Sakurai 现代量子力学——散射
8. 9�
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q0p��Mq3�
• 0p
q0p��MH �f�
• �m�wwJ*qvI V (~r1 − ~r2) �
Ue*�����mvI V (~x) hke4� µ <0p��vhk�
8;KH�T{1�
hk0p|� dn/dt = Fσ(θ, φ)dΩ �Of σ(θ, φ) ��Vhk�C�
σ =
∫
σ(θ, φ)dΩ
1
�rhk�C�wC�<4�� 1Æ = 10−4cm2 ��3�5Æ0p<�CC��
KP5ExS'
(
p2
2m
+ V
)
|ψ...
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8.1 �A
iPhkMtL<�,r9�
hkr9<%I�lM�
1) �6-hk�0p
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��
• 0p
q0p��Mq3�
• 0p
q0p��MH �f�
• �m�wwJ*qvI V (~r1 − ~r2) �
Ue*�����mvI V (~x) hke4� µ <0p��vhk�
8;KH�T{1�
hk0p|� dn/dt = Fσ(θ, φ)dΩ �Of σ(θ, φ) ��Vhk�C�
σ =
∫
σ(θ, φ)dΩ
1
�rhk�C�wC�<4�� 1Æ = 10−4cm2 ��3�5Æ0p<�CC��
KP5ExS'
(
p2
2m
+ V
)
|ψ〉 = E|ψ〉
(
− h¯
2
2m
∇2 + V (x)
)
ψ(x) =
h¯2k2
2m
ψ(x)
O��<+t���4 kˆ = zˆ �
ψ(x) =
1
(2πh¯)3/2
[
eikz + f(θ, φ)eikr/r
]
Of)p-R="�ek��v="�*k����+t�
(∇2 + k2)e
ikr
r
= 0 [ �V�Q]
f(θ, φ) G�PwM� �z
1
E −H0 →
1
E −H0 ± iε
K Lippman-Schwinger S'%�;:�`X#
|ψ〉 = |φ〉+ 1
E −H0V |φ〉+
1
E −H0V
1
E −H0 V |φ〉 + · · ·
}gE��`=f 0) �[:;:<�� R �k���)& R ���XC�� ψ = Al(kr)Ylm(xˆ) �Of Al(ρ) w jl(ρ) � nl(ρ) <�-ts�%
�;:
h±l (ρ) = jl(ρ)± inl(ρ) = (−i)±(l+1)
e±iρ
ρ
Of� h+l �uhk�,�� h
−
l �uek��
dl V = 0 �V Al = jl ��u�Mhk�,��KP=� R q<��
Al(ρ) =
C
2
[
h−l (ρ) + e
2iδlh+l (ρ)
]
[`�l!Md�-e�3lK%-�hk�,�fnMek�< l,m ��~Fp y^-�RG
�:yu� e2iδl w-��� δl wr| hk-e
℄KV�<�> e
2iδl
"E�
(�OU
Y�v V (r) �< Schro¨dinger S'� ψl(x) = Al(r)Ylm(xˆ) ������ Al(0) = 0 �v3|�
Al(r) = Ul(r)/r �;:
U ′′l + (k
2 − 2m
h¯2
V − l(l + 1)
r2
)Ul = 0
Y�=�< Schro¨dinger S'�U r = R ,�?s
βl =
r
Al
dAl(r)
dr
Of�
Al =
C
2
[
h−l (ρ) + e
2iδlh+l (ρ)
]
= Ceiδl
[
cos δljl(ρ)− sin δlnl(ρ)
]
Pw;:�
βl = kR
[j′l(kR) cos δl − n′l(kR) sin δl
jl(kR) cos δl − nl(kR) sin δl
]
�
tan δl =
kRj′l(kR)− βljl(kR)
kRn′l(kR)− βlnl(kR)
it%�J*^EKPA=` l �<�> δl �
Y5/}8;�.M
?;:
ψ(x) −−−→
r→∞
1
(2πh¯)3/2
[
eikz + f(θ)
eikr
r
]
EC
f(θ) =
∞∑
l=0
(2l + 1)fl(k)Pl(cos θ)
8
Q ψ �i�VK
jl(kr) −−−→
r→∞
ei(kr−lπ/2) − e−i(kr−lπ/2)
2ikr
�
eikz =
∑
l
(2l + 1)iljl(kr)Pl(cos θ)
;:
ψ(x) −−−→
r→∞
∞∑
l=0
(2l + 1)Pl(cos θ)
1
2ik
[eikr − e−i(kr−lπ) + 2ikfleikr
r
]
T�)pf VQ h−l + e
2iδlh+l ���%;
e2iδl = 1 + 2ikfl
Z�u&
fl =
e2iδl − 1
2ik
=
eiδl sin δl
k
=
1
k cot δl − ik
Pw
f(θ) =
1
k
∑
l=0
∞(2l+ 1)eiδl sin δlPl(cos θ)
OU�Q^
hk�C��
σ =
∫
|f(θ)|2dΩ
=
1
k2
· 2π
∫
−1
1dξ
∑
l,l′
(2l+ 1)(2l′ + 1)ei(δl−δl′ ) sin δl sin δl′Pl(ξ)Pl′ (ξ)
=
4π
k2
∑
(2l + 1) sin2 δl
=
∑
σl
Of
σl =
4π
k2
(2l + 1) sin2 δl
w l V�U
f 0
dl U > 0 � u′′/u W
��J)v�.q δl < 0 �
-�W6KH
$9v�
V (r) =
{ ∞, r ≤ R, (1a)
0, r > R. (1b)
U[gW(�� Al(R) = 0 ��
cos δljl(ρ)− sin δlnl(ρ)
∣∣∣
ρ=kR
= 0
Pw
tan δl =
jl(ρ)
nl(ρ)
, e2iδ = −h
−
l (ρ)
h+l (ρ)
10
KP s �IW(� s �IW(�� r > R q<�o|�
A0 ∝ [cos δ0j0(kr) − sin δ0n0(kr)]
u0 = rA0 ∝ [cos δ0 sin kr + sin δ0 cos kr]
KP k ≪ 1/R � r > R 5
kr ≪ 1 Ihk�C<����FvqJ)vw
V�d��
bn<�FvMd�+t�Xp<\��Dw{[��
14
8; �B�F
�E V (r) ^^,P{[�� Eb = −h¯2α2/2m ≤ 0 �d��
[-� u +�7 k → 0 qhk�
