Sakurai 现代量子力学——量子力学中的对称性

7. =N:GL-/+F 7.1 >O;HM.0,A G �}sB, jW�r~ a ◦ b V�� • a ◦ b ∈ G ∀a, b ∈ G � • (a ◦ b) ◦ c = a ◦ (b ◦ c) ∀a, b, c ∈ G � • ∃1 Æ 1 ◦ a = a ◦ 1 = a ∀a � • ∀a∃a−1 Æ a ◦ a−1 = a−1 ◦ a = 1 w G �jW�O L� G pSn�?_LpSn~X_��?Li_��_�Tx g = 1 + ih+O(h2), g ∼ 1. �Tx g = {h} [hi, hj ] = i~fijkhk = lim ε→0 − 1ε2 [eiεhiejεhje−iεhie−iεhj − 1] � Le�Æ • ~XL – Z2 Æ {1, a} a2 = 1 – Z Æ {n} n ◦m = n+m � • �?L – U(1) Æ {eiθ, θ∈[0,2π]} eiθ ◦ eiφ = ei(φ+θ) �TxÆ R : [h, h] = 0 � – SU(2) * SO(3) �TxÆ R3 : [hi, hj ] = iǫijkhk � L G _(kÆ D(g) : H → H , +:, ∀g ∈ G D(g)D(h) = D(gh) D −1(g) = D(g−1) (D−1 = D†V�(�(k) D(1) = 1 V D−1(g)HD(g) = H, ∀g ∈ G w G {\F��#�_q>:� Vq ∀g pO D(g) 4?vqQj~8j ( D(1) 0 0 D(2) ) w(knpm_��w(k.pm� s0�Æ gAw% ÆFO �?q>#� αi ∂L ∂qi = 0⇒ αi d dt ( ∂L ∂q˙i ) = 0⇒ αiPis0� 1 ���CÆ D(g)HD(g−1) = H g = i+ ih+O(h2)⇒ [h,H ] = 0⇒ 〈h〉s0� �T V H t SU(2) ��3"%.$ ~J s0� L*Æ V H | ψ〉 = E | ψ〉 D−1(g)HD(g) = H HD(g) | ψ〉 = D(g)H | ψ〉 = D(g)E | ψ〉 ��S D(g) | ψ〉 | ψ〉 i_,�_'��G t,�_'�%ÆF�'Es�+��T Z SU(2) ��.$:��9 Q/d ?i�Æ 2p ��_ 3 �'Ei_.�_L*'� 2 7.2 K, 751I� \^ ~x→ ~x �~Xq> ,Z_L� G = Z2, {1, a}, a2 = 1 � Z _(k�Æ D(a)2 = 1 YKO�"v-�sKÆ.pm(k� D(a) = ±1 � 'GÆqad>$7 Π = D(a) � jWÆ Π | ~x〉 =| −~x〉 � ,�mj2��� Π _: Æ Π† = Π, Π2 = 1, (Π~ˆxΠ) ∫ f(~x) | ~x〉 = Π~ˆx ∫ f(~x)~x〉 = Π ∫ f(~x)− ~x | −~x〉 = ∫ f(~x)(−~x) | ~x〉 = −~ˆx ∫ f(~x) | ~x〉. YK Π†~xΠ = −~x = Π~xΠ �,|^ Π~PΠ = Π(−i~α)Π = −~P �YK {Π, ~x} = {Π, ~P} = 0 � ~L = ~x× ~P ⇒ Π~L = ~LΠ, [Π, ~L] = 0 � �x5z qa�� ΠR(nˆ, θ) = R(nˆ, θ)⇒ [Π, ~J ] = 0 �pS2� [Π, ~S] = 0 ��S Π {�#'� k� Y.{�Qk�� (kyÆ • �g�Æt��%}|g�_$7 4d> (~x, ~P ) • �g�Æt��%}|g�_$7 4d> (~L) • '�Æt��%}|'�_$7 )d> (x2, ~x · ~P , ~L · ~S) • $'�Æt��%}|'�_$7 4d> (~S · ~x, ~L · ~P ) � d>$7%_,&xÆ ψ(~x) = 〈~x | ψ〉 td>$7% ψ(~x)→ ψ˜(~x) � ψ˜(~x) = 〈~x | Π | ψ〉 = 〈−~x | ψ〉 = ψ(−~x). V Π | ψ〉 = ± | ψ〉 ψ(x) = ±ψ(−x) ψ td>$7%�);4� k�*Qk�_��Æ Π | ~P 〉 =| −~P 〉 6= ± | ~P 〉 �Y^a [~L,Π] = 0 pS�xqQ4 ~L,Π Π | θ, φ〉 =| π − θ, φ + π〉, Ylm = 〈θ, φ | l,m〉. Y00 =;x Æi_)d>� Y1m = sin θe ±iφ, cos θ i_4d>� ⇒ Ylm i_d> (−1)l ^a�℄ Clebsch-Gordan #xp*,Qk� Ylm ∼ (Y1m)2 � '���Æ I_Æ [H,Π] = 0 V H = P 2 2m + V (x) ΠHΠ = P 2 2m + V (−x) YK V (x) = V (−x) td>$7% �4� V H | ψ〉 = E | ψ〉 Π | ψ〉 i_,�_X �YK 9F( �AwÆ_OOO�Æ (a) }L* 3 Π | ψ〉 = ξ | ψ〉, ξ2 = 1⇒ ξ = ±1 (b) L*�� Πψ〉p',q | ψ〉n+:m�_�V?