. A = (0, Bx, 0) B = (0, 0, B) ( ) 4. z (x, y) 5. 6. A = 1 x B B 7. 1 65

14 14.1 Stern-Gerlach Schirodinger 19 Stern Gerlach ( 18 ) 47 47 46 1s 4d 0 5s 0 ˆµ 0 (306) m = 1, + 1 18: Stern-Gerlach (https://commons.wikimedia.org/wiki/file: SternGerlach.jpg ) 14. (160) Ĵ i [Ĵ i, Ĵ j ] = iħϵ ij kĵ k, i, j, k = x, y, z, (307) 66

11.5 Ĵ j, j z = ħ j(j + 1) j, j z, Ĵ z j, j z = j z ħ j, j z, j = 0, 1, 1, 3,,, j z = j, j + 1,, j 1, j, (308) j (j + 1) j (j + 1) j Stern-Gerlach j = 1 ħ Lie (85) 3 3 (83) 1 3 3 : R = e iθi X i, (X i ) j k = iϵ i j k, (309) Pauli ( ) ( ) 0 1 0 i σ 1 =, σ =, σ 3 = 1 0 i 0 ( ) 1 0, (310) 0 1 [ σi, σ ] j σ k = iϵ ijk, (311) σ i X i : Σ = e iθi σ i, σi :, (31) 3 Σ 1 σ i Σ = R i jσ j. (313) M i (t) Σ t σ i Σ t t dm i (t) dt = Σ t i θ k[σ k, σ i ]Σ t = θ k ϵ ki jm j (t) = ( iθ k X k ) i jm j (t), (314) M i (t) = (e itθk X k ) i jm j (0) = (R t ) i jσ j t = 1 (313) 3 3 ψ α (x)(α = 1, ) ψ α (x) x i = R i jx j, ψ α (x ) = Σ α βψ β (x), (315) 67

ψ σ i ψ ψ σ i i ψ x i = R i jx j x i ϵ i jkθ k x j ψ α ψ α (x ) ψ α (x) ϵ i jkθ k x j i ψ α (x) (316) = Σ α βψ β (x) ψ α (x) i θk (σ k ) α βψ β (x), θ k ħθ k δ k ψ α (x) = iħϵ kij x i j ψ α (x) + ħ (σ k) α βψ β (x), (317) J i = L i + S i, (318) L i S i L i = d 3 x ψ α (x) ( iħϵ ijk x j k) ψ α (x), S i = d 3 x ψ α (x) ħ (σ i) α βψ β (x), (319) α, β ( ) ψ α ψ + (x) (x) = ψ, (30) (x) ψ + (x) 0 0 S = 0, (31) ψ (x) 0 0 S = 0, (3) + ħ ħ (305) ˆµ = eħ ( ˆL m e ħ + g Ŝ ) e, (33) ħ 68

g e g g e g e Dirac g e = Stern-Gerlach 0 ± ħ 14.3 Ŝi = ħ σi (i = 1,, 3) σ i 1. Ŝ. Ŝ 3 ± ħ ± ± 3. Ŝ ± = Ŝ1 ± iŝ Ŝ+ + Ŝ+ 4. Ŝ + Ŝ 5. Stern-Gerlach z Ĥ = ˆp Ŝz m kz ħ k = g e µ B B ψ = (ψ +, ψ ) Ehrenfest 6. Ehrenfest x 0 x L 69

A SI c = 9979458 m s 1 h = 6.6606957(9) 10 34 J s e = 1.60176565(35) 10 19 C ħc = 197.369718(44) MeV fm m e = 0.51099898(11) MeV/c = 9.1093891(40) 10 31 kg m p = 938.7046(1) MeV/c = 1.6761777(74) 10 7 kg a 0 = 4πϵ 0 ħ /(m e e ) = 0.59177109(17) 10 10 m k = 1.3806488(13) 10 3 J K 1 = 8.617334(78) 10 5 ev K 1 µ B = eħ/(m e ) = 5.788381801(6) 10 11 MeV/T 015 4 http://pdg.lbl.gov/ B Bloch 1 V (x) = V (x a) ϕ(x) 1 ϕ(x) = m(e V (x)) ϕ(x) dx ħ = m (E V (x a)) ħ 1 d = ϕ(x a), (34) ϕ(x a) dx d 0 = d { ϕ(x a) dϕ(x) dx dx a) } ϕ(x)dϕ(x, (35) dx C = ϕ(x a) dϕ(x) dx dϕ(x + a) = ϕ(x) dx ϕ(x)dϕ(x a) dx ϕ(x + a) dϕ(x) dx, (36) 1 d(ϕ(x + a) + ϕ(x a)) = 1 dϕ(x) ϕ(x + a) + ϕ(x a) dx ϕ(x) dx, (37) 70

