3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H, R] = 0 0 0 (7) R = ψ Rψ (8) R v, v 2, v 3 U = (v, v 2, v 3 ) ψ = ψu = (, 2, 3 ) i H, R 2. [ L/2, L/2] ψ(x) = ψ(x + L) () x = d dx xψ φ = ψ x φ x = x ψ x φ = ψ ( x φ ) = ( ψ x ) φ p x = iħ x
3 : 203 2 (2) p x ψ k (x) = ħk ψ k (x) ψ k ψ k (x) = L e ikx k k = 2π n, n = 0, ±, ±2, L (3) ψ k [0, 2π] 2π n einx = δ(x) (4) H = p2 x 2m ψ k(x) (5) R R : x x = Rx = x H = U R HU R U Rψ(x) = ψ(r x) = ψ( x) (6) H H ψ k (x) H R (7) H H U R ψ ϵ,r (Hψ ϵ,r = ψ ϵ,r ϵ, U R ψ ϵ,r = ψ ϵ,r r ) r (8) r ± 3. () (2) (3) (4)
3 2 : 203 2 :. H = H x = p2 x 2m + 2 mω2 x 2 () (2) [x, p] = iħ a x [a, a ] = a = mω 2ħ (x + i p mω ) (3) H a (4) a 0 = 0 0 ψ(x) ψ(x) ψ ( x) (5) ˆn = a a [ˆn, (a ) k ], k = 0,, 2, (6) n = (n!) /2 (a ) n 0 ˆn n (7) H (8) a n = n + n + (9) a n = n n 2. 2 H 2 = p2 2m + 2 mω2 p 2, r = (x, y), p = (p x, p y ) () [a i, a j ] = δ ij a i = mω 2ħ (r i + i p i mω ) (2) H 2 = H x + H y H 2
3 2 : 203 2 (3) L z = xp y yp x [H 2, L z ] = 0 (4) L z a x, a y (5) α, β [α, α ] = [β, β ] =, [α, β] = 0 α = i a x β 2 i a y (6) H 2 α, β (7) L z α, β (8) H z L z 3. 3 ( H 3 ) () L 2 = L 2 x + L 2 y + L 2 z H 3 (2) a z H 3 a x, a y, a z (3) (4) L ± = L x ± il y, L 2 = (L 2 +L + L L + ) + L z L z (5) L ± α, β, a z (6) n α, n β, n z 000 L 2 000 (7) 00 L 2 00 (8) 00 L 2 00
3 3 : 203 3 :. L = r p () p = iħ [r i, p j ] (2) [L i, L j ] (3) [L i, r j ] (4) [L i, p j ] 2. J [J i, J j ] = iħϵ ijk J k () [J 2, J z ] = 0 (2) [J z, J ± ] (3) [J +, J ] (4) J J = iħj (5) J, J 2 ([J i, J 2j ] = 0) [J J 2, J z + J 2z ] (6) J = J + J 2 [J 2, J z ] 3. x r cos ϕ sin θ () 3 r = y = r sin ϕ sin θ, (Ω = θ, ϕ ) z r cos θ (e r, e θ, e ϕ ) ( r r, θ r, ϕ r) e i e j = δ ij (e r e θ = e ϕ )
3 3 : 203 2 (2) = e r r + e θ r θ + e ϕ r sin θ ϕ, r = re r L = r p (3) L + = ħe iϕ ( θ + i cot θ ϕ ), L = ħe iϕ ( θ + i cot θ ϕ ) (4) L 2 L z lm Y lm (Ω) = Ω lm = Θ lm (θ)φ m (ϕ) L z Y lm = ħmy lm Φ m (ϕ) = 2π e imϕ 2π 0 dϕ Φ m (ϕ) 2 = (5) L + Y ll = 0 Θ ll Θ ll θ (6) C l Θ ll = C l sin l θ l cot θθ ll = 0 π (7) C l = ( ) l C l, C l > 0 dθ sin θ Θ ll (θ) 2 = C l C l = ( ) l (2l + )! 2 2 l l! (8) Y ll L [ ] Y lm (Ω) = ( ) m+ m 2l + (l m )! e imϕ m 2 sin m d θ P l (cos θ) 2 (l + m )! 2π d cos θ Y lm, (m = l,, l) l = 0,, 2 d l P l (t) = 2 l l! dt l (t2 ) l l 0