� | φ±〉 =| ψ〉±Π | ψ〉 Π | φ±〉 = ± | ψ〉+Π | ψ〉 = ± | φpm〉� pS�xqQ4 H,Π YK�_ E _��pS�Aj� Π _��� �Æ�^�� H | ~P 〉 = P 2 2m | ~P 〉 Π | ~P 〉 =| −~P 〉 | φ±〉 = 1√ 2 (| ~P 〉± | −~P 〉)� H,Π _�M��� AvjwÆ o�ÆΠΘΠ = λΘ Π | ψ〉 = ξ | ψ〉 Π | ψ′〉 = ξ′ | ψ′〉 3Æ λ, ξ, ξ′ ∈ {−1, 1} � 〈ψ | Θ | ψ′〉 = 〈ψ | ΠΠΘΠΠ | ψ′〉 = λξξ′〈ψ | Θ | ψ′〉. G} λξξ′ = 1 wZjX � 0 � Θ �) ⇒| ψ〉, | ψ′〉 i_,�d>� Θ �4 ⇒| ψ〉, | ψ′〉 i_,{d>� �Æ E1 o9 〈ψ′~x | ψ〉 .��℄Z | ψ〉 | ψ′〉 i_,{_d>� M1 o9 〈ψ′ | ~L+ g~S | ψ〉 .� �℄Z | ψ〉 | ψ′〉 i_,�_d>� 4 7.3 C51I o� grk|A mx¨ = −∇V (x) x(t) Y ⇒x(−t) Y��_�� g#�lndK{G.$ _Æ qi(t)→ qi(−t) .$� ��#�Æ ψ(x,−t) .��rk|A i~ ∂ψ ∂t = ( − ~ 2 2m ∇2 + V (x) ) ψ(x, t) Y ψ∗(x,−t) � ψ(x, t) = ∑ Cn(0)e − i ~ Ent ψ∗(x,−t) = ∑ C∗n(0)e −(− i ~ )En(−t) = ∑ C∗n(0)e − i ~ Ent. �kdK{G�$ CPX �)� {(�$7Æ 8VÆ(�$7i_: U † = U−1 �C&>.$Æ | α˜〉 = U | α〉, | β˜〉 = U | β〉 ⇒ 〈β˜ | α˜〉 = 〈β | U †Uα〉 = 〈β | α〉 � V�X i_$7.$: w |〈β˜ | α˜〉| = |〈β | α〉| � O $7 θ Æ | α〉 →| α˜〉 = θ | α〉 | β〉 →| β˜〉 = θ | β〉 � {+:$7 V�� θ(C1 | α〉 + C2 | β〉) = C∗1 | α˜〉+ C∗2 | β˜〉, $7� {(�$7 V��{+:= 〈β˜ | α˜〉 = 〈β | α〉∗ � Æj"v-�sK H Æ_O < | α〉 pSjW��) K Æ K( ∑ Ci | αi〉) = ∑ C∗i | αi〉. �Æ K P{a<_Av� jÆO+{(�~� θ pS3? θ = UK 3Æ U �(�_� qa<_.�Av U,K _" ℄K�4_j�� Av< | a〉 ,Z_ K Æ K(∑Ca | a〉) =∑C∗a | a〉 �� θK "℄ | α〉 →| α˜〉 = θK | α〉 = ∑ a θK | a〉〈a | α〉 = ∑ a 〈a | α〉θ | a〉, | β〉 →| β˜〉 = ∑ b 〈b | β〉θ | b〉, ⇒〈β˜ | α˜〉 = ∑ a,b 〈b˜ | a˜〉〈β | b〉〈a | α〉 = ∑ a,b 〈β | b〉δba〈a | α〉 = 〈β | α〉. ⇒θK�(�_� ,�_X�ÆO+ UK �{(�_ �K U �(�_� dK{G~� Θ Æ pS2U Θ �$ K � 5 KDÆ | ψ(−δt)〉f = Θ | ψ(δt)〉r , | ψ(0)〉f = Θ | ψ(0)〉r. | ψ(−δt)〉f 〉 = ( 1 + iH ~ δt ) | ψ(0)〉f = ( 1 + iH ~ δt ) Θ | ψ(0)〉r = Θ | ψ(δt)〉r = Θ ( 1− iH ~ δt ) | ψ(0)〉r ⇒iHΘ = −θH. V Θ �(�_ HΘ = −ΘH ��Æ HΘ | ~P 〉 = −ΘH | ~P 〉 = − P 22mΘ | ~P 〉, E < 0 �le, '� GuT� � Θ �{(�_ ⇒ [H,Θ] = 0 � t Θ %~�_9� Θ �{(�_ A �t�_ w 〈β | A | α〉 = 〈α | A | β〉∗ = 〈α˜ | ΘA | β〉 = 〈α | ΘAΘ−1 | β˜〉. tdK{G% O ~��4;)d> Z ΘAΘ−1 = ±A. 〈β | A | α = ±〈α˜ | A | β˜〉 = ±〈β˜ | A | α˜〉∗. V | α〉 =| β〉 〈α | A | α〉 = ±〈α˜ | A | α˜〉 �dK{GZ�C ~x .$�AvÆ Θ | ~x〉 =| ~x〉 ,��m j� ⇒ Θ~xΘ−1 = ~x �q0&UWZ_,&x | ψ〉 = ∫ ψ(x) | ~x〉 Θ | ψ〉 = ∫ ψ∗ | ~x〉, ψ(x)→ ψ∗(x) �) Θ | ~P 〉 = Θ ∫ 1√ 2π~ ei ~P ·~x/~ | ~x〉 = ∫ 1√ 2π~ e−i~P ·~x/~ | ~x〉 =| −~P 〉. z?