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1 09- B, C

2 B ( ) 3 4 WKB I C ( B ) 7 II 8

3 LS 9 Hartree-Fock 0 Born-Oppenheimer II S Tomonaga-Schwinger 3 4 Bell EPR(Einstein-Podolsky-Rozen) 3

4 E = p c + m c 4 mc + m p + O(mc ( p mc )4 ) () mc p mc p () (3) E pc E mc = m p = E NR (4) Heisenberg x p h t da dt = {A, H} {, } [, ] i h [x, p] = i h i) Parisi-Wu ii) Nelson 4

5 Stefan-Boltzmann U = 8π (E + H ) ν ν + dν : u ν dν U = U = σt 4 0 u ν dν σ = erg/(cm 3 K 4 ) Rayleigh-Jeans ν T u ν dν = 8π c 3 kt ν dν k Boltzmann k = R N (erg K ) Wien Rayleigh-Jeans Wien ν T u ν (ν, T ) = ν 3 f( ν T ) f( ν T ) = 8π c 3 k T ν f( ν T ) = α βν e T c α β. Planck 900 f( ν T ) u ν = 8πhν3 c 3 e hν kt 8πk T ν for c 3 ν 8πh exp ( ) hν c 3 kt for ν T k h T k h α = 8π c h, β = h k h Planck Planck h (erg s) 5

6 ϵ = nhν (n = 0,,, ) Boltzmann Planck ( ) < ϵ > = n=0 nhνe hν kt n n=0 e hν kt n (5) Z (/kt ) = Z hν = e hν kt (6) (7). Einstein 905 Planck Lenard ν ν th ν ϵ = hν E = hν W, W : 3. Bohr 93 pdq = nh 4. de Broglie 94 Bohr E = hν = cp, p = hν c = h λ 6

7 E = hν, p = h λ de Broglie E E = hω ω p p = hk k h = h π k = π λ Young Ψ(x) Ψ a + Ψ b = Ψ a + Ψ b +ReΨ aψ b Ψ a + Ψ b 7

8 Ψ de Broglie p = (E = ) p, E Ψ(r, t) = C exp( ī (p r Et)), h C Schrödinger (i) a, b, c, Ψ abc Ψ a, Ψ b, Ψ c, p a p b a) Ψ abc = Ψ a + Ψ b + Ψ c + Ψ a (r, t) = C a e ī h (pa r Eat) Ψ b (r, t) = C b e ī h (p b r E b t) Ψ a or Ψ b b) Ψ ab (r, t) = C a (t)e ī h (pa r Eat) + C b (t)e ī h (p b r E b t) = B a (t)e ī h p a r + B b (t)e ī h p b r Ψ(r, t) = p B p (t)e ī h p r Ψ(r, t) = (π h) 3 A(p, t)e ī h p r d 3 p A(p, t) = (π h) 3 Ψ(r, t)e ī h p r d 3 r 8

9 Ã(p, E ) = (π h) Ψ(r, t)e ī h (p r E t) d 3 rdt Ψ(r, t) = Ã(p, E )e i h (p r E t) d 3 p de (π h) 4 E p (ii)schrödinger E = m p (π h) 4 (E m p )Ã(p, E ) = 0 (E m p )Ã(p, E )e ī h (p r E t) d 3 p de = 0 E h i t, m p m ( h i ) i h h Ψ(r, t) = t m Ψ(r, t) Schrödinger Ã(p, E ) = π hδ(e m p )a(p ) Ψ(r, t) = (π h) 3 a(p )e ī h (p r m p t) d 3 p i h t Ψ(r, t) = Ĥ0Ψ(r, t) Ĥ 0 = m ˆp, ˆp = i h, 9

10 [x, ˆp x ] = i h(x x x) = i h x [y, ˆp x ] = i h(y y x y) = 0 [, ]/i h H = m p + V (r) Ĥ = m ˆp + V (r) Schrödinger i h Ψ(r, t) = ĤΨ(r, t) t i h h Ψ(r, t) = ( t m + V (r))ψ(r, t) { Schrödinger 3 Born (i) =Schrödinger (ii) (iii) (ii) i h t Ψ = h m Ψ + V (r)ψ () i h t Ψ = h m Ψ + V (r)ψ () () Ψ () Ψ i h t Ψ = h m (Ψ Ψ ( Ψ )Ψ) = h m (Ψ Ψ ( Ψ )Ψ) 0

11 (iii) ρ t + j = 0 ρ = Ψ, j = h im (Ψ Ψ ( Ψ )Ψ) ρ(r, t)d 3 r = v ph = E p = p ρ = Ψ (a) micro macro (b) H. Born ρ = Ψ p m ρ = Ψ(r, t) Parseval Ψ(r, t) d 3 r = A(p, t) d 3 p A(p, t) t r d 3 r = dxdydz Ψ(r, t) d 3 r t p d 3 p = dp x dp y dp z A(p, t) d 3 p

12 Ψ(r, t) d 3 r = A(p, t) d 3 p = Ω P (Ω) P (Ω) = Ψ(r, t) d 3 r Ω n Ω, Ω, Ω n N N Ψ(r, t) N Ω i N i ( n i= N i = N) w i = N i N Ω i Ψ(r, t) d 3 r, N x ni= x i N i < x >= x Ψ(r, t) d 3 r N n p ni= p xi N i (p) < p x >= p x A(p, t) d 3 p N(p) < F >= Ψ (r, t)f (r, i h )Ψ(r, t)d 3 r = A (p, t)f (i h p, p)a(p, t)d 3 p Ψ

13 N. Bohr A. Einstein ρ(r, t) = Ψ(r, t) div j = 0 Schrödinger i h Ψ t = ĤΨ, Ĥ t = 0 Ψ(r, t) = u(r)χ(t) { Ĥu = Eu i h dχ = Eχ χ dt e ī h Et E Ĥ E u E (u, Ĥu) = E(u, u) E = (u, Ĥu) (u, u), E = (u, Ĥu) (u, u) = (Ĥu, u) (u, u) E = (u, Ĥ u) (u, u) Ĥ = Ĥ E = E Ĥu ν = E ν u ν (degenerate) E ν Ψ ν (r, t) = e ī h E νt u ν (r) ρ(r, t) = Ψ ν (r, t) = u ν (r) Schrödinger 3

14 A ν (p, t) = e ī h Eνt a ν (p) a ν (p) = u ν (r)e ī h (p r) d 3 r (π h) 3 ρ ν (r) j ν (r) ρ ν t = 0, j ν = 0 S j ν dσ = 0 u ν (r) u ν (r) δ j u ν (r) : j ν = 0 Ψ(r, t) = ν Ψ(r, t) = c ν u ν (r)e ī h Eνt c ν u ν (r) + ν ν c νc ν u ν(r)u ν (r)e i h (E ν E ν )t t h E t E h 4

15 u p = Ce ik r, k = p h (i) ρ p (r) = C, j p (r) = p m C u p (x + L, y, z) = u p (x, y, z) k x L = πn x p x = π h L n x, n x = 0, ±, ±, p y = π h L n y, n y = 0, ±, ±, p z = π h L n z, n z = 0, ±, ±, L L u p (r) dxdydz = L 3 C = C = L 3 u p = L 3 e ik r, k = π L n, n x, n y, n z = 0, ±, ±, (u p, u p ) = l/ d 3 re i π L 3 L (n n ) r L/ (8) = δ p,p (9) 5

16 (ii)δ () L L u p (r)u p(r)dxdydz (0) L = Cp C p e ī L h (px p x)x dx e ī L h (py p y)y dy e ī h (pz p z)z dz () L L L L C p (π h) 3 δ(p p ) () C p = (π h) 3 (iii) u p (r)u p(r)dxdydz = δ(p p ) N 0 [/sec cm ] = j p = C p m C = ( N 0 v ) d u dx = m (E V (x))u h u(x) u (i)e > V (ii)e < V 6

17 d u ν dx = m h E νu ν, m h E ν κ ν, E ν < 0 u ν Ae κ νx, (x < 0) or Be κ νx, (x > 0) u ν e κ ν x { (i) uν e κν x x (ii) u ν, duν dx Ĥ ˆp node 7

18 Ψ : Ψ p = u p (x) C[e ī h px + Re ī h px ], x CT e ī h px, x + d u p dx + k u p = 0, k = me h Ce ī h p x + CRe ī h p x CT e ī h p x p = m(e V ) p = m(e V ) j (i) = v C = N 0 C = N0 v j (r) = v C R = N 0 R j (t) = v C T = v v N 0 T P (r) = j(r) = R j (i) P (t) = j(t) j (i) = v v T P (r) + P (t) = 8

19 x x H = m p + V (x) V (x) V ( x) V (x) = V ( x) Schrödinger Hu ν (x) = E ν u ν (x) Hu ν ( x) = E ν u ν ( x) u ν ( x) = η p u ν (x), η p : u ν ( ( x)) = η p u ν ( x) = ηpu ν (x) = u ν (x) η p = η p = ± { uν ( x) = u ν (x) u (+) ν (x) + u ν ( x) = u ν (x) u ( ) ν (x) H u (+) ν (x) u (+) ν (x) u ( ) ν (x) u ( ) ν (x) H = { 0 : x < a m p + V (x), V (x) = : x a 9

20 u(x) = A cos kx + B sin kx { u (+) (x) = u (+) (x) u ( ) (x) = u ( ) (x) { u (+) (x) = A cos kx u ( ) (x) = B sin kx { cos k a (+) sin k a = 0 ( ) { u (+) ν { (+) : kν = π (ν + ), a ν = 0,,, ( ) : k ν = π (ν), a ν =,, E ν = h π (ν + ) u (+) ma ν (x) = cos[ π (ν + )x] a a E ν = h π ma (ν), u ( ) ν } { sin, cos} = a = a ν=0 [ u (+) ν ν=0 ν=0 (x)u (+) ν (y) + u ( ) ν u ( ) ν (x) = a sin[ π a (ν)x] (x)u ( ) ν (y) ] (3) [cos π a (ν + )x cos π a (ν + )y + sin π a νx sin π a νy ] (4) [ cos π a (ν + )(x + y) + cos π (ν + )(x y) a (5) cos π a ν(x + y) + cos π ] a ν(x y) (6) = [cos π a ν=0 a ν(x + y) cos π a ν(x + y) + cos π ] a ν(x y) (7) = [ δ( ] a a (x + y)) δ( (x + y)) + δ( a a (x y)) (8) = δ(x y) (9) Heisenberg Schrödinger E 0 = h 8a m x p h x a 0

21 p h a E = m ( p) = h 8a m (V 0 > 0) κ ν = I u ν (x) = Ae κ ν(x+ a ) II u ν (x) = B cos k ν x + B sin k ν x III u ν (x) = De κν(x a ) m Eν m(v0 E ν ) h, k ν = h V ( x) = V (x) (+) u (+) ν ( x) = u (+) (x) x = ± a ν A = D, B = 0 A cos k νa B = 0, κ ν A k ν sin k νa B = 0 κ ν cos kνa k ν sin kνa = 0 k ν sin k νa κ ν cos k νa = 0

22 ξ = k νa = ma (V 0 E ν ) ma V 0 h, w = h tan ξ = ξ w ξ w < π : π w < π : π w < 3π : 3 u (+) ν (x)u (+) ν (x)dx = B = I u (+) ν II u (+) ν III u (+) ν κν a a κ ν a + (x) = B cos k νa e κν(x+ a ) (x) = B cos k ν x (x) = B cos kνa e κ ν(x a ) ( ) u ( ) ν x = ± a ( x) = u ( ) (x) ν A = D, B = 0 A + sin k νa B = 0, κ ν A k ν cos k νa B = 0