3 4 203. σ x = 4 : 0 0, σ y = 0 i i 0 2 E 2, σ z = 0 0 () σ 2 x = σ 2 y = σ 2 z = E 2 (2) σ x σ y = σ y σ x = iσ z (3) σ y σ z = σ z σ y = iσ x (4) σ z σ x = σ x σ z = iσ y (5) S = ħ 2 σ, S [S i, S j ] = iϵ ijk S k (6) S z ħm m (7) S 2 = S 2 x + S 2 y + S 2 z (8) S 2 m m S 2 ħ 2 S(S + ) S > 0 (9) (a σ)(b σ) = a z a x ia y a z + ia y a z b z b x ib y b z + ib y b z (a σ)(b σ) = (a b)e 2 + i(a b) σ 2. 3 ñ = (n x, n y, n z ), ( n = ) e iφn σ = E 2 cos φ + in σ sin φ ( ) () (n σ) 2 = E 2
3 4 203 2 (2) P ± = 2[ E2 n σ) ] P± 2 = P ± (3) P + P = 0, P P + = 0 (4) P + + P (5) (n σ)p ± = ±P ± (6) n σ = P + P n (iφn σ) n = (iφ) n P + + ( iφ) n P (7) e in σ = n=0 (iφn σ)n /n! (*) (8) n n x = cos ϕ sin θ, n y = sin ϕ sin θ, n z = cos θ n σ u N = P +, v N = P 0 0 (9) u N = u N / u N, v N = v N / v N n σ +, (0) u S = P 0 +, v S = P 0 u S = u S / u S, v S = v S / v S u N u S v N v S () P + = u N u N = u Su S, P = v N v N = v Sv S 3. Θ = iσ 2 K (K ) () p = iħ ΘpΘ (2) S = ħσ/2 ΘSΘ (3) Θ 2 (4) ψ, ϕ ψ ϕ = ϕ ψ = Θϕ Θψ (5) ψ, ψ Θ = Θ ψ ψ ψ Θ = 0
3 5 203 5 :. 2 J, J 2, [J iµ, J jν ] = δ ij iħϵ µνλ J iλ, i =, 2 j i m i J 2 i j i m i = ħ 2 j i (j i + ) j i m i, J iz j i m i = ħm i j i m i jm = jm 2 = ħ (j + m)(j m + ) J jm ħ 2 (j + m)(j + m ) (j m + )(j m + 2) J jm 2. jm k = [ ] /2 ħ k (j + m) (j + m k + ) (j m + ) (j m + k) J }{{}}{{} jm k k k () J = J + J 2 J (2) j max = j +j 2, j min = j j 2 j max j=j min (2j+) = (2j +)(2j 2 +) (3) Clebsch-Gordan 2. j = /2, j 2 = /2 Clebsch-Gordan () J z,,, 2 2 2 2 J +,,, 2 2 2 2,,, 2 2 2 2 (2) 0 J (3) J 2 (4) 00 = a, + b 2 2 2 2 2, 0 00 = 0 2 2 2 a, b b > 0 00 00 = a 2 + b 2 =
3 5 203 2 (5) 00 J + 00 = 0 (6) Ψ 0 = ( 0, 00 ), ψ 0 = (,, 2 2 2 2 2 ψ C (7) 2 2 P = ψ C ψ C 2, 2 2 0 = ψ 0ψ C (8) (E 2 P )ψ C = 0 E 2 0 (9) ϕ = ψ = (E 2 P )ϕ (0) ψ = ψ / ψ 00 = ψ 0 ψ ( ) () 2 2 M 0 Ψ 0 = ψ 0 M 0 3. j =, j 2 = Clebsch-Gordan () J z J + 22 (2) 2 (3) 20 (4) 2 (5) 2 2 (6) = a 0 + b 0 2 = b > 0 (7) 0 (8) (9) J
3 5 203 3 (0) Ψ = ( 2, ), ψ = ( 0, 0 ) Ψ = ψ M M () Ψ 0 = ( 20, 0, 00 ), ψ 0 = (, 00, ) 00 = ψ 0 ψ ψ (i) 20 = ψ 0 ψ C ψ C (ii) 0 = ψ 0 ψ C2 ψ C2 (iii) ψ C = (ψ C, ψ C2 ) P = ψ C ψ C 0 (iv) ϕ = 0 ψ = (E 3 P )ϕ E 3 (v) ψ C ψ = ψ C2 ψ = 0 (vi) ψ = ψ / ψ 00 = ψ 0 ψ (a) J z 00 (b) J + 00 (c) J 00 (2) Ψ 0 = ψ 0 M 0 M 0 (3) m m 2 jm