{ _ Θ~PΘ−1 = −~P ⇒ Θ(~x× ~P )Θ−1 = −~x× ~P . �O�_8jÆ Θ ~JΘ−1 = − ~J. ���Aw,O Yn/�M pSyz\��Aw → ;K�C [Ji, Jj] = i~ǫijkJk � qQk���Æ Ylm i_,� e imφ � Θ | l,m〉 = (−1)m | l,−m〉. 6 l ∈ Z Btr1"MÆ�? l ∈ Z+ 1/2 _F~�� dK{G ��Æ o���� 1/2 _�� JzΘ | +〉 = −ΘJz | +〉 = −~ 2 Θ | +〉, YK Θ | +〉 = ξ | −〉 ξ `J,��Yn"<�U\ | −〉 = e−iπSy/~ | +〉 YK Θ | −〉 = ξe−iπSy/~ | −〉 = −ξ | +〉 Θ = ξe−iπSy/~Kqa�� 1/2 #�� '�mjÆ ξ = 1 Θ = ( 0 −i i 0 ) K = σyK. �Æ Θ2 = σyKσyK = −σ2yK2 = −σ2y = −1 �X .t,�Av_[-�qaO+^4d>~�� �?_#� Θ2 = −1 �__~��i_ 1/2 �x����� dK{G.$:_X Æ ��R ���t Θ "℄%~�_9� ΘAΘ−1 = ±A ��_9�t'�AoZ[E P{a ,�_Av�`� [H,Θ] = 0 *.'SKPj Θ �p�4���.Nts0jw;Avjw� �Æo�� H | ψ〉 = E | ψ〉 Θ | ψ〉 =| ψ〉 [H,Θ] = 0 �T����_e,&x� | ψ, t〉 = e− i~Et | ψ〉 Θ | ψ, t〉 = e i~Et|ψ〉6=|ψ,t〉. dK{G69^F3�_X�� � [H,Θ] = 0 H | n〉 = En | n〉 HΘ | n〉 = ΘEn | n〉 = En(Θ | n〉) �YK | n〉 Θ | n〉 i_L* '�n��,�_��V�,�_� Θ | n〉 = eiδ | n〉 � Θ2 | n〉 = Θeiδ | n〉 = e−iδΘ | n〉 =| n〉. YKqa��� 1/2 �x�_� | n〉 Θ | n〉 "M�+:��_� Kraner L*Æ O+�$4x ~��_dK{G.$_#� i_ \ 2 �L*� HU�J998`+�(/� H = ~S · ~B �L*�O ~B "��9I Θ~S = −~SΘ YK [H,Θ] 6= 0 �V���$j ~B L{�� �Æ � + h� H ∼ ~I · ~B ~F = ~I + ~S [H,Θ] = 0. =b$ÆÆ F = 1 qZ 3 �� F = 0 qZV��Ynqa�__� Θ2 = 1 YKnp9_� V I = 1, S = 1/2 F = 3/2 4 ��� F = 1/2 2 ��� z(_ Kraner L*� 7 7.4 PE8B0,.62?J o��2m V (x + a) = V (x) ��Æt�w��Æh�_rk�fY'�1 q>:� .Q~�Æ jW τ(l) Æ! τ(l) | x〉 =| x+ l〉 τ(l)† = τ(l)−1 � τ(l)†xˆτ(l) | x〉 = τ(l)†xˆ | x+ l〉 = τ(l)†(x+ l) | x+ l〉 = (x+ l) | x〉 ⇒τ(l)†xˆτ(l) = xˆ+ l. τ(l) | P 〉 = τ(l) ∫ dx√ 2π~ eiPx/~ | x〉 = ∫ dx√ 2π~ eiPx/~ | x+ l〉 = e−iP l/~ | P 〉. YK τ(l) = e−iPˆ l/~ = e−l ∂ ∂x , τ†(l)Pˆ τ(l) = Pˆ . qaO�UWZ_,&x τ(l) | ψ〉 = τ(l) ∫ dxψ(x) | x〉 = ∫ dxψ(x) | x+ l〉 = ∫ dyψ(y − l) | y〉, Z | ψ′〉 = τ(l) | ψ〉 ψ′(x) = ψ(x − l) = e−l ∂∂xψ(x). qat�2:"�r� V (x+ a) = V (x) Æ_�� H = P 2 2m + V (x) τ†(a)Hτ(a) = τ†(a)V (x+ a)τ(a) + P 2 2m = H, YK [H, τ(a)] = 0 � L�Æ ~X.QL Z^ α�? LhÆ· · · , α−1◦α−1, α−1, 1, α, α◦αα◦α◦α · · · `Ja · · ·α−2, α−1, α0, α1, α2 · · · � {αn} Æ αn ◦ αm = αm+n � �KjL_(kÆqQ4 D(α) .pm(k� 1 �_ D(α) = e−iθ �^a [H, τ(a)] = 0 τ(a) = D(α) pS�x^qQ4 H � τ(a) �3F Θ = ka � | ψk〉 �� τ(a) | ψk〉 = e−ika | ψk〉 ψ(x− a) = e−ikaψ(x) ψ(x+ a) = eikaψ(x). 