ϕ(x + a) + ϕ(x a) = Dϕ(x) ( ϕ(x + a) λ ϕ(x) = λ + ϕ(x) λ ϕ(x a) ), λ ± D ± D 4, φ(x) = λ + φ(x a), φ(x) ϕ(x + a) λ ϕ(x), (38) D φ(x) φ(x) = λ n +φ(x na) λ + = 1 λ + = e iθ Bloch φ(x) = e iθ φ(x a), (39) C 3 ( ) ds = dr + r dθ + r sin θdϕ, (330) g ij 3 3 1 0 0 1 0 0 g ij = 0 r 0, g ij = 1 0 0 0 0 r sin r, (331) 1 θ 0 0 r sin θ i, j = r, θ, ϕ g ij g ij g g = r sin θ = 1 g i ( gg ij j ), (33) = 1 { ( 1 r r (r sin θ r ) + θ (sin θ θ ) + ϕ sin θ = r + r r + 1 r θ + cos θ r sin θ θ + )} sin θ ϕ 1 r sin θ ϕ. (333) 71

D Hermite Hermite d H n dy y dh n dy + nh n = 0, (334) H n (y) = a=0 c ay a 0 = a(a 1)c a y a ac a y a + n c a y a = a= a=1 { } (a + 1)(a + )ca+ + (n a)c a y a, (335) a=0 a=0 (n a) c a+ = (a + 1)(a + ) c a, (336) n 0, 1,, a c a+ a c a H n e y n = 0, 1,, Hermite Hermite Hermite dn H n (y) = ( 1) n e y dy n (e y ), (337) yh n+1 = (n + 1)H n + H n+, dh n dy = yh n H n+1, (338) dye y y m H n (y) = 0, (m < n), dye y y n H n (y) = πn!, (339) dyh m (y)h n (y)e y = δ mn n πn!, (340) 7

E Legendre E.1 Legendre Legendre (1 z ) d P l dz z dp l dz + l(l + 1)P l = 0, (341) P l (z) = a=0 c az a 0 = a(a 1)c a (z a z a ) ac a z a + l(l + 1) c a z a = a=0 a=0 { } (a + 1)(a + )ca+ + (l a)(l + a + 1)c a z a, (34) a=0 c a+ = a=0 (l a)(l + a + 1) c a, (343) (a + 1)(a + ) l 0, 1,, a c a+ c a P l 1 1 z l = 0, 1,, Legendre Legendre d l P l (z) = 1 l l! dz l (z 1) l, l = 0, 1,, (344) z dl+1 dl+1 dl = z + (l + 1) dzl+1 dzl+1 dz l, (z 1) dl+ dl+ dl+1 = (z + 1) (z 1) + (l + )(z + 1) dzl+ dzl+ dz l+1 (345) = dl+ dz l+ (z 1) + (l + ) dl+1 dl z (l + 1)(l + ) dzl+1 dz l, Legendre P l (z) = (l)! l (l!) z l + 1 1 1 1 dzz k P l (z) = 0, dzz l P l (z) = 1 l 1 Legendre 1 1 (k < l), 1 dz(1 z ) l = l+1 (l!) (l + 1)!, (346) dzp l (z)p l (z) = δ ll l + 1, (347) 73

E. Legendre Legendre (1 z ) d Pl m dz z dp l m ( dz + l(l + 1) m ) 1 z Pl m = 0, (348) Legendre Legendre Pl m (z) = (1 z ) m d m P l dz m, (349) Legendre (345) d dz (1 z ) dp l m ( dz + l(l + 1) { mz(1 z ) m d m P l = d dz = (1 z ) m = (1 z ) m = (1 z ) m m ) 1 z Pl m + (1 z ) m +1 dm+1 P } l dz m+1 dz m + (l(l + 1) m ) 1 z (1 z ) m d m P l dz m { m dm P l dz m + z d m P l m 1 z dz m (m + 1)z dm+1 P l dz m+1 + (1 z ) dm+ P ( l dz m+ + l(l + 1) m ) d m P } l 1 z dz m { (1 z ) dm+ P l (m + 1)z dm+1 P l dz m+ dz m+1 + ( l(l + 1) m(m + 1) ) d m P } l dz m d m { dz m (1 z ) d P l dz z dp l dz + l(l + 1)P l }, (350) P l Legendre (348) P m l z = 1 l = 0, 1,, m = 0, 1,,, l Legendre dm P l dz = (l)! m l l!(l m)! zl m + Legendre 1 1 1 dzpl m (z)p l m (z) = dz(1 z ) m dm P l 1 dz m (l)! 1 d m P l dz m = δ ll l dz( 1) m z l+m dm P l l!(l m)! 1 dz m (l + m)! = δ ll (l + 1)(l m)!, (351) 74

F Laguerre F.1 Laguerre Laguerre ρ d L h dρ + (1 ρ)dl h dρ + hl h = 0, (35) L h (ρ) = a=0 c aρ a 0 = a(a 1)c a ρ a 1 + ac a (ρ a 1 ρ a ) + h c a ρ a = a=1 a=1 { (a + 1) } c a+1 + (h a)c a ρ a, (353) a=0 a=0 c a+1 = h a (a + 1) c a, (354) h 0, 1,, a c a+1 1 a c a L h e ρ h = 0, 1,, c a = ( 1) a h! (a!) (h a)! c 0 = ( 1) a hc a a!, (355) c 0 = 1 h = 0, 1,, Laguerre Laguerre L h (ρ) = eρ d h h! dρ h (e ρ ρ h ) = h ( 1) a hc a a! ρa, (356) Laguerre L h (ρ) = ( 1)h h! ρ h + 0 a=0 dρ e ρ ρ g L h (ρ) = 0, Laguerre 0 0 (g < h), dρ e ρ ρ h L h (ρ) = ( 1) h h!, (357) dρ e ρ L g (ρ)l h (ρ) = δ gh, (358) 75

F. Laguerre Laguerre ρ d L k h dρ + (k + 1 ρ)dlk h dρ + hlk h = 0, (359) L k h (ρ) = a=0 c aρ a 0 = a(a 1)c a ρ a 1 { + ac a (k + 1)ρ a 1 ρ a} + h c a ρ a = a=1 a=1 { } (a + 1)(a + k + 1)ca+1 + (h a)c a ρ a, (360) a=0 a=0 h a c a+1 = (a + 1)(a + k + 1) c a, (361) h 0, 1,, a c a+1 1 a c a L k h eρ h = 0, 1,, c a = ( 1) a k!h! a!(a + k)!(h a)! c 0 = ( 1) a (h + k)! a!(a + k)!(h a)! = h+kc a+k ( 1)a, (36) a! c 0 = h+k C h h = 0, 1,, Laguerre Laguerre L k h (ρ) = eρ ρ k h! d h dρ h (e ρ ρ h+k ) = h a=0 ( 1) a h+kc a+k a! ρ a = ( 1) k dk L h+k dρ k, (363) Laguerre 0 0 dρ e ρ ρ k ρ g L k h (ρ) = 0, Laguerre 0 0 0 (g < h), dρ e ρ ρ k ρ h L k h (ρ) = ( 1)h (h + k)!, (364) dρ e ρ ρ k L k g(ρ)l k h (ρ) = δ gh dρ e ρ ρ k+1 ρ h L k h (ρ) = ( 1)h (h + k + 1)!(h + 1), dρ e ρ ρ k+1 L k h (ρ)lk h (h + k)!, (365) h! (h + k)! (ρ) = (h + k + 1), (366) h! 76

G 3 Schrodinger (179) α = 0 ( d dρ + d ) { l(l + 1) } µe R(ρ) + 1 ρ dρ ρ R(ρ) = 0, ρ r, (367) ħ Bessel Neumann 13 Bessel R l (r) = j l (ρ), (368) G.1 Bessel (367) ρ = 0 j l (ρ) = ( 1) l ρ l( 1 d ) l sin ρ ρ dρ ρ, (369) Bessel (369) Bessel (367) [ ρ, 1 d ] =, ρ dρ ( 1 d (ρ ( 1 d ) sin ρ sin ρ) = ρ dρ) ρ dρ ρ sin ρ ρ, (370) Bessel j l+ (ρ) = ( 1) l ρ l ρ ( 1 d ) l+ sin ρ ρ dρ ρ = ( 1) l ρ l( 1 d ) l+(ρ sin ρ) ( 1) l (l + )ρ l( 1 d ) l+1 sin ρ ρ dρ ρ dρ ρ = 1 ρ j (l + ) l+1(ρ) j l (ρ) + j l+1 (ρ) ρ = l + 3 j l+1 (ρ) j l (ρ), (371) ρ Bessel 1 d ρ dρ j l(ρ) = l ρ j l(ρ) 1 ρ j l+1(ρ), ρ d 1 d dρ ρ dρ j l(ρ) = l ρ j l(ρ) + l d ρ dρ j l(ρ) + 1 ρ j l+1(ρ) d dρ j l+1(ρ) (37) = j l+ (ρ) l ρ j l+1(ρ) + l l ρ j l (ρ), 13 3 Bessel Neumann Hankel 77

(371) ( d dρ + d ) j l (ρ) = j l+ (ρ) l + 3 j l+1 (ρ) + ρ dρ ρ = j l (ρ) + l(l + 1) ρ j l (ρ) l(l + 1) ρ j l (ρ), (373) Bessel (367) G. Neumann (367) ρ = 0 n l (ρ) = ( 1) l ρ l( 1 d ) l cos ρ ρ dρ ρ, (374) Neumann (374) Neumann (367) [ ρ, 1 ρ d ] =, dρ ( 1 d (ρ ( 1 d ) cos ρ cos ρ) = ρ dρ) ρ dρ ρ cos ρ ρ, (375) Neumann n l+ (ρ) = ( 1) l ρ l ρ ( 1 d ) l+ cos ρ ρ dρ ρ = ( 1) l ρ l( 1 d ) l+(ρ cos ρ) + ( 1) l (l + )ρ l( 1 d ) l+1 cos ρ ρ dρ ρ dρ ρ = 1 ρ n (l + ) l+1(ρ) n l (ρ) + n l+1 (ρ) ρ = l + 3 n l+1 (ρ) n l (ρ), (376) ρ Neumann 1 d ρ dρ n l(ρ) = l ρ n l(ρ) 1 ρ n l+1(ρ), ρ d 1 d dρ ρ dρ n l(ρ) = l ρ n l(ρ) + l d ρ dρ n l(ρ) + 1 ρ n l+1(ρ) d dρ n l+1(ρ) (377) = n l+ (ρ) l ρ n l+1(ρ) + l l ρ n l (ρ), (376) ( d dρ + d ) n l (ρ) = n l+ (ρ) l + 3 n l+1 (ρ) + ρ dρ ρ = n l (ρ) + l(l + 1) ρ n l (ρ) l(l + 1) ρ n l (ρ), (378) Neumann (367) 78

G.3 Hankel Hankel Bessel Neumann h (1) l (ρ) j l (ρ) + in l (ρ), h () l (ρ) j l (ρ) in l (ρ), (379) 79

H 3 H.1 3 3 ( ˆp Ĥ = i m + mω ) ˆx i, (380) i=1 Ψ(x i ) 3 ( ħ m i + mω ) x i Ψ = EΨ, (381) dx i i=1 Ψ(x i ) = ϕ 1 (x 1 )ϕ (x )ϕ 3 (x 3 ) ( ħ d + mω ) m x i ϕ i (x i ) = E i ϕ i (x i ), i = 1,, 3, (38) E = E 1 + E + E 3 ϕ i (x i ) 1 ϕ i (x i ) = B n ini! H n i (Ax i )e A ( mω x i, A ħ ) 1, B ( mω πħ ) 1 4, (383) ( E n = ħω n + 3 ), n = n 1 + n + n 3, n i = 0, 1,, 3,, (384) (n 1, n, n 3 ) E 0 = 3 ħω Ψ (0,0,0) 1 E 1 = 5 ħω Ψ (1,0,0), Ψ (0,1,0), Ψ (0,0,1) 3 E n n 3 n+c = (n + 1)(n + ), (385) Ψ (0,0,0) = B 3 e A r = 4πB 3 e A r Y 0 0, (386) l = 0 H 1 (Ax) = Ax 1 Ψ (1,0,0) = AB 3 re A 4π r sin θ cos ϕ = 3 AB3 re A r ( Y 1 1 + Y1 1 ), Ψ (0,1,0) = AB 3 re A 4π r sin θ sin ϕ = i 3 AB3 re A r (Y 1 1 + Y1 1 ), (387) Ψ (0,0,1) = AB 3 re A 8π r cos θ = 3 AB3 re A r Y 0 1, 80

1 l = 1 H (Ax) = 4A x Ψ (1,1,0) = A B 3 r e A 8π r sin θ cos ϕ sin ϕ = i 15 A B 3 r e A r ( Y + Y ), Ψ (0,1,1) = A B 3 r e A 8π r sin θ cos θ sin ϕ = i 15 A B 3 r e A r (Y 1 + Y 1 ), Ψ (1,0,1) = A B 3 r e A 8π r sin θ cos θ cos ϕ = 15 A B 3 r e A r ( Y 1 + Y 1 ), (388) Ψ (,0,0) = 1 B 3 e { A r A r sin θ(cos ϕ + 1) 1 } { = B 3 e A 4π r 15 A r (Y + Y ) 1 8π 3 5 A r Y 0 + ( } π 3 A r 0 1) Y 0, Ψ (0,,0) = 1 B 3 e { A r A r sin θ( cos ϕ + 1) 1 } = B 3 e A 4π { r 15 A r (Y + Y ) 1 8π 3 5 A r Y 0 + ( } π 3 A r 0 1) Y 0, Ψ (0,0,) = 1 B 3 e A r (A r cos θ 1) { = B 3 e A 8π r 3 5 A r Y 0 + ( } π 3 A r 0 1) Y 0, l = 0 l = H. Ψ(r, θ, ϕ) = R(r)Y lm (θ, ϕ) R(r) ( d dr + r d ) { m ( R(r) + dr ħ E mω r) l(l + 1) } r R(r) = 0, (389) r R e kr r r 4k m ω ħ = 0 R exp( mω ħ r ) r 0 R r a r r a(a + 1) l(l + 1) = 0 R r l R(ρ) = F (ρ)ρ l e ρ, ρ mω ħ r = A r, (390) F (ρ) ρ d F ( dρ + l + 3 ) df ( E ρ dρ + ħω l + 3 ) F = 0, (391) 4 81

F (ρ) = a=0 c aρ a a n r c a+1 = (a + 1)(a + l + 3 )c a, n r E ħω l + 3, (39) 4 n r = 0, 1,,, F (ρ) n r r n r ( E n = ħω n + 3 ), n = n r + l, (393) n l = 0,, 4,, n n+ 1 + 5 + 9 + + (n + 1) = n + n + = (n + 1)(n + ), (394) n l = 1, 3, 5,, n n+1 3 + 7 + 11 + + (n + 1) = n + 4 n + 1 = (n + 1)(n + ), (395) E n (n+1)(n+) (n, l) = (0, 0) (39) (n, l) = (1, 1) (39) (n, l) = (, ) (39) (n, l) = (, 0) R(r) = c 0 e A r, (396) R(r) = c 0 Are A r, (397) R(r) = c 0 A r e A r, (398) ( R(r) = c 0 3 A r + 1 )e A r, (399) 8