3 6 203 6 :. j =, j 2 = 2 Clebsch-Gordan 2. S = /2 3 S i, (i=,2,3) (J > 0) H 3 = J(S S 2 + S 2 S 3 + S 3 S ), S 2 i = ħ 2 2 3 2 = 3 4 ħ2 (i) s s 2 s 3 = s s 2 s 3, s i = (ii) S = S + S 2 + S 3 H = JS 2 9 4 Jħ2 (iii) 2 2 = 0 2 2 2 = 2 2 3 2 (iv) H (v). H 3. 2 ( )S S i, (i=,2) (J > 0) H S = J S S 2, S 2 = ħ 2 S(S + ) (i) S = S + S 2 H S S (ii) S S = 2S 2S 0 H S
3 6 203 2 (iii) m H S m = J m (S S 2 ) m 4. S /2 N N H N = J i<j S i S j, S 2 i = 3 4 ħ2 (i) H N S = N i= S i (ii) s,, s N, s i = ± 2 (iii) S z s,, s N = ħm s,, s N, M = N i= σ M = N/2 m D M (m = 0,, N) ( :N m S z ħm ) (iv) S 2 s,, s N = ħ 2 S(S+) s,, s N, S z s,, s N = ħs s,, s N,, d S, D S S = N/2 d S = D S D S+ S = 0,, N/2 (v) N/2 (2S + )d S = 2 N (vi) H N 0
3 7 : 203 7 :. R k D (q) (R) 2k + O (k) q O q (k) RO (k) q R = q O (k) q D (k) q q (R) (a) 2k + T (k) q k (b) R R = e iδθn J O k q k ( Dq k q (e iδθn J ) = δ q q iδθn D (k) q q (J) ) (c) 2 A, B A B = 2 (A +B + A B + ) + A z B z (A ± = A x ± ia y ) (d) k T k q J 2 T k q = ħ 2 k(k + )T k q 2. (a) jm, jm J 2, J z jm jm m (b) Wignet-Eckart (c) O S jm O S j m = 0, j j 0 (d) O V jm O V j m = 0, j j 0, ±
3 7 : 203 2 3. 2 U, V T (k) q = T j q T j 2 q 2 j q, j 2 q 2 kq (a) U (b) U ± = U ± / 2, U 0 = U z (c) 2 (d) 0 U V (e) U V (f) 2 U ±, U 0, V ±, V 0 CG j m m m 2 j m, j 2 m 2 jm 2 2 2 0 / 2 2 0 / 2 2 0 - / 6 2 0 0 0 2/ 6 2 0 - / 6 2 - - 0 / 2 2-0 - / 2 2-2 - - j m m m 2 j m, j 2 m 2 jm 0 / 2 0 / 2 0 - / 2 0 0 0 0 0 - / 2 - - 0 / 2-0 - / 2 0 0 - / 3 0 0 0 0 / 3 0 0 - / 3
3 8: 203 8 :. 3 () 3 R det R = (2) 3 (3) v R(v) Q R = QRQ R (4) (α, β, γ) ()z R α (z) ( (x, y, z = z) ) (2) y β R β (y ) ( (x 2, y 2 = y, z 2 ) ) (3) z 2 γ R γ (z 2 ) R(α, β, γ) = R γ (z 2 )R β (y )R α (z) R(α, β, γ) = R α (z)r β (y)r γ (z) 2. H E ψ i, (i =,, d) d ( ) R H [H, R] = 0 () R ψ j = ψ i D ij (R) d D: {D(R)} ij = D ij (R) (2) ψ = ( ψ,, ψ d ) Rψ = ψd(r) (3) (R 2 R )ψ = ψd(r 2 )D(R ) (4) (R )ψ = ψ[d(r)]
3 8: 203 2 3. H(x) = ħ2 d 2 ψ 2m dx 2 ψ(x) ψ(x + L/2) = ψ(x L/2) ψ k = L /2 e ikx, k = 2π n, n = 0, ±, ±2, L P ψ(x) = ψ( x) () {P, P 2 = } ( ) (2) k > 0 ψ = (ψ k, ψ k ) R = P, Rψ = ψd(r) D(R) (3) ψ ± = (ψ k ± ψ k )/ 2 P ψ + = ψ +, P ψ = ψ (4) ψ = (ψ +, ψ ) ψ = ψ U U (5) Rψ = ψ D (R) D (R) R =, P (6) D(R) D (R) 4. ẑ R(α, β, γ) ψ(r(α, β, γ)ẑ) Q Qψ(R(α, β, γ)ẑ) = ψ(q R(α, β, γ)ẑ) l l DMN l (R) M = l,, l (N ) ψ M (R(α, β, γ)ẑ) = [DMN(R(α, l β, γ))] ψ M D l Qψ M (R(α, β, γ)ẑ) = [DMN(Q l R(α, β, γ))] = [DMK(Q l )] [DKN(R(α, l β, γ))] = [[D l (Q)] ] MK[DKN(R(α, l β, γ))] = [DKN(R(α, l β, γ))] DKM(Q) l = ψ K (R(α, β, γ)ẑ)dkm(q) l