3F ψ(x) = eikxψ˜(x) eik(x+a)ψ˜(x+ a) = eik(x+a)ψ˜(x) ψ˜(x+ a) = ψ˜(x) ��S Yt x→ x+ a Æn��2_� 8 /�.jÆ �Æi_�jsKK�_�Cm tFe 1 Æ_�Cmje�� H | nk〉 = En | nk〉, τ(a) | nk〉 =| (n+ 1)k〉, 3Æ n �a��� k �'E�(kÆ | θk〉 = 1√ 2π ∞∑ n=−∞ einθ | nk〉, H | θk〉 = Ek | θk〉, τ(a) | θk〉 = 1√ 2π ∞∑ n=−∞ einθ | (n+ 1)k〉 = e−iθ | θk〉. �O4ÆV 〈nk | ml〉 = δnmδkl 〈θk | θ′l〉 = δklδ(θ − θ′). t��Æ �_'ElnL*_ �*�� �Æ�^�� V = 0 �� o��� | P 〉 � H | P 〉 = P 22m | P 〉 τ(a) | P 〉 = e−iPa/~ | P 〉 � E _1�? y�L*� 0&_8jÆ[a�^ j*je j�K_ÆK j� \w�_|Æ O�LV_ 7Æ • I_ta��Km|�� Y.Ea�C • O� | n〉 � Fe_<��S� ÆFa� jÆ 〈n | n′〉 = δnn′ τ | n〉 =| n+ 1〉. � \w�_| �Æ 〈n | H | n′〉 = 0 G}n′ ∈ {n− 1, n, n+ 1}. 9 jW 〈n± | H | n〉 = −∆ jW [τ,H ] = 0 � YK H =   E0 −∆ ∆ E0 −∆ 0 −∆ E0 −∆ 0 −∆ E0 . . . . . . . . .   �Æ�s_$Wt��Æ�2��� jWÆ | θ〉 =∑ einθ | n〉 τ | θ〉 =e−iθ | θ〉 H | n〉 =E0 | n〉 −∆ | n− 1〉 −∆ | n+ 1〉 H | θ〉 =E0 | θ〉 −∆ ∑ einθ(| n+ 1〉+ | n− 1〉) = [ E0 −∆(eiθ + e−iθ) ] | θ〉 = (E0 − 2∆cos θ) | θ〉. YK L*� ∆ �a� t/�gH&^\�?'S E0 − 2∆ ≤ E ≤ E0 + 2∆ � a'��Æ | θ = 0〉 � '��Æ | θ ± π〉 0 =∑(−1)n | n〉 � 0&A8%_'�1Æ fY|A H | ψ〉 = E | ψ〉 − ~ 2 2m ψ′′(x) + V (x)ψ(x) = Eψ(x), V (x+ a) = V (x). 10 wU|AÆNt +:��Y ψ1 ψ2 �Y�2: ψ1(x+ a) ψ2(x+ a) TpYF�( ψ1(x+ a) ψ2(x+ a) ) = ( A11 A12 A21 A22 )( ψ1(x) ψ2(x) ) , A = ( A11 A12 A21 A22 ) $7d~� ψ1 ψ2 t φ �e ⇒A �e� qQ4 A Æ φ1(x+ a) = λ1φ1(x) φ2(x+ a) = λ2φ2(x) λ1 λ2 � A _��� λ _|AÆ det(A− λ1) = 0 (A11 − λ)(A21 − λ)−A12A21 = 0 λ2 − (A11 +A22)λ+ (A11A22 −A12A21) = 0 λ2 − (TrA)λ + detA = 0 ⇒λ = [TrA± √ (TrA)2 − 4detA]/2. YK�'n �AwÆ_O�Æ a.λ1, λ2 T�ex b.λ1 = λ ∗ 2 � d dx (φ1φ ′ 2 − φ2φ′1) = φ1φ′′2 − φ′′1φ2 = 0 (φ1φ ′ 2 − φ2φ′1)x+a = (φ1φ′2 − φ2φ′1)x = λ1λ2(φ1φ′2 − φ2φ′1)x YKÆ λ1λ2 = 1 � V λ1 λ2 T�ex λ1 = 1 λ2 �G} a b m�� w φ1 φ2 B�xy<��}�.p�O 4Y� V λ1 = λ ∗ 2 w φ1 φ2 ���2_���Y Tk��OJ�O4� λ � E _&x Æ! A Kj� a. Z� a Awd λ+ 1λ = TrA ≥ 2 b. Z� b Awd λ1 + λ2 = TrA = e iα + e−iα = 2 cosα ≤ 2 � YK q='S�aFe TrA ≤ 2 Æ�P6fÆ A = ±1 φi(x+ a) = ±φi(x) F��2Y;{� 2Y� |m&_j:�vÆ 11 dO'SÆ a'��Æ λ = 1 �2� �sK C Æ λ _�℄ '�� λ = −1 ��<�_z�8j dw'SÆ a'��Æ λ = −1 O� m&Æ_�&x�8 S '��Æ λ = +1 '�1 SZ�OU�2mF~X _0&8j�\#:�� �o��Vh�Aw pS�$\s h�8j�q=SI1E�Ælk��q=�1�EÆ[�� 12 7.5 �D9O <9O� gAw% h�pS�% �'G�� T�k TptT%!AÆ%F t���CÆ*}TK �!Am_�)� 2 ��"v-�sK H12 = H1 ×H2 = H ×H qaI���� • 1- ��< {| n〉} • 2- ��< {| n,m〉 =| n〉⊗ | m〉} !2Aw% | n〉 | m〉 �� eHZqaI����y% | n,m〉 * | m,n〉 P7L*�� 8VhJ9_��4Æ 2- ��� a†k,αa † k′α′ | 0〉 = a†k′,α′a†kα | 0〉 ts�� Fock sKÆ�,�_�� �7~�Æ P12 | n,m〉 =| m,n〉 P7��� P12 = P21 P 2 12 = 1 � P12 `?