23 + sin k νa k ν cos k νa κ ν = 0 k ν cos k νa + κ ν sin k νa = 0 cot ξ = ξ w ξ I II III w < π : 0 π w < 3π : 3π w < 5π : u ( ) ν (x)u ( ) ν (x)dx = B = u ( ) ν u ( ) ν u ( ) ν κν a a κ ν a + (x) = B sin kνa (x) = B sin k ν x (x) = B sin k νa e κ ν(x+ a ) e κ ν(x a ) (i)e > V 0 (V 0 ) I C(e ikx + Re ikx ) II C(B e ik x + B e ik x ) III CT e ikx 3

24 k = me m(e h, k V0 ) = h x = 0 { + R = B + B ik( R) = ik (B B ) x = a { B e ik a + B e ik a = T e ika ik (B e ik a B e ik a ) = ikt e ika B = k(k + k ) (k + k ) (k k ) e ik a B = k(k k )e ik a (k + k ) (k k ) e ik a R = k {(k k )B + (k + k )B } = (k k )( e ik a ) (k + k ) (k k ) e ik a T = k k + k ei(k k)a B R = = [ 4kk e i(k k)a (k + k ) (k k ) e ik a + 4(kk ) ] (k k ) sin k a T = [ + (k k ) sin k a 4(kk ) R + T = ] k a = nπ (n =,, ) R = 0, T = (i)e < V 0 4

25 k = I C(e ikx + Re ikx ) II C(B e κ x + B e κ x ) III CT e ikx me h, κ m(v0 E) = h [ R 4(kκ ) = + (k + κ ) sinh κ a [ T = + (k + κ ) sinh κ a 4(kκ ) E > V 0 E < V 0 ] ] E = h m k C = 0, CR = finite x < 0 k = iκ, CRe ikx CRe κx κ = m h E > 0 CT e ikx CT e κx C R T R T (iκ + k ) (iκ k ) e ik a iκ + k = ±(iκ k )e ik a { tan k k a = κ cot k a κ k a = ξ, κa = w ξ 5

26 cot ξ = ξ w ξ tan ξ = ξ w ξ Heisenberg S universal length () Schrödinger eq. : H = m p + mω x, h d u ν m dx ξ = ω = k m + mω x u ν = E ν u ν ( ) mω / x h Schrödinger eq. [ d H ν d u ν dξ ξ u ν = ϵ ν u ν, ϵ ν = E ν hω ξ, u ν ξ u ν = 0 = u ν e ξ / u ν (ξ) = CH ν (ξ)e ξ / dξ ξ d dξ + (ϵ ν ) ] H ν = a j ξ j j=0 H ν (ξ) = 0 Schrödinger eq. ξ j (j + )(j + )a j+ (j + ϵ ν )a j = 0 { a0, a, a 4, (+) a, a 3, a 5, ( ) a j+ a j = (j + ϵ ν) (j + )(j + ) j, j 6

27 a n+ a n = ξ N e ξ e ξ = n=0 n! ξn n! (n + )! = n + n = n 0 n + ϵ n = 0 E n = hω (n + ) = hω(n + ), n = 0,,, d H n dξ ξ dh n dξ + nh n = 0 H n (ξ) = ( ) n dn ξ e dξ n e ξ n e t +tz = n=0 t n n! H n(z) H n (z) = dn +tz dt n e t = ( ) n dn z e dz n e z t=0 (0) () d dz H n(z) = nh n () zh n (z) = nh n (z) + H n+(z) (3) n H 0 = (4) H = ξ (5) H = 4ξ (6) H 3 = 8ξ 3 4ξ (7) H 4 = 6ξ 4 48ξ + (8) 7

28 n n H n (ξ)e ξ ( ) n dn ξ e dξ n e ξ dξ = H n (ξ)h n (ξ)e ξ dξ = n n! πδ n n H n (ξ)( ) n dn dξ n e ξ dξ (9) = ( ) n+ H dn n (ξ) e ξdξ dξn (30) (3) = ( ) n+m H (m) n dn m (ξ) e ξ n m dξ dξ (3) m > n 0 m > n 0 n = n m = n H n (n) (ξ) = dn dξ n (ξ)n = (n)!! = δ n n(n)!! π ( ) mω /4 ( ) mω u n (x) = Hn n n! hπ h x e mω h x n = 0 u n (x)u n(x)dx = δ nn E 0 = hω ( ) mω /4 u 0 (x) = e mω h x < x >= 0 (33) hπ ( x) = < (x < x >) >=< x >= h (34) mω a 0 (p) = π h u 0 (x)e ipx h dx (35) ( ) /4 = e p hmω < p >= 0 (36) mω hπ ( p) = < (p < p >) >=< p >= hmω (37) x p = h E = m ( p) + mω ( x) = hω = E 0 8

29 Ψ(r, t) = (π h) 3/ a(p )e ī h (p r E p t) d 3 p a(p ) p E(p ) = E(p) + E (p p) i + E (p p) i (p p) j + p i p i p j Ψ(r, t) e ī h (p r Ept) Φ Φ(R) = (π h) 3/ R = r E p t ( r E ) p t a(p )e ī h (p p) R d 3 p Ψ R = r E t p Φ(R) Riemann-Lebesque R Φ 0 v g = E p = ω k 9

30 R = r v g t [ (p p) ] v g t p 3 p Φ(r, t) p p E p = p m = p m + p m (p p) + m (p p) = p m + v gq + m q, q = (p p) Ψ(x, 0) = (πσ ) a(q) = π h = x e 4σ /4 Ψ(x, 0)e ī h qx dx (πσ ) /4 e σ q h π h Φ(x, t) = = a(q)e ī h [qx (v gq+ m q )t] dq (38) { (π) /4 (σ + i h exp (x v gt) } t)/ 4σ mσ + i h t m (39) Ψ(x, t) = { } (x vgt) i p (π) /4 (σ + i h exp{ t)/ 4σ mσ + i h exp (px t} h m t) m 30

31 Ψ(x, t) = π (σ + h t 4m σ ) / exp < x >= v g t, x = σ + h t < p >= p, p = h (x v gt) (σ + h t 4m σ σ + h t 4m σ ) 4m σ x p = h Ψ(r, t) = A(p, t) = (π h) 3/ (π h) 3/ A(p, t)e ī h p r d 3 p Ψ(r, t)e ī h p r d 3 r Ψ(r, t) A(p, t) r ˆpφ p (r) = p φ p (r), ˆp = i h φ p (r) = (π h) 3/ e ī h p r ˆrχ r (r) = r χ r (r), χ r (r) = δ (3) (r r ) (φ, Ψ) φ (r)ψ(r, t)d 3 r Ψ(r, t) = (χ r, Ψ) Ψ(r, t) = (χ r, Ψ) A(p, t) = (φ p, Ψ) A(p, t) = (φ p, Ψ) Dirac bra ket 3

32 n ( { e i }) (φ, Ψ) < φ, Ψ > n A = A i e i i= A i = ( e i A) n A = e i ( e i A) i= Ψ >= d 3 r r >< r Ψ > d 3 r r >< r = < r r >= δ(r r ) ˆr r > = r r > (40) ˆp p > = p p > (4) d 3 r r >< r ˆr r > = r d 3 r r >< r r > (4) = r r > (43) < r ˆr r > = r δ(r r ) (44) d 3 p p >< p ˆp p > = p d 3 p p >< p p > (45) < p ˆp p > = p δ(p p ) (46) ˆp d 3 r d 3 r r >< r ˆp r >< r p > = p d 3 r >< r p > (47) < r ˆp r > = δ(r r )( i h r ) (48) < r p >= 3/ e ip r π h < p ˆr p >= δ(p p )(i h p ) 3

33 Ψ φ (φ, Ψ) {χ r } {φ p } (χ r, χ r ) = δ(r r ) χ r (r)χ r (r )d 3 r = δ(r r ) (φ p, φ p ) = δ(p p ) φ p (r )φ p (r )d 3 p = δ(r r ) Ψ(r, t) = χ r (r)(χ r, Ψ)d 3 r = φ p (r)(φ p, Ψ)d 3 p F (r, p) p i h ˆF F (r, i h ) < F >= Ψ (r, t)f (r, i h )Ψ(r, t)d 3 r p ˆp φ p (p) = p φp (p) φ p (p) = δ(p p ) Fourie ˆpφ p (r) = p φ p (r) ˆp φ p (p) = p φp (p) φ p (p) = ( ) e ī h p r e ī h p r d 3 r (π h) 3/ = δ(p p ) ˆr χ r (p) = r χ r (p) χ r (p) = (π h) 33 i e h p r 3/

34 Fourie χ r (p) = δ(r r )e ī (π h) 3/ h p r d 3 r Parseval ( ϕ, ψ) = ˆr i h p ϕ (p) ψ(p)d 3 p = (ϕ, ψ) A(p, t) = e ī (π h) 3/ h p r Ψ(r, t)d 3 r { χ r } { φ p } = Ψ(p, t) A(p, t) = ( φ p, A(p, t)) = ( φ p, Ψ) Ψ(r, t) = ( χ r, A(p, t)) = ( χ r, Ψ) ( χ r, χ r ) = δ(r r ) χ r (p) χ r (p )d 3 r = δ(p p ) ( φ p, φ p ) = δ(p p ) φp (p) φ p (p )d 3 p = δ(p p ) F (r, p) p Fourie {F (r, i h )Ψ(r, t)} = (π h) 3/ e ī h p r F (r, i h )Ψ(r, t)d 3 r = e ī (π h) 3 h p r F (r, i h )e ī h p r Ψ(p, t)d 3 p d 3 r = F (r, p )e ī (π h) 3 h (p p ) r Ψ(p, t)d 3 p d 3 r = (π h) 3 F (i h p, p ) Ψ(p, t)e ī h (p p ) r d 3 p d 3 r = F (i h p, p) Ψ(p, t) ˆF = F (i h p, p) 34

35 Ψ(r, t) Ψ(p, t) = A(p, t) r r i h p χ r (r) = δ 3 (r r ) χ r (p) = e ī (π h) 3/ h p r p i h p r φ p (r) = e ī (π h) 3/ h p r φ p = δ 3 (p p ) F (r, p) F (r, i h ) r F (i h, p) p (χ r, Ψ) ( χ r, Ψ) (φ p, Ψ) ( φ p, Ψ) r i h { x x } x x ψ(x, t) = i hψ(x, t) {x y y } x i h ψ(x, t) = 0 ψ xˆp x ˆp x x = i h, xˆp y ˆp y x = 0, p { } i h p x p x ψ(p, t) = i h ψ(p, t) p x p x { } i h p y p y ψ(p, t) = 0 p x p x ψ ˆxp x p xˆx = i h, ˆxp y p y ˆx = 0, ( ) ˆx α ˆp β ˆp β ˆx α = i hδ αβ, (α, β =,, 3) ˆx αˆx β ˆx β ˆx α = 0 ˆp αˆp β ˆp β ˆp α = 0 35