3 9 : Schwinger Boson 203 9 : Schwinger Boson. a, a [a, a ] = ˆn = a a () aˆn = (ˆn + )a (2) aˆn 2 = (ˆn + ) 2 a (3) aˆn k = (ˆn + ) k a (k =, 2, ) (4) x f(x) af(ˆn) = f(ˆn + )a (f(x) x = 0 ) (5) e iθˆn ae iθˆn = e iθ (6) k = 0,, 2, [a, (a ) k ] = k(a ) k (7) [a, f(a )] = a f(a ) (8) af(a ) 0 = a f(a ) 0 2. J = 2 a σa, a = a + () [J i, J j ] = iϵ ijk J k a (2) J 2 = 2 ˆn( 2 ˆn + ), ˆn = a a [a ξ, a η] = δ ξη (3) n = 3 h = n σ ± σ (4) h ± ψ ± U = (ψ +, ψ ) a = a + a = Ua [a ξ, a η ] = δ ξη
3 9 : Schwinger Boson 203 2 (5) n J = 2 (ˆn + ˆn ) (6) P ± = ψ ± ψ ± h ± e iθn J ae iθn J = Ue iθn J a e iθn J () = Udiag (e i θ 2, e i θ 2 )a (2) = Udiag (e i θ 2, e i θ 2 )U a (3) = (ψ +, ψ )diag (e i θ 2, e i θ 2 ) ψ + ψ a (4) = (e i θ 2 P+ + e i θ 2 P )a (5) = 2 (ei θ 2 (E2 + (n σ)) + e i θ 2 (E2 (n σ)))a (6) = (E 2 cos θ 2 + in σ sin θ )a (7) 2 (7) u = E 2 cos θ + in σ sin θ SU(2) 2 2 3. u SU(2) σ = uσu = σ σ 2 σ 3 = Qσ = Q σ σ 2 σ 3 () σ α = σ α (2) Trσ = 0 (3) σ α = Q αβ σ β, Q αβ R (4) {σ α, σ β } Q αγq βγ = δ αβ {A, B} = AB + BA (5) Q Q = E 3 (6) σ σ 2σ 3 = 2iE 2 2 det Q =
3 0 : 203 0 :. () f(z) z C f(z) = dξ f(ξ) 2πi ξ z n f (n) (z) = n! f(ξ) dξ 2πi C (ξ z) n+ (2) P l (t) = d l 2 l l! dt l (t2 ) l P l (t) = [ ] ξ 2 l dξ 2πi C t 2(ξ t) ξ t (3) ζ = ξ2 2(ξ t) ξ = R, R = 2tζ + ζ ζ 2 ) (4) ζ 0 R ζ 0 ξ t ζ C C + t ξ C + 0 R ζ = 0 (5) dξ = ξ t dζ ζr (6) P l (t) = d l l! dζ l R ζ=0 (7) R ζ = 0 ( ) 2tζ + ζ 2 = P l (t)ζ l l=0
3 0 : 203 2 (8) (r > = max( r, r ), r < = min( r, r ) ) r r = l=0 r< l r> l+ P l (ˆr ˆr ) = 4π lm 2l + r< l r> l+ Y lm (ˆr)Y lm( ˆr ) (9) ρ(r) ϕ(r) ϕ(r) = 4πϵ 0 lm 4π q lm 2l + r Y lm(ˆr), q l+ lm = d 3 r (r ) l ρ(r )Y lm(ˆr ) 2. 3 () R R j m, j 2 m 2 = (R j m ) (R j 2 m 2 ) = j m, j 2 m 2 D j m m (R)Dj 2 m 2 m 2 (R) = jm D j m m (R) jm j m, j 2 m 2 (2) j D j m m D j m m Dj 2 m 2 m 2 = j m, j 2 m 2 jm D j m m jm j m, j 2 m 2 2l + (3) Y lm (β, α) = 4π [Dl m0(α, β)] π 0 dβ sin β 2π 0 dα Y lm(β, α)y l m (β, α)y l2 m 2 (β, α) [ ] (2l + )(2l 2 + ) = l0 l 0, j 2 0 lm l m, l 2 m 2 4π(2l + )