� Z2 q>L P12 = D(a), a 2 = 1 � Z2 _.pm( kÆ P12 = ±1 tO��sKÆ� ��Æ | n,m〉s = 1√2 (| n,m〉± | m,n〉), n 6= m �qa m = n P12 | n,m〉 = + | n, n〉 YK.n {q>�� qaI��� "�r�t�7 1←→ 2 %q>� H = P 21 2m + P 22 2m + Vext(x1) + Vext(x2) + V (|x1 − x2), P12HP12 = H. �MZÆNt ���Æ • +Y�Æ P12 = +1 ,Y - �Y{��F��Æ��� • ~��Æ P12 = −1 ~��F��Æh� >�� uq� ���Fj�Æ pt,q���9�ÆF��� Sje:i�: �� 13 • ����x_���+Y� • ��� 1/2- �x_���~��� j�%��<���*,?�� e− �~�� ?i��+Y��� �Æo� h�Ia� S = 1 m = 1 � �,��� 180◦ eimπ = −1 ���P7h� ^~��FÆF -1 � Æj<���_j pS^\,?�_X � P (H) 12 = P (e) 12 P (p) 12 = (−1)(−1) = +1. ,�.,RiÆ ~��.'It,�_�Z� YÆ P12 | n, n〉 = + | n, n〉 � Yn+Y�pSIt,�_�Z���[ �7XhR_�X � • ��Æ_~��Æh� • +Y - �Y{�(f • d����~�6�``� Æ�6� s���# �x\ N I����#��F�v�j\ N ! ��ÆO���Æq 3 +Y� / ~���# | n,m, p〉s/A = 1√ 6 [| n,m, p〉± | n, p,m〉+ | m, p, n〉± | m,n, p〉+ | p, n,m〉± | p,m, n〉], 3Æqa P12, P23, P13 i_�� ±1 �1�Æ O_�a N > 2 _�s��� 2- h�#�Æ H = (H1 ⊗H2)A *�t P12 _ -1 �sKÆ�pS3F�&xÆ ψ = ∑ φmm′(x, x ′) | m,m′〉, m,m′ ∈ {−1 2 ,+ 1 2 }, H1 ⊗H2 = H (x)1 ⊗H (x)2 ⊗H (s)1 ⊗H (s)2 � H (s)1 ⊗H (s)2 , (s = s1 + s2) _< χ11 =| ++〉 χ10 = 1√ 2 (| +−〉+ | −+〉) χ1−1 =| −−〉  P (s) 12 = +1 S 2 = 1W�� q>� 14 χ00 = 1√ 2 (| +−〉− | −+〉) } P (1) 12 = −1 S2 = 0V� {q>� qa ��Aw pSAv H _<Æ ψ (x) A ψ (spin) S ψ (x) S ψ (spin) A ��YtKj N > 2 ��_< d J9.Æ O�H�s� qaW��Æ ψ = ∑ m=±1,0 φm(x, x ′)χ1m, φm(x, x′) = −φm(x′, x). qaV� ψ = φ(x, x′)χ00, φ(x, x′) = +φ(x′, x). W��Æ��q> ��{q>����� K#n� V�Æ��{q> ��q> ���pSIt,�_�Z� V.Nt,3"℄ φ = 1√ 2 (ωA(x1)ωB(x2)± ωA(x2)ωB(x1)) |φ|2 = 1√ 2 [|ωA(x1)|2|ωB(x2)|2 + ωA(x2)|2|ωB(x1)|2 ± Re(ωA(x1)ωB(x2)ω∗A(x2)ω∗B(x1))] 3Æ ωA(x1)ωB(x2)ω ∗ A(x2)ω ∗ B(x1) �P7�o�Z x1 = x2 d qW�� |φ|2 → 0 qV� |φ|2 → i zy� ,�����y;�� �Æqal~_�� P7�o → 0 ~��F��� 2- h�i�ÆH− He Li+ �� H = P 21 2m + P 22 2m − Ze 2 r1 − Ze 2 r2︸ ︷︷ ︸+ e2 r12︸︷︷︸ H0 V V.Nth�K,3"℄ wi_�&x 2E (0) 0 { | 1s, 1s〉S =| (100)(100)〉Sχ00 E (1) 1 ·E(1)2   | 1s, 2s〉A =| (100)(200)〉Sχ00 | 1s, 2s;m〉A =| (100)(200)〉Aχ1m | 1s, 2p;µ〉S =| (100)(21µ)〉Sχ00 | 1s, 2p;µ,m〉A =| (100)(21µ)〉Aχ1m ^ah�_fB_Em sKq>� V��Mt�*D^\�s_'��pS�℄�N��~ '�� #i�Æ �,3"℄<�'�Æ E0 = 2 ( −Z 2e2 2a0 ) ∼ −109eV(8 · (−13.6eV)). HU,3"℄' 〈 e2 r12 〉 ⇒ −74.8eV [ (−Z2 + 5 8 Z) ( e2 a0 )] . 