36 (quantum condition) (commutator): [â, ˆb] = âˆb ˆbâ (commutation relation): Poisson ( ) (quantization) [ˆx α, ˆp β ] = i hδ αβ, [ˆx α, ˆx β ] = 0, [ˆp α, ˆp β ] = 0, (α, β =,, 3) {x α, p β } = δ αβ, (49) {x α, x β } = 0, {p α, p β } = 0, (50) (α, β =,, 3) (5) {a, b} [â, ˆb] i h [â, ˆb] = [ˆb, â] [â, C] = 0, [Câ, ˆb] = C[â, ˆb] (C ) [â + ˆb, ĉ] = [â, ĉ] + [ˆb, ĉ] [âˆb, ĉ] = â[ˆb, ĉ] + [â, ĉ]ˆb [â, [ˆb, ĉ]] + [ˆb, [ĉ, â]] + [ĉ, [â, ˆb]] = 0 (Jacobi ) 36

37 I (Ia) a = (a, a,, a n ) (quantum number) (Ib) R a H a ψ(a) P = (ψ, ψ ) (ψ, ψ ) = ψ(a)ψ (a)ρ(a)da () ( ) (ψ, ψ ) = ( ψ, ψ ) II (IIa) Hermite H: : Ĥ(ψ a + ψ b ) = Ĥψ a + Ĥψ b ˆp: ψ(r, t + δt) = U H (δt)ψ(r, t), U H (δt) = e ihδt/ h (5) = ψ ī h Hψδt + O((δt) ) (53) = ψ + ψ t δt + O((δt) ) (54) ψ t δt = ( ī h Hδt)ψ ˆp(ψ a + ψ b ) = ˆpψ a + ˆpψ b ψ(r + δr, t) = U p (δr)ψ(r, t), U p (δr) = e i δr ˆp h (55) δr ˆp = ψ + i h ψ + O((δr) ) (56) = [ +! (δr r ) +! (δr r ) + ]ψ(r, t) (57) 37

38 ˆr: ψ δr = iδr ψ r ˆp h ˆr(ψ a + ψ b ) = ˆrψ a + ˆrψ b ψ(p + δp, t) = Ûr(δp) = e ī h δp ˆr Ĥ ˆp ˆr ψ δp = iδp ψ p ˆr h F (ˆr, ˆp)(ψ a + ψ b ) = F ψ a + F ψ b Hermite : Hermite : ˆF u ν = λ ν u ν ˆF = ˆF λ ν = λ ν, (u ν, u ν) = 0, (λ ν λ ν) u ν u ν Â = Â, ˆB = ˆB (Â + ˆB) = Â + ˆB Â = Â, ˆB = ˆB (Â ˆB) = ˆBÂ Â ˆB Weyl Fourier F (ξ, η) = (π) 3 F (ˆr, ˆp) = (π) 3 F (r, p)exp[ i(ξ r + η p)]d 3 rd 3 p F (ξ, η)exp[i(ξ ˆr + η ˆp)]d 3 ξd 3 η 38

39 xp F = π δ(ξ)δ(η) (58) ξ η F = dξdηe i(ξˆx+η ˆp) δ(ξ)δ(η) (59) ξ η ( ) = dξdη ξ η ( )(ξˆxηˆp + ηˆpξˆx) δ(ξ)δ(η) (60) = (ˆxˆp + ˆpˆx) (6) : ψ(r) = c ν u ν (r) c ν = (u ν, ψ) (6) ν ψ(r) = (u ν, ψ)u ν (r) (63) ν = u ν (r)u ν (r )ψ(r )d 3 r (64) ν u ν (r)u ν (r ) = δ(r r ) (65) ˆF = F (ˆr, ˆp) linear (66) ˆF = ˆF (67) ˆF u ν > = λ ν u ν >, λ ν = λ ν (68) < u ν u ν > = δ ν,ν, u ν >< u ν = (69) ν Ψ > H, c ν (70) Ψ > = c ν u ν > (7) ν = ν u ν >< u ν Ψ > (7) c ν = (u ν, ψ) (73) w ν = c ν ν c ν ˆF λ ν (74) F = lim λ ν N ν (75) N N ν < ˆF > = ν λ ν c ν λ ν w ν = (expectation value) (76) ν ν c ν = ψ (r, t) ν λ ν u ν (r)u ν(r )ψ(r, t)d 3 rd 3 r ψ (r, t) ν u ν (r)u ν(r )ψ(r, t)d 3 rd 3 r (77) 39

40 = = ψ (r, t) ˆF (r, h i r ){ ν u ν (r)u ν(r )}ψ(r, t)d 3 rd 3 r ψ (r, t)ψ(r, t)d 3 r ψ (r, t) ˆF (r, h )ψ(r, i r t)d3 r ψ (r, t)ψ(r, t)d 3 r (78) (79) = (ψ, ˆF ψ) (ψ, ψ) ν λ ν c ν ν c ν = < ψ uν > λ ν < u ν ψ > < ψ u ν >< u ν ψ > = < ψ ˆF u ν >< u ν ψ > < ψ ψ > = < ψ ˆF ψ > < ψ ψ > (80) (8) (8) (83) ( F ) =< ( ˆF < ˆF >) >=< ˆF > < ˆF > ( F ) = < ( ˆF < ˆF >) > = 0 c ν λ ν c ν c ν λ ν λ ν = 0 ν ν ν (84) c ν ( c ν )λ ν c ν λ ν λ ν = 0 ν ν<ν (85) c ν = c ν (86) ν ν c ν c ν (λ ν λ ν ) = 0 (ν,ν ),ν ν (87) ν ν c ν 0 (88) (89) ˆF = ˆF ψ Ĝ = Ĝ ξ (< i[ ˆF, Ĝ] > ) ψ = [ (ξ( ˆF < ˆF >) + i(ĝ < Ĝ >)] ψ (90) (ψ, ψ ) = ξ ( ˆF ) + ξ < i[ ˆF, Ĝ] > +( Ĝ) (9) { = ( ˆF ) (ξ ξ 0 ) D } 4( ˆF 0 (9) ) ξ 0 = < i[ ˆF, Ĝ] > /( ˆF ) (93) D =< i[ ˆF, Ĝ] > 4( ˆF ) ( Ĝ) 0 40

41 ( ˆF )( Ĝ) < i[ ˆF, Ĝ] > (i)[ ˆF, Ĝ] = 0 ˆF u νρ = λ ν u νρ Ĝu νρ = µ ρ u νρ ( ˆF ) = ( Ĝ) = 0 (ii) ξ = ξ 0 ψ ψ (ξ 0 ( ˆF < ˆF >) + i(ĝ < Ĝ >))ψ 0 = 0 ˆF = ˆx α, Ĝ = ˆp α (i) < i[ ˆF, Ĝ] >= h x α p α h (α =,, 3) (ii) ξ 0 = h ( x α ), x α p α [ ī ] x α h < p α > + ( x α ) (x α < x α >) ψ 0 (r) = 0 ψ 0 (r) = [(π) 3 Π α ( x α ) ] /4 exp [ i h < p > r α (x α < x α >) ] 4( x α ) Schrödinger ψ(x) = + A(k)exp(ikx)dk π A k 0 A(k) 0, for k k 0 k k = k 0 + ξ ψ(x) = exp(ik 0 x)φ(x) 4

42 Φ(x) = π + k k A(k 0 + ξ)exp(iξx)dξ Φ(x) =, for x ( k) Φ(x) = 0, for x ( k) (94) (95) x k = x p = h t ω = t E = h t E (IIb) F, G ˆF, Ĝ {F, G} i h [ ˆF, Ĝ] III Schrödinger Heisenberg Schrödinger Scrödinger i h ψ S(t) t = Ĥψ S(t) < ˆF > (t) = (ψ S(t), ˆF S ψ S (t)) (ψ S (t), ψ S (t)) Heisenberg t = 0 Schrödinger ψ 0 = ψ S (t = 0) ψ H 4

43 t Schrödinger ψ S (t) = Û(t)ψ 0 Û i h Û t = ĤÛ, Û(0) = Û(t) = exp( ī hĥt) Schrödinger < ˆF > (t) = (ψ H, Û ˆFS (t)ûψ H) (ψ H, ψ H ) = (ψ H, ˆF H (t)ψ H ) (ψ H, ψ H ) (96) (97) ˆFH (t) = Û (t) ˆF S Û(t) (98) Heisenberg d ˆF H (t) dt = i h [ ˆF H (t), Ĥ] Heisenberg ˆx H (t) = Û (t)ˆx S Û(t) ˆp H (t) = Û (t)ˆp S Û(t) U H H H = Û HÛ = H (99) H H (ˆr H (t), ˆp H (t)) = H(r, p) (00) [ˆx Hα (t), ˆp Hβ (t)] = i hδ αβ (0) [ˆx Hα (t), ˆx Hβ (t)] = 0 (0) [ˆp Hα (t), ˆp Hβ (t)] = 0 (03) Heisenberg Ĥ(ˆr, ˆp) = m ˆp + V (ˆr) 43

44 Heisenberg dtˆr(t) d [ˆr(t), Ĥ(ˆr(t), ˆp(t))] i h (04) d dt ˆp(t) = [ˆp(t), Ĥ(ˆr(t), ˆp(t))] i h (05) [ ] ˆx α (t), i h m ˆp = = ˆp α m[ ] V (r) i h [ˆp α, V (ˆr)] = x α i hm {ˆp β[ˆx α, ˆp β ] + [ˆx α, ˆp β ]ˆp β } (06) r=ˆr(t) (07) (08) dtˆr(t) d = ˆp(t) (09) m[ ] d V (r) dt ˆp(t) = (0) r r=ˆr(t) Ehrenfest d dt < ˆr > = < ˆp > () m [ ] d V (r) dt < ˆp > = () r r=ˆr dtˆr(t) d = ˆp(t) m (3) d ˆp(t) dt = 0 (4) ˆx(0) = ˆx 0, < ˆx 0 > = 0, x = σ (5) ˆp(0) = ˆp 0, < ˆp 0 > = p, p = h (6) σ < ˆx 0ˆp 0 + ˆp 0ˆx 0 > = 0 (7) ˆp(t) = ˆp 0, ˆx(t) = m ˆp 0t + ˆx 0 44

45 < ˆx(t) >= pt, < ˆp(t) >= p m ( x) = < ˆx (t) > < ˆx(t) > (8) = m {< ˆp 0 > < ˆp 0 > }t + < ˆx 0 > (9) = h 4m σ t + σ (0) ˆp p = h/(σ) x p = h + h t 4m σ 4. Heisenberg V (r) = mgz dtˆr(t) d = ˆp(t) m () d ˆp(t) dt = mgk () ˆr(0) = ˆr 0, < ˆr 0 > = 0, x = y = z = σ (3) ˆp(0) = ˆp 0, < ˆp 0 > = p, p x = p y = p z = h (4) σ < ˆx i0ˆp i0 + ˆp i0ˆx i0 > = 0 (5) ˆp(t) = mgtk + ˆp 0 (6) ˆr(t) = gt k + m tˆp 0 + ˆr 0 (7) < ˆp(t) > = mgtk+ < ˆp 0 >= mgtk + p (8) < ˆr(t) > = gt k + m t < ˆp 0 >= gt k + tp (9) m ˆx i (t) = 4 g t 4 δ i,z + m tˆp i0 + ˆx i0 gt3 m ˆp i0δ i,z + m t(ˆp i0ˆx i0 + ˆx i0ˆp i0 ) gt ˆx i0 δ i,z (30) < ˆx i (t) > = 4 g t 4 δ i,z + m t < ˆp i0 > + < ˆx i0 > gt3 m p iδ i,z (3) 45

