![;T arXiv:2004.12155v2 [hep-ph] 23 May 2020 · L;T R toSM,whichisdubbed asVLQTmodel. TheLagrangiancanbewrittenas[21] L= L SM+ LYukawa T + L gauge T; LYukawa T = i T Q i L eT R M T](https://static.fdocument.org/doc/165x107/5fc6f89706f746179e1ee992/t-arxiv200412155v2-hep-ph-23-may-2020-lt-r-tosmwhichisdubbed-asvlqtmodel.jpg)
nm ' Q Sm L[yN Z ^ & Q i L[yN Z Z ' ,-5 j # L[ L[ ~ o q k^@ C (' L[ k^@ 1+5603647 3.Ic XP Q OL[ Vl...
12
qqTNEO[oSfF[oS(<^"tG HN&dvHNI%X&fF[oS(P@} ^9)-0*2jW[oS}fF Z(A:me!V(~&"aqLNHNg ?(z \dvHNGk$ &HN` 1 >$&5:u&5:u[oS 7 A &%UFnQF({kp[oS]Y& &UFdvHN4F({&$ yK&nQF&B&Dqw_='fF[o S − 2 2μ ∇ 2 + V ( r) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Ψ = EΨ (5.1.1) ]&ΨfFZE )-0*2&μ Ys %NNMc!Nbr 83R(h| m (< V(r)HN`Nn 6;l#&.,1+/0)-0*2%nQ r (< V ( r) = − 1 4 πε 0 e 2 r (5.1.2) &83tG! k = (4πε 0 ) -1 (<&&~& fF[oS − 2 2 m ∇ 2 − k e 2 r ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Ψ = EΨ (5.1.3) & dvHNiC[Jhx$'&
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Transcript of nm ' Q Sm L[yN Z ^ & Q i L[yN Z Z ' ,-5 j # L[ L[ ~ o q k^@ C (' L[ k^@ 1+5603647 3.Ic XP Q OL[ Vl...
1
A
−
2
2μ∇2 +V (r)
⎛
⎝ ⎜
⎞
⎠ ⎟ Ψ = EΨ (5.1.1)
Ψ E μ
m V(r)
r
V (r) = −1
4πε0
e2
r (5.1.2)
k = (4πε0)-1
−
2
2m∇2 − k
e2
r
⎛
⎝ ⎜
⎞
⎠ ⎟ Ψ = EΨ (5.1.3)
r
r f(r) x
∂f∂x
=df
dr
∂r∂x
(5.1.4)
r = x 2 + y 2 + z2
∂r∂x
=x
r (5.1.5)
∂f∂x
=df
dr
x
r (5.1.6)
x
∂2 f∂x 2
=∂∂x
df
dr
x
r
⎛
⎝ ⎜
⎞
⎠ ⎟ =
d2 f
dr2x
r
⎛
⎝ ⎜ ⎞
⎠ ⎟ 2
+df
dr
1
r2r −
x 2
r
⎛
⎝ ⎜
⎞
⎠ ⎟ (5.1.7)
∇2 f (r) =∂2
∂x 2+∂2
∂y 2+∂2
∂z2⎛
⎝ ⎜
⎞
⎠ ⎟ f (r)
=d2 f
dr2+ (3r − r)
1
r2df
dr=d2 f
dr2+2
r
df
dr (5.1.8)
−
2
2m
d2
dr2+2
r
d
dr
⎛
⎝ ⎜
⎞
⎠ ⎟ − k
e2
r
⎡
⎣ ⎢
⎤
⎦ ⎥ Ψ(r) = EΨ(r) (5.1.9)
C
(5.1.9) 1 2
(5.1.9)
(5.1.9)
(5.1.9) Ψ(r) = Aexp(−ar)
d
drΨ(r) = −aAexp(−ar) = −aΨ(r)
d2
dr2Ψ(r) = a2Ψ(r) (5.1.10)
−
2
2ma2 −
2a
r
⎛
⎝ ⎜
⎞
⎠ ⎟ − k
e2
r= E (5.1.11)
E = −
2
2ma2
ke2 =
2
ma (5.1.12)
a =
mke2
2 (5.1.13)
E = −
mk 2e4
2 2 (5.1.14)
n = 1
1s
1s 1s 1s
n = 2 2s
Ψ(r) = B(1− br)exp(−cr) (5.1.9) b
c E
1s 2s
1s 2s
0
(5.1.9) A B 0
A B
A
B r = (x, y, z) dV = dx dy dz
|Ψ(r) |2 dxdydz (5.1.15)
1
|Ψ(r) |2 dxdydz∫ =1 (5.1.16)
1 A B
1s
C
dV = r2 sinθdθdϕdr (5.1.17)
1 = dθ0
π
∫ dϕ dr0
∞
∫0
2π
∫ r2sinθ • A2 exp −2r
a0
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
= 4π dr r2A2 exp −2r
a0
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
0
∞
∫ (5.1.18)
A
A = ±1
πa03
(5.1.19)
D
1s 2s r 5.1
1s 2s
1s
2s
r
4πr2
r 1s 2s
5.2 1s
5.1 1s 2s
5.2 1s 2s
1s
5.1.3)
Ψ(x,y,z) = Czexp(−dr) r = x 2 + y 2 + z2 n = 2
2s
∂Ψ∂x
= −dCzexp(−dr)∂r∂x
= −dx
rΨ (5.2.1)
x
∂2Ψ∂x 2
= −dx
r−d
x
rΨ
⎛
⎝ ⎜
⎞
⎠ ⎟ +
1
r2r − x
x
r
⎛
⎝ ⎜
⎞
⎠ ⎟ Ψ
⎡
⎣ ⎢
⎤
⎦ ⎥ =
d
r3x 2(dr +1) − r2[ ]Ψ (5.2.2)
y x y z
∂Ψ∂z
=1
z− d
z
r
⎛
⎝ ⎜
⎞
⎠ ⎟ Ψ (5.2.3)
∂2Ψ∂z2
=d
r3z2 dr +1( ) − 3r2[ ]Ψ (5.2.4)
∂2
∂x 2+∂2
∂y 2+∂2
∂z2⎛
⎝ ⎜
⎞
⎠ ⎟ Ψ = (dr − 4)
d
rΨ (5.2.5)
(5.1.3) 1s
d =
mke2
2 2 =1
2a0 (5.2.6)
E = −
mk 2e4
8 2 (5.2.7)
n = 2
2pz
2pz z z x
(5.1.3) 2pz
z 2pz
2px
2py 2p
2p
2p
2p
2s
2p
2pz z z =
0 z
5.3 xy z = 0 py
2pz z
2py 5.3 y
(5.1.3)
n n
E = −
mk 2e4
2 2
1
n2 (5.2.8)
n n =∞ 0
n
E∞ − E1 =
mk 2e4
2 2 (5.2.9)
2s 2p
0
0
l 0 n – 1 l
L2 = l(l +1)
l = 0, 1, 2, 3, 4, …
s, p, d, f, g …
1s 2s 2p
1s n = 1 l = 0
2p n = 2 l = 1
2p 3
m
–l l p l = 1 m = –1 0 1
px py pz
d
l = 2 5
5.1 nm –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
H 0.11 N 0.15 O 0.140 F 0.135 P 0.19 S 0.185 Cl 0.180 As 0.20 Se 0.200 Br 0.195 Sb 0.22 Te 0.220 I 0.216
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
0.053 nm 1s
5.1
0.170 nm
5.2
5.2 nm ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Li+ 0.060 O2- 0.140 F- 0.136 Na+ 0.095 S2- 0.184 Cl- 0.181 K+ 0.133 Se2- 0.194 Br- 0.195 Rb+ 0.148 Te2- 0.221 I- 0.216 Cs+ 0.169
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
A B
AB
A2 B2
5.3
5.3 nm
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– H 0.030 C 0.077 N 0.070 O 0.066 F 0.064 Si 0.117 P 0.110 S 0.104 Cl 0.099 Ge 0.122 As 0.121 Se 0.117 Br 0.114 Sn 0.140 Sb 0.141 Te 0.137 I 0.133 ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
1976
D.A. J.D.
1999
L. 3
1974
© K. Saito, 06/05/2010