15 eH�Æ −78.8eV ��KÆF��s_2x �℄$ypSbKF~\ 10−6 � r1"MÆ�℄ �N$yF~O�e��� #i�?x�Æ φS A (x1, x2) = 1√ 2 (ψ1(x1)ψ2(x2)± ψ1(x2)ψ2(x1)),〈 e2 r 〉 =e2 ∫∫ d3x1d 3x2[ψ1(x1)ψ2(x2) 1 r12 ψ∗1(x1)ψ ∗ 2(x2)± ψ1(x1)ψ2(x2) 1 r12 ψ∗2(x1)ψ ∗ 1(x2)] =VD ± VE �Æ a. VD ≥ 0 b. ∫ |ψ1(x1)ψ2(x2)± ψ1(x2)ψ2(x1)|2/r12 = 2VD ± 2VE ⇒ VD > |VE | c. ��N$7Æ 1 r12 = ∫ d3k ei ~k·(~x1−~x2) k2 VE = ∫ d3k k2 (∫ d3x1e i~k·~x1ψ1(x1)ψ∗2(x1) )(∫ d3x2e i~k·~x2ψ2(x2)ψ∗1(x2) ) ≥ 0 YKpS^\'E-Æ E (1) 1 + E (1) 2 � � �6 ? VD � �� @ @@ 6 VE ? VE V� �#� W�� �#� -Æ1ZlRm�A�℄*, M"�r�n����_ ZpS�v���,�,3"℄� 〈V 〉S A = VD − 1 2 (1 + ~σ1 · ~σ2)VE . ~σ1 · ~σ2 = 2(s2 − s21 − s22) = 2s2 − 3 s2 − 12 (1 + ~σ1 · ~σ2) W��Æ 2 -1 V�Æ 0 +1 ��V�Æ�#���V�Æ�#� pS 3� 2- h�i���Æ H− _w��� �j�Æ�N� ⇒ −0.4726 e2a0 > (−0.5 + 0) e 2 a0 �Yn$yF~ ⇒ −0.528 e2a0 � Æ59_|Æ 16 qa N ≥ 2 h�i�#� .NtR�Y Y�pS�℄Æ59_| =n�N�ÆI_"℄a� h�__1m9mzja)m9H3bh�_h(/m9� LV_8jÆ Hartree �89_|Æ qa N h�#� I_"℄ad i h�_m9^S% M8`Æ a. )m9 −Ze2/r b. 3bh�_h(/ ∑ k 6=i −e|φk|2 � O,&x3?�=8jÆ ψ(x1, x2, . . . , xN ) = φ1(x)φ2(x) · φN (xN ), Hartree |A Hiφi = −1 2 ∇2iφi − Ze2 ri φi + ∑ k 6=i (∫ dxk |φk(xk)|2e2 rki ) φi = εiφi. Æj ∫ φ∗i (xi)φi(xi)dxi = 1 〈ψ | Hi | ψ〉 = εi, 〈ψ | H | ψ〉 = 〈ψ | ∑ i ( −1 2 ∇2i − Ze2 ri ) + ∑ i�� ψ(~x1, ~x2) = φ(~x1)φ(~x2), Hartree |A −1 2 ∇2φ(~x)− Ze 2 |x| φ(~x) + ∫ d~y e2 |~x− ~y|φ(~y) 2φ(~x) = εφ(~x). }nO �RAq_> - �|A�pSe�YÆ ^p�Y&x φ0(~x) ni ℄zF~ V (~x) = ∫ d3~y e2 |~x− ~y|φ(~y) 2. 5U\Bj%|A��YF φ1(~x) · · · . 〈H〉 = 2ε− 〈 e2 |~x1 − ~x2| 〉 �pSF�Y^O� 7 r1"M�� 17 7.6N > 2 3�D9O40,A ��YsI����# N h}�7_q>L SN n"K_�i� �7L SN Æj N _>q/ a, b, c, . . . OO�7�q/>_0&�*��Æ P _9�℄P{q/_�� 'f��� p�℄D5X��vOU�7� 3" (1342)(5) � Æ;b��2� 1 _D5 ⇒ (1342) �� "℄t N q/Z_ N ! �7�?�7L SN � SN �}��vL O�Aw% P1P2 6= P2P1 � �Æ P1 = (123) P2 = (12) 7�~� P(ij) �7 i, j 3� (ij)�OO�7TpS3�O� P(ij) _�>��7_d>� δp = (−1)k 3Æ k � P _�_P7��_Lx� SN _(k� o�^ {1, . . . , N} _�_�7�?_ N ! �g�sK��Æqa N = 3 | 123〉 | 132〉 | 231〉 | 213〉 | 312〉 | 321〉 � OO�7S$7d~_8j"�O <� �O9��O Æm_O 1 3bh}� 0 ���Æ P(13) →   0 1 1 0 0 1 1 0 0 1 1 0   | 123〉 | 132〉 | 231〉 | 213〉 | 312〉 | 321〉 K� �w(k �$I1.pm(k� 18 I�Æ N _ Æ λ1 + · · ·+ λn = N λ1 ≥ λ2 ≥ · · · ≥ λn � N _ ←→ �)�!_Æt SN Æ g ∼ h−1gh qa�O� N _ TNtI� Yλ Æ �Æ N = 2 Æλ = (2) λ = (1, 1) N = 3 Æλ = (3) λ = (2, 1) λ = (1, 1, 1) I+Æ ÆjO I� S�x 1, 2, . . . , N 'G� '�+Æ9* _'G�x).`*%yH� �Æ → 1 2 3 → 1 2 3 + 1 3 2 qO I� '�+_x�Æ Dλ = N !∏ i,j h(i, j) . h(i, j) ���<� D^.`*.%__.+pVG_-�x���Æ h(1, 2) = 4 ��Æ λ = (2, 12) Dλ = 4! 4 · 2 = 3   1 23 4 1 3 2 4 1 4 2 3   SN L_.pm(kÆ �O SN _.pm(k O I�,qZ� Dλ = (k_�x TDF(t�w(kÆ_(k_L x� ⇒ N ! = ∑ λ D2λ � F��N SN _.pm(kÆ ÆjO I� Yλ pST%�N.pm(kÆ 19 qa�O �'�+� �℄�_+:�,��9q>�, {q>�, �℄�� Tp�℄' '0j* �� �Æ N = 3 λ = (2, 1) 1 2 3 ⇒| 123〉+ | 213〉− | 321〉− | 312〉 (A) 1 3 2 ⇒| 132〉+ | 231〉− | 312〉− | 321〉 (B) 8?� S3 w�(k_O�<�H�Æ (123)A =| 231〉+ | 132〉− | 213〉− | 123〉 = B −A (12)A =| 213〉+ | 123〉− | 231〉− | 132〉 = A−B ... S3 _.pm(k q>D = 1 (×1)1 :,D = 2 (×2)4 {q>D = 1 (×1)1 6 = 3! .pm(k_<Æ | 123〉 | 132〉 | 231〉 | 213〉 | 312〉 | 321〉 ψS = 1√ 6 [ 1 1 1 1 1 1 ] ψA = 1√ 6 [ 1 -1 1 -1 1 -1 ] ψM1,1 = 1 2 [ 1 0 0 1 -1 -1 ] ψM1,2 = 1 2 √ 3 [ -1 2 2 -1 -1 -1 ] ψM2,1 = 1 2 [ 1 0 0 -1 -1 -1 ] ψM2,2 = 1 2 √ 3 [ 1 2 -2 -1 1 -1 ] pS}| �NOU SN L_(k��Æ • ψM1 tP7 1 � 2 'o%q> • ψM2 tP7 1 � 2 'o%{q>� YKÆI�'�� SN _.pm(k�'�+ÆF�.pm(k_<� I+_Z℄Æ A. �N SN _.pm(k *�v3�� 20 B. �v�t (Hk) N Æt SN %_s���_�� C. �v� SU(k) _.pp(k� *t (Hk) N �Z�N�.pm(k� t SN %_s���Æ o� N )i_ k �"v-�sK Hk ��"v-�sK H = (Hk) N dimH = kN � � TÆ k = 2 �� -1/2 _��< | ± ± · · · ±〉 �� xJO (Hk) N Y� SN .pm(k� P�Æqa� ��_I� pS^\O�.pm(k�qa� ��_�'� k- +� ��Æ • x� ≤ k • 9n}yH_ • yH� ZM .pm(k_�oQMn Dλ �(kÆ D k λ � λ I�_'� k- +_x�� Dkλ _|AÆ (k δi = λi − λi+1, i = 1, . . . , k − 1 Dkλ = (1 + δ1)(1 + δ2) · · · (1 + δk−1) × ( 1 + δ1 + δ2 2 )( 1 + δ2 + δ3 2 ) · · · ( 1 + δk−2 + δk−1 2 ) × ( 1 + δ1 + δ2 + δ3 3 ) · · · ( 1 + δk−3δk−2δk−1 3 ) × · · · × ( 1 + δ1 + · · ·+ δk−1 r − 1 ) pAv_(OÆ 8V��<� h(i, j) LpSjW D(i, j) = j − i = M-�Fx_ x� - M-�Fx_9x� Dkλ = ∏ i,j k +D(i, j) h(i, j) SZX�,`� jÆ ∑ DkλD k λ = k N � �Æ 3 ��� 1/2 _��Æ 8 �"v-�sK�.pm(kÆ − − − − − + − + + + + +   Dλ = 1 q>�(D 2 λ = (1 + 3) 2 3 · 32 41 = 4[δ1 = 3]), − − + − + +  Dλ = 2 :,�D2λ = (1 + 1) = 2[δ1 = 1] 1× 4 + 2× 2 = 8 21 pS^\�&x O'�+5U\�Æ ��p^\Sb_�+:,�_;� 0 _�� �K_(O − − + ψM1,1 = 1√ 2 (| − −+〉− | +−−〉) ψM1,2 = 1√ 6 (| − −+〉+ | +−−〉 − 2 | −+−〉) ψM2,: = 0. (t��pSYÆ SN _.pm(k�xJO�w(k* (Hk) N Y\ SN _�w(kÆ� SU(k) _�w(k}Æ 1OC2 q SU(2) _.pm(k}ÆqOO j ∈ Z/2 {| j,m〉, m = −j, . . . , j} � SU(k) _:kj�� SU(k) _9��q>X��j Y [SU(k), SN ] = 0 � jÆSU(k) _.pm(kqZ9x0a k _I� .pm(k λ _�x`a Dkλ � λ t (Hk) N Æ λ F(_Lx� Dλ � �oÆ • ∑ λ DkλDλ = k N ,O • Kj_(k^ SU(k) t�Z_9��? '� k- +,�# • i_ k -�_ → �I{q> 9�}|V� pS�b�� �Æ SU(2) (k (j = 1) D2λ = 2 (j = 1) D2λ = 3 (j = 3/2) D2λ = 4 ... · · · (2j -�) (j =O+x�) D2λ = 2j + 1 �dÆ ∼ • ∼ ∼   t (H2) N ÆF( ;KFU Y Dkλ .�_`J�,�� �ÆY (H2) 38�sK t SU(2)�S3 L% 3 �� -1/2_��⇒ (j = 3/2)×1, (j = 1/2)×2�� 22 Dλ = 1 D 2 λ = 4 (4 D = 1S3 (k 1 D = 4SU(2) (k�) Dλ = D 2 λ = 2 (2 D = 2S3 SU(2) (k�) |���(kÆ r&I SU(N) L� ft(/Æ"|���Y��Æ qa SU(2) � ⊗ = • + + (j = 1) (j = 1) (j = 0) (j = 1) (j = 2) 3 × 3 = 1 + 3 + 5 0&:jwÆ 1. �℄ a, b, c, · · · )tdO�dw�dW9Æ'�dw I� a a a a b b b c c ... 2. ℄OnF�" Mu^\> aaba · · · �℄7T%AwÆt .!G�_!AÆ NtO+�� F(_Lx b sa a c sa b ``�� �Æ SU(2) ⊗ a a = a a ⊕ · · a a ⊕ · · a a = • + + . 23 �UÆ (Hk) N _Y℄℄n ⊗ ⊗ · · · ⊗ N ����jwÆ�LyHO -� ÆF�_ i_0a`a k 9_'�I� '� = -���_>��pS�� Dλ � Yλ t (Hk) N ÆF(_L x� n�Nt SN (k_|���,}|_|A Mut SN _(kÆÆF Yλ ⊗ Yλ _Y� .NtZ℄a0&Aw_LV~y� �u_8jÆ ⊗ Y = Y ⊗ = + + 2 × 2 = 1 + 1 + 2 pMS%_�gÆ��Æ SU(2)(k_x� S3(k_x� (H2) 3 ⇒ 1 1 2 2   (1 · 4 + 2 · 2 = 8) (H4) 3 ⇒ 1 20 2 20 1 4   (1 · 20 + 2 · 20 + 1 · 4 = 64) ^a H4 = H2 ⊗H2 "MNt^O0_ S3 (kÆ( 4 + 2 ) ⊗ ( 4 + 2 ) = 16 ⊕ 16 ⊕ 4 ( ⊗ ) = 20 ⊕ 20 ⊕ 4 ⇒ ⊗ = ⊕ ⊕ . pS�!^�℄�&x ψS = ψM ( 1 0 0 1 ) ψ˜M ψA = ψM ( 0 −1 −1 0 ) ψ˜M z�N}O!A� ψS(r1, r2, r3; s1, s2, s3) = ψM,1(r1, r2, r3)ψM,1(s1, s2, s3) + ψM,2(r1, r2, r3)ψM,2(s1, s2, s3) ψA(r1, r2, r3; s1, s2, s3) = ψM,1(r1, r2, r3)ψM,1(s1, s2, s3) + ψM,2(r1, r2, r3)ψM,2(s1, s2, s3) 24 0&X�Æ{q>(k℄F(t Y ⊗ Y˜ Æ Y˜ = ��(Y ) ��Æ ⊗ = + + + � Z℄e�Æ 1. Xi�Æ (2p)3 ����xÆ ( 6 3 ) = 6! 3!3! = 6 · 5 · 4 3 · 2 · 1 = 20. sK - ��:,,&x"Mn{q>_ S Ψspace ⊗Ψspin �<3F�sKÆ j = 1(H3) ��Æ j = 1/2(H2) � ;K^\t Yspace ⊗ Yspin _|���Æ_I��p9:Æ sK �� D2λ = 0 4 � D3λ = 1 D 2 λ = 4 l = 0, s = 3 2 ⇒4 S3/2 16 � D3λ = 8 D 2 λ = 2 l = 2, 1, s = 1 2 ⇒2 D5/2,2D3/2,2 P3/2,2 P1/2. �Æ�N 2D5/2,m = 5/2 _�&x "uqÆ u s d �Nt SU(3) 25 �s��Æ q t Æ q¯ t Æ� [� qq¯ �Æ ⊗ = + 3 × 3 = D3λ = 8 + D3λ = 1 (���) (V�) �� qqq �Æ ⊗ ⊗ = ⊕ ⊕ ⊕ D3λ = 10 8 8 1 �� 1/2+ ����� �Æ �� 3/2+ �� �� �Æ u2_% Æ�� �� ∆++ · · · i_� i_ S = 3/2 �� � IasK<�,&x ⇒ s