46 ( x) = ( y) = ( z) = h t 4m σ + σ (3) ( ) h ( p x ) = ( p y ) = ( p z ) = (33) σ x i p i = h + h 4m σ 4 t (34) H = m ˆp (t) + mωˆq (t) Heisenberg dˆq(t) dt dˆp(t) dt = ˆp(t) [ˆq(t), Ĥ] = i h m, (35) = i h [ˆp(t), Ĥ] = mωˆq(t) (36) ( ) d ˆq = dt ˆp ( ) ( ) 0 m ˆq mω 0 ˆp A A = A = ( ) 0 m mω A = 0 ( ) mω i m hω mω i ( ) h mω imω imω ( ) iω 0 0 iω (37) (38) (39) ( ) â(t) â = A (t) ( ) ˆq(t) = ˆp(t) ( ) mωˆq + iˆp m hω mωˆq iˆp ( ) ( ) ( ) d â(t) iω 0 â(t) dt â = (t) 0 iω â (t) â(t) = âe iωt, â(t) = â e iωt 46

47 ( ) ( ) ( h ˆq(t) â(t) = A mω ˆp(t) â = [âe iωt + â e iωt ) ] (t) i m hω [âe iωt â e iωt ] ˆq(0) = ˆq, ˆp(0) = ˆp h ˆq = mω [â + â ], m hω ˆp = i [â â ] ˆq(t) = ˆq cos ωt + ˆp sin ωt (40) mω ˆp(t) = ˆp cos ωt mωˆq sin ωt (4) [ˆq(t), ˆp(t)] = i h, [ˆq(t), ˆq(t)] = [ˆp(t), ˆp(t)] = 0 [â(t), â (t)] = [â(t), â(t)] = [â (t), â (t)] = 0 Schrödinger Ĥ = hω[â â + ] : ˆN = â â [ ˆN, â] = [â â, â] = [â, â]â = â (4) [ ˆN, â ] = [â â, â ] = â [â, â ] = â (43) ˆN ˆN n >= λ n λ n > â â : ˆNâ λ n >= â( ˆN ) λ n >= (λ n )â λ n > : ˆNâ λ n >= â ( ˆN + ) λ n >= (λ n + )â λ n > 47

48 â λ n > λ n > (44) â λ n > λ n + > (45) λ n λ n =< λ n ˆN λn >= dq < q â λ n > 0 λ n minλ n = λ 0, ˆN λ 0 >= λ 0 λ 0 > â λ 0 > â λ 0 >= 0 λ 0 =< λ 0 ˆN λ0 >=< λ 0 â â λ 0 >= 0 λ 0 = 0, λ 0 > 0 > ˆNâ 0 > = â 0 > (46) (47) ˆN(â ) n 0 > = n (â ) n 0 > (48) λ n = n, n = 0,,, (49) n > (â) n 0 > (50) < n n >= â n > = c n+ n + > (5) â n > = d n n > (5) c n+ = < n ââ n > (53) = < n (â â + ) n > (54) = n + (55) c n+ = n +, d n = n â n >= n + n + >, â n >= n n > 48

49 n > = n! (â ) n 0 > (56) ˆN n > = n n > (57) < n m >= δ n,m (58) [ ] Ĥ n > = hω ˆN + n > (59) ( = hω n + ) n > (60) ( E n = hω n + ) (6) : < q 0 >= φ 0 (q) â 0 >= { } dqdq q >< q (iˆp + mωˆq) q >< q 0 >= 0 m hω q < q (iˆp + mωˆq) q >= [ h ddq ] + mωq δ(q q ) h â 0 >= dq q > mω φ 0 = N 0 exp [ d dq + mωq ] φ 0 = 0 h ( mωq h ) [ d dq + mωq ] φ 0 = 0 h, N 0 = ( ) mω /4 hπ : < q n >= φ n (q) φ n (q) = ( ) ( h n d ) n n! m hω dq + mωq φ 0 (q) 49

50 f = e ξ ξ = βq, β = mω h [ φ n (q) = N n ξ d ] n β exp( ξ /), N n = dξ π / n! n [ [ [ ] ( ξ d e ξ ξ f(ξ) = e d f (6) dξ dξ (63) ξ d ] n e ξ f(ξ) = ( ) n e ξ d n dξ dξ f (64) n ξ d dξ Hermite ] n exp( ξ /) = ( ) n e ξ ) d n dξ n e ξ (65) = e ξ Hn (ξ) (66) φ n (q) = N n H n (ξ) exp( ξ /) (67) H n (ξ) = ( ) n dn ξ e dξ n e ξ (68) h [ < m ˆq(t) n >= < m â n > e iωt + < m â n > e iωt] mω < m â n >= n + δ m,n+, < m â n >= nδ m,n < m ˆN n >= nδm,n h [ < m ˆq(t) n >= nδm,n e iωt + n + δ m,n+ e iωt] mω m hω [ < m ˆp(t) n >= i nδm,n e iωt n + δ m,n+ e iωt] 50

51 < 0 ˆq(t) 0 > = 0 (69) < 0 ˆp(t) 0 > = 0 (70) < 0 ˆq (t) 0 > = < 0 ˆq(t) >< ˆq(t) 0 >= h mω (7) < 0 ˆp (t) 0 > = < 0 ˆp(t) >< ˆp(t) 0 >= m hω (7) q(t) p(t) = h N ψ(r, r,, r N, t) t r d 3 r r d 3 r N r N d 3 r N ψ(r, r,, r N, t) d 3 r d 3 r d 3 r N ψ(r, r,, r N, t) d 3 r d 3 r d 3 r N = Fourier A(p, p,, p N, t) = ) N ( (π h) 3 ψ(r, r,, r N, t) (73) [ exp ī h ] N p i r i d 3 r d 3 r d 3 r N (74) i= A(p, p,, p N, t) d 3 p d 3 p d 3 p N H(r, r,, r N, p, p,, p N, ) Schrödinger i h ψ ( t = H r, r,, r N, i h, i h,, i h ) ψ r r r N 5

52 ψ(r, r, t) ν ν C(ν, ν, t) = u ν (r)u ν (r )ψ(r, r, t)d 3 rd 3 r {u νi } ψ(r, r, t) = ν,ν C(ν, ν, t)u ν (r)u ν (r ) ν ν C(ν, ν, t) ϵ = C(ν, ν, t) = ϵc(ν, ν, t) (75) = ϵ C(ν, ν, t) (76) ϵ = ± (77) C(ν, ν, t) = ±C(ν, ν, t) (78) ψ(r, r, t) = ±ψ(r, r, t) (79) ˆP : ± ˆP ψ(r, r, t) = ψ(r, r, t) (80) ˆP =, = ±ψ(r, r, t) (8) ˆP = ˆP Schrödinger ˆP Ĥψ = ( ˆP Ĥ ˆP ) ˆP ψ (8) = Ĥ ˆP ψ (83) 5

53 ˆP Ĥ ˆP = Ĥ ˆP Ĥ = Ĥ ˆP (84) [ ˆP, Ĥ] = 0 (85) ψ(t) = exp( iĥt/ h)ψ 0 ˆP ψ(t) = exp( iĥt/ h) ˆP ψ 0 ν = a, ν = b C(a, a), C(b, b), C(a, b), C(b, a) (i) C(a, a), C(b, b), C(a, b) = C(b, a) (ii) C(a, a) = C(b, b) = 0, C(a, b) = C(b, a) (i) Bose-Einstein Bose boson ψ(, r i,, r j, ) = ψ(, r j,, r i, ) (ii) Fermi-Dirac Fermi fermion ψ(, r i,, r j, ) = ψ(, r j,, r i, ) W. Pauli (S = 0,,, ) Bose Einstein (86) (S = /, 3/, ) Fermi Dirac (87) 53

54 < ˆp > p = O( h) h ψ = A exp( ī h W ), A W Schrödinger A t + ( x W t + m W A m x ( W x ) = 0 (88) ) + V + h m A A x = 0 (89) Schrödinger ρ = A, j x = ( ) W A m x W t + m ( W x ) + V + O( h ) = 0 p = W x W t + m p + V = 0 W t + H(x, p) = 0 W (x, t) = S(x) + F (t) ( ) df dt = m p + V = const. = E F = Et + const. p = ds dx S = pdx 54

55 W = pdx Et + constant Schrödinger ( ī ) ψ = A exp h ( pdx Et) ˆpψ = i h ψ x = ( p i h A ˆp = p + O( h) V Q = h m ) A ψ = (p + O( h))ψ x A A x lim h 0 i h [ ˆF, Ĝ] = {F, G} Weyl : F (x, p) = G(x, p) = F (ξ, η) exp[i(ξx + ηp)]dξdη (90) G(ξ, η ) exp[i(ξ x + η p)]dξ dη (9) : ˆF (ˆx, ˆp) = Ĝ(ˆx, ˆp) = F (ξ, η) exp[i(ξˆx + ηˆp)]dξdη (9) G(ξ, η ) exp[i(ξ ˆx + η ˆp)]dξ dη (93) Poisson Fourier : { F G {F, G} = x p G } F (94) x p = F (ξ, η) G(ξ, η )(ηξ ξη ) (95) exp[i(ξx + ηp)] exp[i(ξ x + η p)]dξdηdξ dη (96) i h [ ˆF, Ĝ]ψ = F (ξ, η) G(ξ, η) (97) i h [exp(i(ξˆx + ηˆp)), exp(i(ξ ˆx + η ˆp))]ψdξdηdξ dη (98) 55

56 x exp(i(ξx + ηˆp)) x exp( i(ξx + ηˆp)) = x + hη exp(i(ξx + ηˆp)) ˆp exp( i(ξx + ηˆp)) = ˆp hξ exp(i(ξx + ηˆp)) exp( i(ξ x + η ˆp)) exp( i(ξx + ηˆp)) = exp( i(ξ (x + hη) + η (ˆp hξ))) exp(i(ξx + ηˆp)) = exp( i(ξ (x + hη) + η (ˆp hξ))) exp(i(ξx + ηˆp)) exp(i(ξ x + η ˆp)) { exp(i(ξ x + η ˆp)) exp( i(ξ (x + hη) + η (ˆp hξ)))} exp(i(ξx + ηˆp)) exp(i(ξ x + η ˆp)) = { exp( i h(ηξ ξη ))} exp(i(ξx + ηˆp)) exp(i(ξ x + η ˆp)) h 0 { exp( i h(ηξ ξη ))} = i h(ηξ ξη ) + O( h ) ˆp p i h [ ˆF, Ĝ]ψ {{F, G} + O( h)}ψ Max Born Bell Born G. Wentzel, A. Krammers and L. Brillouin A ψ(x, t) = A(x)e ī h W (x,t), W = S(x) Et S (A x ( S x ) m(e V ) h A x ) = 0 (99) A x = 0 (00) 56

57 A = A 0 (S ) / (S ) = m(e V ) + h [ 3 4 (S S ) S ] S S h S 0 S 0 = O( h) S = S 0 + h S + (S 0) = m(e V ) S 0S = 3 4 ( S 0 S 0 ) (a) E V (x) > 0 S 0 S 0 k(x) = h m(e V (x)) (0) ( x ) S 0 = ± h k(x )dx + φ (0) x ψ(x, t) = exp ( ī ) h Et u 0 (x) (03) [ A x ] 0 u 0 (x) = k(x) cos k(x )dx + φ (04) x A 0 A h 0 (f) E V (x) < 0 κ(x) = h m(v E) u 0 = [ x x ] A 0 exp( κ(x )dx ) + B 0 exp( κ(x )dx ) κ(x) x x h S < h S 0 h S < (5) 57

58 S 0 = ± hk(x) (for E V > 0) S = ± 4 h = 4 h [ 3 [ k ] k k 3 k ( k k ) + k k( k ) ] (05) (06) hs = 4 [ k k + ] k( k x k ) dx k k < E V < 0 k κ x m h V < (6) [m E V (x) ] 3/ (turning point) k(x) = 0 (6) Schrödinger V (x) E h m c (x x ) x x (07) h ( ) dv m c = (08) dx x=x (a) : allowed (f) : forbidden Schrödinger (a) (f) k (x) = m h (E V (x)) = c (x x ) x > x (09) κ (x) = m h (E V (x)) = c (x x ) x < x (0) d u dx + c (x x )u = 0 x > x d u dx c (x x )u = 0 x < x 58

59 u ± a = A ± y /3 J ±/3 (y) x > x u ± f = B ±z /3 I ±/3 (z) x < x y = z = x x k(x )dx = 3 c(x x ) 3/ x > x () x x κ(x )dx = 3 c(x x) 3/ = 3 c x x 3/ x < x () (y/) ±/3 ( ) π ( J ±/3 (y) y 0 Γ( ± ), y y cos y π 6 π ) 4 3 (3) (z/) ±/3 I ±/3 (z) z 0 Γ( ± ), z (πz) ( e z + e z e ( ± )πi) 3 (4) 3 (i) x x u + /3 (c/3) /3 a A + (x x ) (5) Γ(4/3) u a A /3 Γ(/3) u + f B + /3 (c/3) /3 Γ(4/3) u f B /3 Γ(/3) (6) x x (7) (8) u + a u + f = A + = B + (9) u a u f = A = B (0) A + = B + = A, A = B = A u + a x x A ( ) / ( cos y 5π ) πk(x) E V > 0 () u + f x x A(πκ(x)) / [e z + e z e iπ5/6 ] E V < 0 () ( ) / ( u a x x A cos y π ) E V > 0 (3) πk(x) u f x x A (πκ(x)) / [e z + e z e iπ/6 ] E V < 0 (4) 59

60 Airy function Φ(z) = 3 z π (I 3 3 πz(j 3 ( z 3 ) I ( z 3 )) z > 0 ( z 3 )) 3 z < 0 ( 3 z 3 ) + J 3 z > 0 A = A 3 (u + f + u f ) x x 3 A(πκ) (e i π 6 e i 5π 6 )e z ( ) / [ ( A cos y 5π ) + cos(y π ] 3 πk(x) ) (5) = A πκ e z (6) = A ( ) / ( cos y π ) πk(x) 4 e z ( cos y π ) κ k 4 u(x) a = k(x) cos(y + φ ) u + a u a (u + a + u a ) ( (cos y 5π ) 3 3 ( u + a u a cos y 5π ) cos ( + cos )) ( = cos y π ( y π ) ( = sin y π ) 4 y π 4 ( cos y π ) ( cos φ sin y π ) sin φ = cos (y π ) φ φ = φ π 4 ) (7) (8) ( cos φ(u + a + u a ) sin φ(u + a u a ) y A 3 πk cos y π ) 4 + φ (u + f + u f ) x x A e z 3 πκ (u + f u f ) x x A πκ [ e z i e z ] 60

61 cos φ(u + f + u f ) sin 3 φ(u+ f u f ) z A πκ [sin φe z + e z iφ ] φ 0 φ = 0 cos (y π ) k 4 + φ sin φ e z (9) κ ( cos y π ) k 4 e z (30) κ y z (7) x = x u 0 = u 0 = [ A κ(x) exp x ] κ(x )dx x [ A x k(x) cos k(x )dx π ] x 4 x < x (3) x < x < x (3) x = x u 0 = u 0 = [ B x ] κ(x) exp κ(x )dx x > x (33) x [ B k(x) cos x k(x )dx π ] x < x < x (34) x 4 [ B k(x) cos x k(x )dx π x 4 + x ] k(x )dx x (35) = [ B k(x) cos x k(x )dx + π x 4 x ] k(x )dx x (36) = [ B x k(x) cos k(x )dx π x 4 ( x k(x )dx π ] x ) (37) 6

62 x x k(x )dx π = nπ, B = ( )n A k(x) = h m(e V ) = π h p x ( pdx = n + ) h x x x pdx = ( n + ) h h Bohr-Sommerfeld Schrödinger - k(x) cos( x x kdx π) 4 ψ I = = [ k(x) exp x i( k(x )dx π ] x 4 ) + R k(x) [ ( exp x i k(x )dx π )] x 4 ) k(x) [ ( + R) cos ( x x k(x )dx π 4 ( x i( R) sin k(x )dx π x 4 (38) )] (39) sin φ = π/ II ψ II = [ ( x ) ( x )] κ(x) ( + R) exp κ(x )dx + i( R) exp κ(x )dx x x III k(x) exp [ i( x x kdx π 4 )] ψ III = [ ( x T k(x) exp i k(x )dx π )] x 4 = [ ( x T cos k(x )dx π ) ( x + i sin k(x )dx π )] k(x) x 4 x 4 (40) (4) 6

63 II sin φ = π/ [ ( ψ II = T κ(x) exp x ) ( κ(x )dx x )] i exp κ(x )dx x x [ S ( x ) = T κ(x) exp κ(x )dx i ( x )] x S exp κ(x )dx x ( x ) S = exp κ(x )dx x (4) (43) (44) ( + R) = i T, (45) S i( R) = S T (46) T = is + is, (47) S 4 R = 4 S S + S 4 (48) P transmission = S (49) [ exp h x ] m(v (x) E)dx (50) x 63

64 I H = m p + m p + V (r r ) R = m r + m r m + m, r = r r H = M P + µ p + V (r) M = m + m, µ = m m m + m [R i, P j ] = i hδ ij, [x i, p j ] = i hδ ij Schrödinger Ψ(R, r, t) i h t = ] [ h M R h µ r + V (r) Ψ(R, r, t) [ Ψ(R, r, t) = exp i h ( )] P R P M t ψ(r, t) (5) ] ψ(r, t) i h = [ h t µ r + V (r) ψ(r, t) (5) r L x = (yp z zp y ) = y p z + z p y yp z zp y zp y yp z (53) = y p z + z p y yzp z p y zyp y p z + i h(yp y + zp z ) (54) L = r p r(r p) p + i h(r p) p i h 64

65 r p i hr r L = h r + h r ( r r [r, L] = 0 r ) ( r + h r ) r [ ] h = h r + r r ( = h r ) r r r Schrödinger ( ψ(r, t) i h = [ h r ) ] + L t µ r r r µr + V (r) ψ(r, t) + L r (55) + L r (56) x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ r = x + y + z cos θ = z x + y + z cos ϕ = x x + y r x θ x ϕ x ( r y θ y ϕ y r z θ z ϕ z ) = L x = i h y z z y ( { r = i h y z r + θ z θ + ϕ } z z ϕ [ = i h sin ϕ ] cot θ cos ϕ θ ϕ sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ cos θ sin ϕ sin θ r r r sin ϕ cos ϕ 0 r sin θ r sin θ { r y r + θ y θ + ϕ }) y ϕ (57) (58) (59) L y = i h [ L z = i h ϕ cos ϕ θ ] cot θ sin ϕ ϕ (60) (6) L = h [ sin θ θ (sin θ θ ) + sin θ ] ϕ 65

66 ds = h dx + h dx + h 3dx 3 = h h h 3 [ h ( + h x ( ) h h 3 x x ) h 3 h x + h 3 x 3 ( )] h h x 3 (6) (63) ds = dr + r dθ + r sin θdϕ h =, h = r, h 3 = r sin θ = ( r ) + ( sin θ ) + r r r r sin θ θ θ = ( r ) r r r r sin θ ϕ (64) L h r (65) ψ(r, t) = exp[ ī Et]ψ(r) h ( [ h r ) ] + L µ r r r µr + V (r) ψ(r) = Eψ(r) ( h d µ r dr r dr dr ψ(r) = R(r)Y (θ, ϕ) R(r) = u(r)/r L Y (θ, ϕ) = h λy (θ, ϕ) (66) ) ( h ) λ + µr + V (r) R(r) = ER(r) (67) h d ( u h ) µ dr + λ µr + V (r) u(r) = Eu(r) 66

67 [ sin θ ( sin θ ) + θ θ sin θ Y (θ, ϕ) = Θ(θ)Φ(ϕ) d Φ Φ dϕ = sin θ Θ sin θ [ sin θ ( d sin θ dθ dθ dθ ] Y (θ, ϕ) = λy (θ, ϕ) ϕ ( d sin θ dθ ) ] + λθ = m ( ) dθ dθ d Φ dϕ + m Φ = 0 (68) ) + m sin Θ = λθ (69) θ Φ(ϕ) = exp[imϕ] Φ(ϕ + π) = Φ(ϕ) m = 0, ±, ±, z = cos θ [ d ( ) ] z d dz dz + λ m Θ = 0 z Legendre P m l (z) = ( )l l l! λ = l(l + ) l = 0,,, ( z ) m d l+ m dz l+ m (z ) l, m = l, l +,, l, l L L z h l(l + ) hm Yl m (θ, ϕ) = ( ) m+ m l + (l m )! 4π (l + m )! eimϕ Pl m (cos θ) L Y m l (θ, ϕ) = l(l + ) h Y m l (θ, ϕ) (70) L z Y m l (θ, ϕ) = m hy m l (θ, ϕ) (7) Y m l (θ, ϕ)y m l (θ, ϕ)d(cos θ)dϕ = δ ll δ mm ] [ h d d µ r r dr dr + h l(l + ) + V (r) R µr l (r) = ER l (r) (7) [ h d ] µ dr + h l(l + ) + V (r) u µr l (r) = Eu l (r) (73) 67

68 m a V (r) = { V0 (V 0 > 0) r < a 0 r a R l (r) [ h d R l m dr + r ] dr l ( h ) l(l + ) + + V (r) R dr mr l = E R l α = m h (V 0 E ), ρ = αr (i) r < a [ d dρ + ( d ρ dρ + R l (ρ) = f l (ρ)/ ρ )] l(l + ) R ρ l = 0 [ d dρ + ( d ρ dρ + (l + )] ) ) f ρ l = 0 (l + ) Bessel J l+ (ρ) Neumann N l+ (ρ) R l (ρ) Bessel j l (ρ) Neumann n l (ρ) j l (ρ) = n l (ρ) = π ρ J l+ (ρ) (74) π ρ N l+ (ρ) (75) n l (ρ) = ( ) l+ j l (ρ) (76) ( ) l ( ) j l (ρ) = ( ρ) l d sin ρ (77) ρ dρ ρ ( ) l ( ) n l (ρ) = ( ρ) l d cos ρ ρ dρ ρ (78) Schrödinger ρ l j l (ρ) ρ 0 (l + )!! (79) n l (ρ) ρ 0 (l )!!ρ l (80) 68

69 R l (0) = finite R l (ρ) = Aj l (ρ) (ii)r a [ d dr + r ( d l(l + ) dr + m )] r h E R l = 0 β = [ d dρ + ( d ρ dρ + m E h, ρ = iβr )] l(l + ) R ρ l (ρ) = 0 Bessel Neumann Hankel h () l = j l (ρ) + in l (ρ) (8) h () l = j l (ρ) in l (ρ) (8) ( [ h () l ρ ρ exp i ρ h () l ρ ρ exp ( i [ ρ ]) (l + )π (l + )π ]) (83) (84) R l (r) = Bh () l (iβr) [ ] djl (αr) /j l (αr) = dh() l (iβr) /h () l (iβr) dr dr r=a r=a α β ξ = αa, η = βa ξ + η = mv 0a β( α) E = h m β l h (a)l = 0 ξ cot ξ = η 69

70 (b)l = cot ξ ξ ξ = η + η ( ) u 0 (r) Schrödinger [ h d ] µ dr + V (r) u 0 (r) = Eu 0 (r) (85) u 0 (0) = 0 (86) Schrödinger 70

71 I Schrödinger eq. Schrödinger eq. i) ii) H 0 H = H 0 + λh λ : real (E < 0) λh kk E k i) ii) i) { H0 u k = E k u k (u k, u j ) = δ kj Hψ = Eψ { ψ = ψ0 + λψ + λ ψ + E = E (0) + λe () + λ E () + λ λ 0 : H 0 ψ 0 = E (0) ψ 0 ψ 0 u k, E (0) = E k (87) λ : H 0 ψ + H ψ 0 = E () ψ 0 + E (0) ψ (88) λ : H 0 ψ + H ψ = E () ψ 0 + E () ψ + E (0) ψ (89) ψ {u n } ψ = c () n u n 7 (90)

72 () u j H u k + c () n E n u n = E () u k + E k c () n (u j, H u k ) + c () n E n (u j, u n ) = E () (u j, u k ) + E k c () n (u j, u n ) (u i, u j ) = δ ij (H ) jk + c () j E j = E () δ jk + E k c () j u n j = k E () = (H ) kk j k E k E j j k c () j = (H ) jk E k E j c () k = (ψ, ψ) (9) = + λ[(ψ, ψ 0 ) + (ψ 0, ψ )] + λ [(ψ, ψ 0 ) + (ψ, ψ ) + (ψ 0, ψ )] + (9) λ (ψ, ψ 0 ) + (ψ 0, ψ ) = 0 (93) ( c () n u n, u k ) + (u k, c () n u n ) = 0 (94) c () k + c () k = 0 (95) c () k = iα α = real (96) ψ = ψ 0 + λψ (97) = ( + iαλ)u k + λ (H ) nk u n (98) n k E k E n ( + iαλ) e iαλ c () k = 0 u k 0 (ψ 0, ψ) = 7

73 E () ψ {u n } ψ = m c () m u m () u k c () k E k + (H ) km c () m = E () + E () c () k + E k c () k E () = m k (H ) km (H ) mk E k E m ψ = u k + λ (H ) nk u n (99) n k E k E n +λ { (H ) nm (H ) mk n m (E k E n )(E k E m ) (H ) nk (H } ) kk u (E k E n ) n (300) E = E k + λ(h ) kk + λ (H ) nk (30) n k E k E n anharmonic oscillator H = H 0 + λx 4 (30) H 0 = m p + kx (303) E 0 = hω, ω = k m (304) ψ 0 = ( ) k 4 e kx hω (305) π hω E () = (ψ 0, H ψ 0 ) (306) = ( ) k λ x 4 e kx hω dx π hω (307) = 3λ 4 ( hω k ) (308) E = hω [ + 3λ ( hω )] (309) k ii) Rayleigh-Schrödinger λ (H ) lk < E (0) l E (0) k, l k 73

74 E () k = l k (H ) lk E (0) k E (0) l E (0) l = E (0) k (H ) lk = 0 H = H 0 + λh H 0 E (0) k g k H 0 k, m > (0) = E (0) k k, m > (0) (30) k, > (0), k, > (0),, k, g k > (0) (3) H ψ kn > = m a k nm k, m > (0) (3) (0) < k, j H ψ kn > = E kn ψ kn > (33) a k nmh k, m > (0) = E kn a k nm k, m > (0) (34) m (0) < k, j H k, m > (0) H k jm m (H jm k E knδ jm )a k nm = 0 m det(h k jm E knδ jm ) = 0 a k nm H E = E0 k + E kn ψ kn >, n =,,, g k advanced cource k l g l k, m > () = () a km,n a km,ln g k n g l a km,n ψ kn > + a km,ln ψ ln > n l k, m > () = g k E () g l km = g l p m n= l k (E (0) g l n= l k + ψ kp >< ψ kp H ψ ln >< ψ ln H ψ km > k E (0) l )[< ψ km H ψ km > < ψ kp H ψ kp >] ψ ln >< ψ ln H ψ km > n= l k < ψ km H ψ ln > E (0) k E (0) l E (0) k E (0) l (35) (36) (37) 74

75 s-p state (Stark) p, s g = 4 H n = l = 0 m = 0, > (0) = u 00 l = m = 0, > (0) = u 0 m =, 3 > (0) = u m =, 4 > (0) = u >= c, > (0) +c, > (0) +c 3, 3 > (0) +c 4, 4 > (0) H E () H H 3 H 4 H H E () H 3 H 4 H 3 H 3 H 33 E () H 34 H 4 H 4 H 43 H 44 E () = 0 z H = eez = eer cos θ (38), > = 4 3 π a 0 ( r ) e r a 0 (39) a0, > = 4 3 r π a 0 e r a 0 cos θ a 0 (30), 3 > = 4 3 r π a 0 e r a 0 sin θe iφ a 0 (3), 4 > = 4 3 r π a 0 e r a 0 sin θe iφ a 0 (3) H ij = ee r dr sin θdθdφ < i r cos θ j > (33) φ θ δ mm, H ii = 0 E () H 0 0 H E () E () E () = 0 H = H = ee 3πa 3 0 ( r a0 ) r a0 e r a 0 r 3 cos θ sin θdrdθdφ 75

76 θ φ r a 0 = ξ H = eea ( ξ)ξ 4 e ξ dξ = 3eEa 0 ξ n e ξ dξ = Γ(n + ) = n! H, + >= E () = 3eEa (u 00 + u 0 ) 0, >= (u 00 u 0 ) E = E n= 3eEa 0, + > E n= + 3eEa 0, > E n= u E n= u 76

77 II Schrödinger equation V (r) = Ze r [ d d r dr r dr + µ ] l(l + ) (E V (r)) R h r l (r) = 0 β = 8µE h (E < 0) (34) λ = Zµe h (35) β ρ βr (36) [ d d ρ dρ ρ dρ + λ l(l + ) ] R ρ ρ l (ρ) = 0 4 [ d ρ 0 d ] dρ ρ l(l + ) R l (ρ) 0 (37) dρ ρ R l ρ l, ρ (l+) (38) [ d d ] dρ ρ dρ ρ R l (ρ) 0 (39) 4 R l ρ l e ρ Fl F l [ ρ d dρ + (l + ρ) d ] (l + λ) F l (ρ) = 0 dρ 77 R l e ± ρ (330)

78 F l = a ν ρ ν ν=0 a ν+ = n ν λ (l + ν + ) (ν + )(l + ) + ν(ν + ) aν λ = l + ν + = n n total quantum number ν radial quantum number = n β = µze h λ = µze n h E = β h 8µ E n = µz e 4 h n l = 0,, n = 0,, n = n + l + n l m n l m E n l = 0,,,, (n ) (33) m = l, (l ),, l = (l + ) (33) n l=0 (l + ) = n n l F nl (ρ) Laguerre L l+ n+l L l+ n+l (ρ) = [(n + )!] F (l + n, l + ; ρ) (l + )!(n l )! F Kummer F (α, γ; z) = n=0 α(α + ) (α + n ) z n γ(γ + ) (γ + n ) n! 78

79 u nlm (r) = = [ 3 (n l )! β n[(n + )!] 3 ] Y m ( ) µze 3 (n l )! n h n[(n + )!] 3 l (θ, φ)e ρ ρ l L l+ n+l Y m (ρ) (333) l (θ, φ)e ρ ρ l L l+ (ρ) (334) n+l u n l m u nlmr drd(cos θ)dφ = δ nn δ ll δ mm n a B = h µe = cm R 0 (r) = R 0 (r) = R (r) = ( Z ab ( Z ) 3 e Zr a B (335) a B ( ) Z 3 ab ) 3 ( Zr a B 3 ( Zr a B ) e Zr a B (336) ) e Zr a B (337) 79

80 G ϵ W G δw = ϵ{g, W } δw = iϵ h [G, W ] G : z ϵ x x = x cos ϵ + y sin ϵ x + ϵy y y = y cos ϵ x sin ϵ y ϵx z z = z iϵ h [L z, x] = iϵ h [xp y yp x, x] = iϵ h y[p x, x] (338) = ϵy = δx (339) iϵ h [L z, y] = iϵ h [xp y, y] (340) = ϵx = δy (34) iϵ h [L z, z] = 0 = δz (34) L z z L x L y [L x, L y ] = i hl z [L y, L z ] = i hl x [L z, L x ] = i hl y [L, L] = 0 L z L L Y m l (θ, φ) = l(l + ) h Y m l (θ, φ) (343) L z Y m l (θ, φ) = m hy m l (θ, φ) (344) Y m l m (θ, φ)yl (θ, φ)d(cos θ)dφ = δ ll δ mm 80

81 D h = D x (ϵ) = iϵj x D y (ϵ) = iϵj y D z (ϵ) = iϵj z D J x, J y, J z n θ D n (ϵ) = iϵn J N θ N D n (θ) = n = [ ( D n ( θ N )] N (345) i θ N n J ) N (346) N e iθn J (347) D n (θ) = e iθn J e iηj y e iϵj x e iηj y e iϵj x = e iϵηj z x y x y ϵ η ϵ η z ϵη 0 ϵη 8

82 J x J y J y J x = i hj z J y J z J z J y = i hj x (348) J z J x J x J z = i hj y (349) 3 [J i, J j ] = i h ϵ ijk J k 3 ( ) O(3) SU() k= J [J, J] = 0 k m { J a (m) = ka (m) J z a (m) = ma (m) J ± J x ± ij y [J z, J ± ] = ±J ± b J a (m) J z b = J z J a (m) = J (J z )a (m) (350) = (m )J a (m) = (m )b (35) b = ca (m ) a (m) a (m) = (m ) a (m ) = c J a (m) = c c (J a (m) ) (J a (m) ) (35) = c a(m) (J + J )a (m) (353) = (k m + m) c a (m) a (m) (354) 8

83 c = k m(m ) J ± J = J x + J y ± i[j y, J x ] = J J z ± J z k m a ( j) J + J a ( j) = 0 J + J a ( j) = 0 k j(j + ) = 0 k = j(j + ) m j J + a (j ) = 0 k = j (j + ) j = j j m m = j, (j ),, j, j = j ( j) + = j + = j k c = (j + m)(j m + ) j, m > j j, m > = (j + m)(j m + ) j, m > (355) j + j, m > = (j m)(j + m + ) j, m + > (356) () j = 0 () j = (3) 3 j = () m =, a (/) = ( 0 ), a ( /) = 83 ( 0 )

84 J ( ) ( ) ( ) 0 J =, J + = ( ) ( ) ( ) 0 0 J = 0, J + = 0 ( ) ( ) ( ) ( ) 0 0 J z =, J z = 0 0 J = ( J z = ), J + = ( 0 0 ) ( σ z ) J x J y J x = (J + + J ) = ( 0 0 ) σ x J y = i (J + J ) = ( 0 i i 0 ) σ y σ x = ( 0 0 ), σ y = ( 0 i i 0 ), σ z = ( 0 0 ) σ i = J i [σ i, σ j ] = iϵ ijk σ k [σ i, σ j ] + = δ ij σ i σ j = δ ij + iϵ ijk σ k ( A σ)( B σ) = A B + iϵ ijk A i B j σ k (357) = A B + i( A B) σ (358) e i θ n σ = cos θ + i sin θ n σ (359) 84

85 J J [J, J ] = 0 J, J z : j (j + ), m = j,, j, = (j + ) (360) : j, m > (36) J, J z : j (j + ), m = j,, j, = (j + ) (36) : j, m > (363) J = J + J j, j ; m, m > j, m > j, m > = (j + )(j + ) [J x, J y ] = ij z, [J y, J z ] = ij x, [J z, J x ] = ij y [J, J z ] = [J, J z ] = 0 (364) [J, J ] = [J, J ] = 0 (365) [J iz, J ] = [J iz, J J ] 0 (366) J, J, J, J z J j, j ; j, m > = j(j + ) j, j ; j, m > (367) J z j, j ; j, m > = m j, j ; j, m > (368) : J ± J x ± ij y = J ± + J ± (369) J i± J ix ± ij iy (370) [J z, J ± ] = ±J ± J ± j, j ; j, m >= (j m)(j ± m + ) j, j ; j, m ± > j, m >, j, m > j j m = j m = j j, j ; m, m >< j, j ; m, m = j, j ; j, m > = = j j m = j m = j j, j ; m, m >< j, j ; m, m j, j ; j, m > (37) j j m = j m = j C(j m j m j j jm) j, j ; m, m > (37) 85

86 C(j m j m j j jm) < j, j ; m, m j, j ; j, m > Clebsch-Gordan j, j j, m >= m m, m m m, m >< m, m j, m > J z j, m > = (J z + J z ) m, m >< m, m j, m > m m (373) m j, m > = (m + m ) m, m >< m, m j, m > m m (374) m m (m + m m) m, m >< m, m j, m >= 0 < m, m C-G < m, m j, m > 0 m +m = m m + m = m j jmax = (j z )max = (m + m )max = j + j j min j > j m m m j j j j j + j j j + j j + j j j j + j + j + j (j j ) + j j + j j + j j + j j + j j + (j + j ) + j j j j j j j + j j (j + j ) + j, m > j +j i=j j (i + ) = j +j i= (i + ) j j i= (i + ) (375) = (j + j )(j + j + ) + (j + j ) (j j )(j j ) (j j (376) ) = (j + )(j + ) (377) j j = j + j, j + j,, j j 86

87 m j, m > j +j j= j j j, m >< j, m = m, m > = j j, m >< j, m m, m > (378) = j +j j= j j C (j m, j m ; j j jm) jm > (379) j = j = j m m+ 0 m m m+ m =, > = α > α > j =, m = 0, 0 > = { α > β > + β > α >} m =, >= β > β > 3 j = 0, m = 0 0, 0 >= { α > β > β > α >} j = l, j = l j = l ± l 0 j m l + l l+m+ l+ l m+ l+ l m+ l+ l+m+ l+ l +, m l, m = = l + m + l + l m + l + Y m l m + l α + Y m l α + l + Y m+ l β (380) l + m + l + Y m+ l β (38) 87

88 ( + ) J = S + S, S = 3 4 S J = S + S + S S = S S + = S + S + S S + + S z S z + (38) (383) (384) S j(j + ) = { S χ t = χ t { j = j = 0 S S χ s = χ s P P Ψ(r, S ; r, S ) = ηψ(r, S ; r, S ) P P =, η = η = ± { φ(r, r ) χ(s, S ) P Ψ = Ψ } Ψ = φχ Ψ = : { φs χ A, χ A = 0, 0 > φ A χ S, χ S =, m > φ S (r, r ) = (u a (r )u b (r ) + u b (r )u a (r )) (385) φ A (r, r ) = (u a (r )u b (r ) u b (r )u a (r )) (386) 88

89 Slator determinant φ A ({r i }) = N! u a (r ), u b (r ),, u c (r ) u a (r ), u b (r ),, u c (r ),,, u a (r N ), u b (r N ),, u c (r N ) doublet E n (n, l, m) l, m s (l = 0) : (singlet) p, d, (l =,, ) : (doublet) 95 Uhlenbeck, Goudsmit j = l + s l 0 : j = l ± = (387) l = 0 : j = = (388) LS (spin-orbit interaction term) i) B = v c E 89

90 loop current: d µ = ids/c i = ev (e > 0) πr µ = ev πrc πr = epr m e c µ = e m e c L, L = r p L int = H int = e c A v = e (B r) v (389) c = e (r p) B = µ B (390) m e c H int = µ B S µ = g( e m e c S) g g- gyromagnetic ratio e H int = g m e c B S? e h m e c Zeeman g µ = e m e c S ii)thomas factor H int H int 90

91 ee = V (r) (39) = dv dr r (39) r H int = ( µ B) = µ ( v E) (393) c e = S (E p) (394) m ec H int = m ec r V = Ze r H = m e p + V (r) + H int dv dr L S m ec r dv dr L S L S = (J L S ) (395) = { j(j + ) l(l + ) 3 } h (396) 4 (n, l, j) n, l j E nlj = h 4m ec j = l ± { dv j(j + ) l(l + ) 3 } h r dr 4 nl E doublet = Ze h 4m ec (l + ) r 3 nl α e hc = (6) A = B r 9

92 896 Zeeman, H = m e ( p ( e) c A ) + V (r) (397) = p m e + V (r) + e m e (p A + A p) + e m e c A (398) p Aψ = (p A)ψ + A (pψ) = A (pψ) (399) A p = (B r) p = (r p) B = B L (400) H = p m e + V (r) + e m e c A p (40) = p + V (r) + e m e m e c B L (40) B = (0, 0, B) ψ nlm E nlm = e hb m e c m H = p + V (r) + e m e m e c B L + dv m ec r dr L S + e m e c B S (403) = H 0 + e B (L + S) (404) m e c = H 0 + e B (J + S) (405) m e c H 0 = p m e + V (r) + m ec r dv dr L S 9

93 B = (0, 0, B) H = H 0 + ω(j z + S z ) ω = e m e c B E = ω < J z + S z >= m hω + ω < S z > < S z >= h { m j = l + l + m j = l E ljm = m hω j + l + m = j, j,, j j = l + j + m ; j, m > = l + Y m j m l α + l + Y m+ l β (406) j = l j + m ; j, m > = l + Y m j + + m l α + l + Y m+ l β (407) < S z >= l + ( ) j+m j m = m ( j+ m j++m j = l + ) l+ = m j = l l+ 93

94 n, l, m, S z n, l, m, S z n,, 3, 4, 5, 6, 7 K L M N O P Q l : 0,,, 3, 4, 5, 6, 7,, n : s p d f g h i k m = l, l,, l = (l + ) S z =, = s : () = p : ( + ) = 6 (IA, IIA,, VIIIA) d : ( + ) = 0 (IB, IIB,, XB) f : ( 3 + ) = 4 La, Ac 94

95 n l s s p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 7s H IA He VIIIA 3 Li IA 4 Be IIA 5 B IIIA 6 C IVA 7 O 3 VA 8 N 4 VIA 9 F 5 VIIA 0 Ne 6 VIIIA Na 6 IA Mg 6 IIA 3 Al 6 IIIA 4 Si 6 IVA 5 P 6 3 VA 6 S 6 4 VIA 7 Cl 6 5 VIIA 8 A 6 6 VIIIA 9 K 6 6 IA 0 Ca 6 6 IIA Sc 6 6 IIIB Ti 6 6 IVB 3 V VB 4 Cr VIB 5 Mn VIIB 6 Fe VIIIB 7 Co VIIIB 8 Ni VIIIB 9 Cu IB 30 Zn IIB 3 Ga IIIA 3 Ge IVA 33 As VA 34 Se VIA 35 Br VIIA 36 Kr VIIIA 95

96 n l s s p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 7s 37 Rb IA 38 Sr IIA 39 Y IIIB 40 Zr IVB 4 Nb VB 4 Mo VIB 43 Tc VIIB 44 Ru VIIIB 45 Rh VIIIB 46 Pd VIIIB 47 Ag IB 48 Cd IIB 49 In IIIA 50 Sn IVA 5 Sb VA 5 Te VIA 53 I VIIA 54 Xe VIIIA 55 Cs IA 56 Ba IIA 57 La IIIB 58 Ce IIIB 59 Pr IIIB 60 Nd IIIB 6 Pm IIIB 6 Sm IIIB 63 Eu IIIB 64 Gd IIIB 65 Tb IIIB 66 Dy IIIB 67 Ho IIIB 68 Er IIIB 69 Tm IIIB 70 Yb IIIB 7 Lu IIIB 7 Hf IVB 73 Ta VB 74 W VIB 75 Re VIIB 76 Os VIIIB 77 Ir VIIIB 78 Pt VIIIB 79 Au IB 80 Hg IIB 8 Tl IIIA 8 Pb IVA 83 Bi VA 84 Po VIA 85 At VIIA 86 Rn VIIIA

97 n l s s p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 7s 87 Fr Ra Ac Th Pa U Np Pu Am Cm La- (5 ) La(57),Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu(7) Ac- (5 ) Ac(89), Th, Pa, U, Np, Pu, Am, Cm, Bk, Cf, Es, Fm, Md, No, Lr(03) 97

98 n Ψ(r S, r S,, r n S n ) Ψ(r S,, r i S i,, r j S j,, r n S n ) (408) { +Ψ(r S,, r j S j,, r i S i,, r n S n ) = (409) Ψ(r S,, r j S j,, r i S i,, r n S n ). He Ψ(r S, r S ) = 0, for r = r, S = S L S, S S H = H 0 + e (40) r H 0 = p + p e e (4) m e m e r r H 0 (n i, l i, m li ) i =, ( E n,n = m e c α + n n ), α e hc u ni l i m li (r i ) i =, { χms :triplet χ Sms = :singlet χ 00 98

99 H, P, L = L + L, S = S + S H, P, L, S, L z, S z E, A, l(l + ), S, m l, m s n = n e r Ψ nlsml m s = { [u00 ()u nlml () + u 00 ()u nlml ()]χ 00 [u 00 ()u nlml () u 00 ()u nlml ()]χ ms nlsm l m s e nlsm l m s r = A ± B { singlet triplet A B ( ) u 00 ()u nlml (), e u 00 ()u nlml () (4) r ( ) u 00 ()u nlml (), e u 00 ()u nlml () > 0 (43) r B E(singlet) > E(triplet) r r r r He (n = n ) 99

100 i)n = 3 S u 00 ()u 00 () u 00 ()u 00 () = 0 ii)e(s) > E(t) iii)n = S iv)n ++ A,B:,: H = m e p + m e p ( e ) + e + e + e r A r A r B r B + e r AB + e r u A (r ): u B (r ): A B dr u A(r )u A (r ) = dr u B(r )u B (r ) = Ψ(r S ; r S ) = [u A (r )u B (r ) ± u A (r )u B (r )] χ s,t S A A E c E c E e E e dr u A(r )u B (r ) overlap integral (44) [ dr u A(r ) ] e u A (r ) (45) m e r A [ dr u A(r ) ] e u B (r ) (46) m e r A e dr u A (r ) r B (47) e dr dr u A (r ) u B (r ) r (48) e dr u r A(r )u B (r ) A (49) e dr dr u r A(r )u B(r )u A (r )u B (r ) (40) 00

101 exchange integral singlet E = < H > (4) = A + A S (E c + E e S) (E c + E e) + e (4) + S + S r AB r AB 3 / F sc = [f(θ) ± f(π θ)] χ s,t dσ dω = (S + )(S + ) = final spin F sc (43) { f(θ) + f(π θ) + 3 } f(θ) f(π θ) (44) 0

102 = { f(θ) Ref (θ)f(π θ) + f(π θ) } (45) θ π θ θ = π/ 0 F sc = [f(θ) + f(π θ)] (46) dσ dω = { f(θ) + Ref (θ)f(π θ) + f(π θ) } (47) θ π θ θ = π/ 0

103 II Et i ψ(r, t) e h { e ik r + f eikr r + } H = p m + V (r) Hψ(r) = Eψ(r), E > 0 (E ) k = me h, U(r) = m h V (r) ( + k )ψ(r) = U(r)ψ(r) : r U(r) 0, r Coulomb ru(r) const., r ψ(r) = e ik r + ψ scatt (r) j = i h m (ψ ( ψ) ( ψ )ψ) 03

104 j inc = hk m = v R ds dω ds = R dω j scatt ˆrds = i h ( ψ ψ scatt sactt m r ) ψ sactt ψ scatt R r r=r dω R indep. ψ sact ψ scatt r const r=r R, R lim ψ scatt(r) = f(k, k ) eikr R R f(k, k ) = f(k, θ) ψ(r) = e ik r + f(k, θ) eikr r ds j scatt ˆrds dσ = lim = f(k, θ) dω R j inc dσ dω = f(k, θ) 04

105 Green ( + k )ψ(r) = U(r)ψ(r) Green ( + k )G(r, r ) = δ(r r ) Fourier G(r r ) = δ(r r ) = (k q ) G(q) = e iq (r r ) G(q)d 3 q (48) (π) 3/ e iq (r r ) d 3 q (49) (π) 3/ (430) (π) 3/ c G(q) = [ ] P (π) 3/ q k + cδ(q k ) x iϵ = x x + ϵ ± i ϵ x + ϵ P ± iπδ(x), ϵ 0 x c = ±iπ G(q) = (π) 3/ q k iϵ G ± (r r ) = lim ϵ 0 (π) 3 d 3 q = q dqd(cos θ)dφ (43) e iq (r r ) q (k ± iϵ) d3 q (43) φ G ± (r r ) = lim ϵ 0 (π) 0 i = lim ϵ 0 (π) r r i = lim ϵ 0 4π r r q dq 0 e iq r r cos θ d(cos θ) q (k ± iϵ) (433) qdq q (k ± iϵ) (eiq r r e iq r r ) (434) e iq r r qdq (435) q (k ± iϵ) 05

106 G + (r r ) = e ik r r 4π r r G (r r ) = e ik r r 4π r r (436) (437) Schrödinger ψ (±) (r) = ψ (0) (r) + G ± (r r )U(r )ψ (±) (r )d 3 r ( + k )ψ (0) = 0 ψ (0) = e ik r ψ (+) (r) = e ik r m e ik r r h 4π r r V (r )ψ (+) (r )d 3 r Born m h V L < ψ (+) (r) = ψ (0) (r) + d 3 r G + (r r )U(r )ψ (0) (r ) (438) + d 3 r d 3 r G + (r r )U(r )G + (r r )U(r )ψ (0) (r ) + (439) ψ (0) = e ik r ((first) Born approximation) ψ (+) (r) = e ik r m e ik r r h 4π r r V (r )e ik r d 3 r r r r = ( r + r rr cos α ) ( / = r ) / r r cos α + r (440) r r r cos α, r (44) r r = r ( ) r l P l (cos α) r r + r r cos α l=0 06

107 e ik r r r r r exp(ik(r r cos α)) ψ (+) (r) e ik r m e ikr π h r V (r )e i(k r kr cos α) d 3 r (44) = e ik r + eikr f(k, θ) (443) r f(k, θ) = m π h kr cos α = k r V (r )e i(k k ) r d 3 r V (r ) q k k q = k + k kk cos θ, k = k = k ( cos θ) = 4k sin θ f(k, θ) = m h k sin θ V (r ) sin(kr sin θ )r dr (444) = m h V (r ) sin(qr )r dr (445) q (screened Coulomb) V (r) = V 0 e µr r f(k, θ) = mv 0 e µr +iq r π h r dr d(cos θ)dφ (446) r = mv 0 iq h = mv 0 iq h 0 [ dr e µr [e iqr e iqr ] (447) µ iq µ + iq ] = mv 0 h q + µ (448) = mv 0 h 4k sin θ + µ (449) 07

108 ( ) dσ dω = mv0 h (4k sin θ + µ ) µ = 0 Z e Z e V 0 = Z Z e, E = h k dσ dω = Z Ze 4 6E sin 4 θ (Rutherford) m e ikr h V (r )e ik r d 3 r e ik r 4π r r=0 = m V : a : i)pa/ h e ik r m h V a ii)pa/ h V (r ) V cos θ m h V a h pa m h V a pa h sin pr h pr h < h pr (eikonal) ka, V E ( + k )ψ k (r) = m h V (r)ψ k(r) 08

109 φ k (r) ψ k (r) = e ik r φ k (r) [ik + ]φ k (r) = m h V φ k(r) k = (0, 0, k) φ k z = i hv V φ k φ k, z [ φ k (x, y, z) = exp i z hv [ ψ k (x, y, z) = exp ikz i hv V (x, y, z )dz ] z V (x, y, z )dz ] (450) (45) f(k, θ) = m π h e ik r V (r)ψ k (x, y, z)d 3 r (45) = m [ π h e i(k k ) r V (r)exp i z ] V (x, y, z )dz d 3 r (453) hv d 3 r = d () bdz (454) (k k ) r = (k k ) b + [(k k ) ê z ]z θ γ π hv V 0 d = min(a, hv V 0 ) [(k k ) ê z ]z kdθ θ kd 09

110 r = b + ê z z f(k, θ) = m π h d () b e i(k k ) b V (b + ê z z) (455) [ exp i z ] V (b + ê z z )dz dz (456) hv = m ( hv ) [ π h d () be i(k k ) b exp i z V (b + ê z z )dz ] (457) i hv [ ] f(k, θ) = k d () be i(k k ) b [e iχ(b) ] (458) πi χ(b) = V (b + ê z z )dz (459) hv V z γ = π (b, φ) (k k ) b = b sin γ cos φ b cos φ (460) π 0 k k = k sin θ (46) χ(b) = V (b, z )dz (46) hv e ib cos φ dφ = πj 0 (b ) (463) f(k, θ) = k i 0 J 0 (b )[e iχ(b) ]bdb (partial wave expansion) L = bp, b ˆL, ˆL l(l + ) l i) ii) i) L z = i h φ [ L = h sin θ θ sin θ θ + ] sin θ φ (464) (465) 0

111 { Lz Yl m (θ, φ) = hmyl m (θ, φ) L Yl m (θ, φ) = h l(l + )Yl m (θ, φ) Y m l (θ, φ) = [ (l + )(l m )! 4π(l+ m )! d(cos θ) π 0 ] P m l (cos θ)e imφ (466) dφy m l (θ, φ)y m l (θ, φ) = δ ll δ mm (467) e ikr cos θ = a lm = l l=0 m= l d(cos θ) a lm (r)y m l (θ, φ) (468) π 0 dφe ikr cos θ Y m l (θ, φ) (469) π 0 dφe imφ = πδ m0 Yl 0 = l+ P 4π l(cos θ) a lm (r) = δ m0 π(l + ) d(cos θ)e ikr cos θ P l (cos θ) (470) = δ m0 π(l + )i l j l (kr) (47) e ikr cos θ = (l + )i l j l (kr)p l (cos θ) l=0 j l (kr) ( kr sin kr lπ ), r (47) [ = e i(kr l π) e i(kr l π)] (473) ikr e ikr cos θ ikr (l + )i l [e i(kr l π) e i(kr l π) ]P l (cos θ) l=0 ii) Schrödinger ( + k )ψ (+) (r) = U(r)ψ (+) (r)

112 ψ (+) (r) = b lm = l l=0 m= l d(cos θ) b lm (r)y m l (θ, φ) (474) π 0 dφψ (+) (r)y m l (θ, φ) (475) U(r) = U(r) k z dφ δ m0 b lm = δ m0 π(l + ) d(cos θ)ψ (+) (r)p l (cos θ) ( ) i l c l χ l (kr) χ l d(cos θ)ψ (+) (r)p l (cos θ) ψ (+) (r) = (l + )i l c l χ l (kr)p l (cos θ) l=0 Schrödinger P l (cos θ) d(cos θ) :u l = rχ l [ ] d l(l + ) + U(r) + rχ dr r l = k rχ l r 0 d dr (rχ l(l + ) l) + rχ r l = 0 χ l { r (l+) : singular r l : nonsingular χ l (kr) (kr) l r ψ (+) (r) e ikr cos θ + eikr f(k, θ) r f(k, θ) (l + )T l (k)p l (cos θ) k l=0 e ikr cos θ ikr (l + )i l [e i(kr l π) e i(kr l π) ]P l (cos θ) l=0

113 ψ (+) ikr (l + )i [ l ( + it l )e i(kr l π) e i(kr l π)] P l (cos θ) l=0 S l + it l S l = S l S l (k) e iδ l(k) δ l (k) (phase shift) δ l (k) = Reδ l (k) + iimδ l (k) S l = e Imδ l Imδ l (k) 0 Imδ l (k) 0 (Imδ l (k) = 0) ψ (+) kr ( (l + )i l e iδ l sin kr l ) l=0 π + δ l P l (cos θ) r χ l (kr) ( kr sin kr l ) π + δ l 3

7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E

7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E B 8.9.4, : : MIT I,II A.P. E.F.,, 993 I,,, 999, 7 I,II, 95 A A........................... A........................... 3.3 A.............................. 4.4....................................... 5 6..............................

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M ω f ω = df ω = i ω idx i f x i = ω i, i = 1,..., n f ω i f 2 f 2 f x i x j x j x i = ω i x j = ω j x i, 1 i, j n (3) (3) ω 1.4. R 2 ω(x, y) = a(x, y

M ω f ω = df ω = i ω idx i f x i = ω i, i = 1,..., n f ω i f 2 f 2 f x i x j x j x i = ω i x j = ω j x i, 1 i, j n (3) (3) ω 1.4. R 2 ω(x, y) = a(x, y 1 1.1 M n p M T p M Tp M p (x 1,..., x n ) x 1,..., x n T p M dx 1,..., dx n Tp M dx i dx i ( ) = δj i x j Tp M Tp M i a idx i 1.1. M x M ω(x) Tx M ω(x) = n ω i (x)dx i i=1 ω i C r ω M C r C ω( x i ) C

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