9. 0�����5
9.1 -61��
|
#�Bw,
xµ = (ct, x, y, z), µ = 0, 1, 2, 3
_Bl��
s2 = ηµνx
µxν = c2t2 − |~x|2
Ra
ηµν =
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
��>v�
x′µ = Λµνx
ν
>;g�r
%_Bl s2 ���?k
ηµσΛ
µ
νΛ
σ
τ = ηντ
>;g�r Λ ℄">;g\�R detΛ = ±1 �8E +1 7!�W >;g�r�8E −1 7!�N
W >;g�r�T\�"�
e?�
(
1 0
0 R
)
�Ra R ∈ SO(3) �b^ z
7e?��
1 0 0 0
0 cos θ sin θ 0
0 − sin θ cos θ 0
0 0 0 1
.
�?�b1 x
7�?�
γ − vcγ 0 0
− vcγ γ 0 0
0 0 1 0
0 0 0 1
.
Ra γ = 1√
1−v2/c2
�
1
N�V>;g�r�"b��Ke�
1
−1
−1
−1
.
(*$� (*:�
z�o1 V µ BzXO1� Λµν �rb��
(V ′)µ = ΛµνV
ν
b?1� pµ = (E/c, ~p) �
p2 = pµpµ = E
2/c2 − |~p|2 = m2c2
B`�q�
pµ = ηµνp
ν , pµ = ηµνpν , η
µνηνλ = δ
µ
λ
z�P1�rb��
T ′µν = ΛµσΛ
ν
τT
στ
b� Fµν = ∂µAν − ∂νAµ
9.2 Klein-Gordon �{
�B�464�D<'q7 Schro¨dinger M$�LN�D<'q��B
E =
p2
2m
+ V (x), E = ih¯
∂
∂t
, pj = −ih¯ ∂
∂xj
Schro¨dinger M$�
ih¯
∂
∂t
ψ(x, t) =
(
− h¯
2
2m
~∇2 + V (x)
)
ψ(x, t)
9z�D7 p �M17
B0X E = ±|E| �SXR7M1 −|E| �T
#t�Ql�&7�0tDE'"�ÆvH0t�*7�L�s���DERM16�4�>
�
-h�t= E → −∞ �
DEU96��s^ Schro¨dinger M$�B
ψ∗
[
ih¯ψ˙
]
= ψ∗
[
− h¯
2
2m
∇2ψ + V ψ
]
[
−ih¯ψ˙∗
]
ψ =
[
− h¯
2
2m
∇2ψ∗ + V ψ∗
]
ψ
0q���64
∂
∂t
ρs = −~∇ ·~js
Ra ρs = |ψ|2 tU9DB� ~js = − ih¯2m (ψ∗~∇ψ − ψ~∇ψ∗) tU96�
DE Klein-Gordon M$���37MJ�964�
ψ∗ � ψ − ψ � ψ∗ = −∂µ[ψ∗∂µψ − ψ∂µψ∗] = 0
64z�U96�
jµ = A (ψ
∗∂µψ − ψ∂µψ∗)
Ra At6X x�VB ∂µjµ = 0�FN�D<7U96Z66^��5 A =
ih¯
2m �)l�
~j = ~js �
V
ρ = j0/c =
ih¯
2mc2
(
ψ∗
∂
∂t
ψ − ψ ∂
∂t
ψ∗
)
LN�DtW7�}9�M
v�/-h7M$�SX��tAI*
O
M$7�`>T7� ψ, ψ˙ L t = 0 l��9_;Z>�AESX!J� Klein-Gordon M$)ne�
W_m��me� ±E 7��t�D<�!>T7�*4?�F-h&< (1h�<) � Dirac Q
42GwB`1-h76
OM$�
L Klein-Gordon M$a7RM1metK-h� Pauli k Weisskopf L 1934 OXH Klein-
Gordon M$u�E�1��V ρ �r<
γ0 = β =
(
1 0
0 1
)
, γi = βαi =
(
0 σi
−σi 0
)
64 Dirac M$��
ih¯γ0∂0ψ = (mc− ih¯γi∂i)ψ
{
(ih¯γµ∂µ −mc)ψ = 0
�F3&6$ �7`��
γ0
2
= β2 = 1
γ0γi + γiγ0 = β(βαi) + βαiβ = 0
γiγj + γjγi = (βαi)(βαj) + (βαi)(βαi)
= −β2(αiαj + αjαi) = −2δij
{�
γµγν + γνγµ = 2ηµν
H Klein-Gordon M$�9F"�
ih¯γµ∂µψ = mcψ
−h¯2(γµ∂µ)(γν∂ν)ψ = m2c2ψ
= −h¯2 1
2
(γµγν + γνγµ)∂µ∂νψ
= −h¯2 � ψ
{ (� +
m2c2
h¯2
)ψ = 0
Dirac �z���&
LM$0�o# ψ∗ 9zLR\tM$0eC# ψ �
ψ∗
[
ih¯ψ˙
]
= ψ∗
[
mc2βψ − ih¯cαj ∂
∂xj
ψ
]
[
ih¯ψ˙∗
]
ψ =
[
mc2β∗ψ∗ + ih¯cα∗j
∂
∂xj
ψ∗
]
ψ
��64�
∂
∂t
(ψ∗ψ) +
∂
∂xj
(cψ∗α˙jψ) = 0
Ra ψ∗ψ �W>7U9DB�
5
γ �;4<
A γ �U7#w��9'" 16 X�)�`7�U�~q ΓM ,M = 1, · · · , 16 �b��
1, γµ, γµγν , γµγνγλ, γµγνγλγσ
6X zX 7X zX 6X
S3�_;6X 4× 4 7�U��9&" M =∑CMΓM �
Pauli �u��
h γµ, γ
′
µ �0m_;7 4× 4 7�U�?k
{γµ, γν} = 2ηµνI, {γ′µ, γ′ν} = 2ηµνI
N�>,L6XNS=�U S �p
γ′µ = SγµS
−1
}9�DzO1�hx ψl 7 Dirac M$B��7�!�9eÆf7t Pauli-Dirac�!�L9?
MW �kOB��
/��4�? "�
d�N�D<1h..4&
SU(2) \�r�H�D<1h..4&
Lorentz/Poincare´ \�
r�
S* Lorentz \Ae? Ji k�? Ki �i�'.x��
[Ji, Jj ] = iεijkJk
[Ji,Kj] = iεijkKk
[Ki,Kj] = −iεijkJk
�6'q�� Jµν , µ ∈ {0, 1, 2, 3}�Ra
J0i = Ki
Jij = εijkJk
[Jµν , Jρσ ] = −i (ηνρJµσ − ηµρJνσ + ηµσJνρ − ηνσJµρ)
9e� Lorentz .x�� �
Q7�ih Pµ �4X
[Pµ, Pν ] = 0
[Pµ, σρσ ] = −i (ηµρPσ − ηµσPρ)
{'" Poincare´ \.x�gB�?�e?kQ7�
�8 Poincare´\7��K�r� Wigner,1939��B Casimir|h C1 = P
µPµ F}B|QD:�
%{E SU(2) \a7 J2 �5 PµPµ = m
2
�A m2 �9YOY�r�HV P 0 L Lorentz �r�Qi
���
�8b��r� p2 = m2 > 0, p0 > 0 �B`17��-M�V� Pµ �B
Pµ|ψ〉 = pµ|ψ〉
6
pµ "\� Poincare´ \7h\�L pµ _>7W ��964 SU(2) \�b Pµ = (mc, 0, 0, 0) �}9
��K�rA m2 ki, S #Gw�i,#i Pauli-Lubanski ro1�
Wµ = −1
2
εµνλσJ
νλP σ
�℄B WµP
µ = 0 �Et�964 Casimir |h
C2 =W
µWµ = −m2s(s+ 1).
S0X Casimir |h C1, C2 �>2 Poincare´\7�r�
*�`1ki,�9YO-h�}9?�\7`1ki,�
Dirac M$Gw7ti,� 1/2 7-h�R�hx ψl tzO1,1� R s = 1/2 7 Poincare´ \.
x�r�(�r�4�i,IbF Klein-Gordon M$7WRM1 ±E ���/��*4`1-E
��
℄)�8�<2 ψj 7 Lorentz �r S �Ra ψ �zO1,1�hx�
S Z&2W Lorentz \7zO1,1�r�
Ji =
i
4
εijkγjγk
Ki =
i
2
γ0γi
?k Lorentz .x
[J, J ] = iεJ
[J,K] = iεK
[K,K] = −iεJ
6�:�B Jµν =
i
4 [γµ, γν ] ≡ 12σµν �
+�
1. 1 z
e?�B θ �
SR = e
i i
2
γ1γ2θ,
D
−γ1γ2 = −
(
0 σ1
−σ1 0
)(
0 σ2
−σ2 0
)
= i
(
σ3 0
0 σ3
)
.
Et
SR = e
i
2 (
σ3 0
0 σ3
)θ.
}9U0XO1 IO1,1�r�n0XO15t63�
2. 1 z
M 7�?�Ra tanhu = v/c �
SB = e
i i
2
γ0γ3u,
D
γ0γ3 =
(
1 0
0 −1
)(
0 σ3
−σ3 0
)
=
(
0 σ3
σ3 0
)
.
Et
SB = e
− 1
2 (
0 σ3
σ3 0
)u.
�?�IO1,1�(ul�
8
3. Kg�*4
Λµνγ
ν = SγµS−1, Λµν =
1
−1
−1
−1
{ Sγ0S−1 = γ0, SγiS−1 = −γi �
DEe?� S tsW7� S† = S−1 �D [γ0, γiγj ] = 0 �}9 γ
0S†γ0 = S−1 �DE�?� S �
tsW7� S† = S �D {γ0, γ0γi} = 0 �}9 γ0S†γ0 = γ0Sγ0 = S−1 �{� γ0S†γ0 = S−1 DEe
?k�?�",�
'.42v�
ψ Lorentz �r� ψ′ = S(Λ)ψ ��9'"y�)%�1�5
ψ¯ = ψ†γ0
ψ¯′ = [S, ψ]†γ0 = ψ
†γ0γ0S
†γ0 = ψ¯S
−1
Et ψ¯ψ t%�7�
>< γ5 = iγ
0γ1γ2γ3 ��964R~y�)%�1��
ψ¯ψ �1
ψ¯γ5ψ r�1
ψ¯γµψ o1
ψ¯γµγ5ψ ro1
ψ¯σµψ P1
R%�'qb�
ψ¯′ψ′ = ψ¯ψ, ψ¯′γ5ψ
′ = det(Λ)ψ¯γ5ψ,
VB γ0γ5γ
0 = −γ5 �
A7�? Dirac �z�
L�[���� pi = 0 �A
(ih¯γµ∂µ −mc)ψ = 0,
�964
ih¯
c
γ0
∂
∂t
ψ = mcψ,
{
i
(
1 0
0 −1
)
ψ˙ =
mc2
h¯
ψ.
9
Et
��
�hx
M1 H i, Sz
ψ(1)(t) = e−i(mc
2)t/h¯
1
0
0
0
mc2 + 1/2
ψ(2)(t) = e−i(mc
2)t/h¯
0
1
0
0
mc2 − 1/2
ψ(3)(t) = e−i(−mc
2)t/h¯
0
0
1
0
−mc2 + 1/2
ψ(4)(t) = e−i(−mc
2)t/h¯
0
0
0
1
−mc2 − 1/2
Dirac �z �x�
Z> E, pi z�hx ψ =
(
uA
uB
)
� Dirac M$��
(
ih¯γ0∂0 + ih¯γ
i∂i −mc
)
ψ = 0,
Ra
γ0 =
(
I
−I
)
, γi =
(
0 σi
−σi 0
)
.
.
64
1
c
EuA − (p · σ)uB −mcuA = 0,
64
uA =
c(σ · p)
E −mc2uB, uB =
c(σ · p)
E +mc2
uA.
10
Ra c2(σ · p)2 = p2c2, E2 = m2c4 + p2c2 ��9
6�
u(1)(p) =
1
0
p3c/E +mc
2
(p1 + ip2)c/E +mc
2
u(2)(p) =
0
1
(p1 − ip2)c/E +mc2
−p3c/E +mc2
u(3)(p) =
−p3c/|E|+mc2
−(p1 + ip2)c/|E|+mc2
1
0
u(4)(p) =
−(p1 − ip2)c/|E|+mc2
p3c/|E|+mc2
0
1
b6p7�d��
ψ = Nu1,2ei~p·~x/h¯−iEt/h¯
ψ = Nu3,4ei~p·~x/h¯+i|E|t/h¯
Me`�
u(r)†(p)u(r)(p) =
|E|
mc2
,
64 N =
√
mc2
|E|V �
�B�Po:=RM1K�
DE ψ(x, 0) ∼
φ(x)
0
0
0
� |φ|2 �HL6X"Y� ∆x L�� �x Cp,i, i = 1, · · · 4 �B
Cp,3
Cp,1
∼ − p3c|E|+mc2 ,
Cp,4
Cp,2
∼ − (p1 − ip2)c|E|+mc2 .
�Y�1 ∆p ∼ mc l�t��9o:7�{1 ∆x ≤ h¯
mc
l�}9��HBmH�7 −E O1�
Klein 2I� E2 = m2c4 + p2c2 ��86�s
V (x) =
0 x < 0,V0 x > 0.
g�� p2c2 = (E+mc2)(E−mc2) > 0�64T2
�s$'N� p2c2 = (E−V+mc2)(E−V−mc2)�
1 mc2 > E − V0 > −mc2 l� p2c2 < 0 �64lN
�{6�7Kg
�H1 E − V0 > mc2 l�B
p2c2 > 0 �t64T2
�{RM1
�)lKgU9-E 1 �/-h
vj
�
A7�Æ Dirac �z
M$ (ih¯γµ∂µ −mc)ψ = 0 �Ra {γµ, γν} = 2ηµν � ψ �zO1,1�hx�Z> ~p �964
zX
� u(i)(p) �Ra0X E > 0 �4�0X E < 0 ��RM1
v�M1 −E 7W<
K = γ0~Σ · ~J − γ0 h¯
2
,
�Y [H,K] = [ ~J,K] = 0. Et�� H 7�VO%��
H : E
J2 : h¯2j(j + 1)
J3 : h¯m
K : −kh¯
A K 7><�6
K = γ0
[
~Σ · (~L+ h¯
2
~Σ)− h¯
2
]
= γ0
(
~Σ · ~L+ h¯
)
.
D [γ0, ~Σ] = 0 �}9
K2 = (~Σ · ~L+ h¯)2 = L2 + h¯~Σ · ~L+ h¯2.
DA
J2 = L2 + h¯~Σ · ~L+ 3
4
h¯2,
�Y K2 = J2 + h¯2/4 �}9 k2 = j(j + 1) + 1/4 = (j + 1/2)2 �Et k = ±(j + 1/2) �RaWRiO
��ri,Q(EqtKQ(Ej�?1�+b� j = 1/2 : k = ±1, j = 3/2 : k = ±2 �
k = γ0(~Σ · ~L) =
(
~σ · ~L+ 1 0
0 −(~σ · ~L+ 1)
)
.
/Q k, J2, Jz 7\��Vhx� ψ =
(
ψA
ψB
)
�
K J2 Jz
ψA −kh¯ j(j + 1)h¯2 mh¯
ψB kh¯ j(j + 1)h¯
2 mh¯
DB L2 = J2 − h¯~σ · ~L− 3h¯2/4 �}9 ψA,B 5t L2 7�Vhx�VB�
−k = j(j + 1)− lA(lA + 1) + 1
4
k = j(j + 1)− lB(lB + 1) + 1
4
13
Et64
k = j + 1/2 ⇒ lA = j + 1/2 lB = j − 1/2
k = −j − 1/2 ⇒ lA = j − 1/2 lB = j + 1/2
℄n ψA, ψB ��t L3 z Σ3 7�Vhx�Et�9&&
ψ =
(
ψA
ψB
)
=
(
g(r)yj3j,lA
if(r)yj3j,lB
)
.
Ra yj3jl tb6p7i,�hx��9&"b�'q�
1 j = l+ 1/2 l�
yj3jl =
√
l + j3 + 1/2
2l+ 1
Yl,j3−1/2
(
1
0
)
+
√
l− j3 + 1/2
2l+ 1
Yl,j3+1/2
(
0
1
)
1 j = l− 1/2 l�
yj3jl = −
√
l − j3 + 1/2
2l + 1
Yl,j3−1/2
(
1
0
)
+
√
l + j3 + 1/2
2l+ 1
Yl,j3+1/2
(
0
1
)
bd4X
s� V (r) a7 Dirac M$�\*4� M$ f(r), g(r) �{
c(~σ · ~p)ψB =
(
E − V (r) −mc2)ψA
c(~σ · ~p)ψA =
(
E − V (r) +mc2)ψB
8B
[
~σ · ~x
r
]
yj3j,lA Z&2 J
2, J3, L
2
7�Vhx��B��7 j, j3 �0t�K7
5G!�Et
(~σ · ~p)ψB = −h¯df
dr
yj3jlA −
(1 − k)h¯
r
fyj3jlA .
~ F (r) = rf(r), G(r) = rg(r) �EtB
h¯c
(
dF
dr
− k
r
F
)
= −(E − V −mc2)G,
h¯c
(
dG
dr
+
k
r
G
)
= (E − V +mc2)F.
5 h¯ = c = 1 � V = −Ze
2
r
= −Zα
r
��6
F = e−ρρs
∑
m=0
amρ
m
G = e−ρρs
∑
m=0
bmρ
m
℄nL- r k" r l����(��E|xL n′ l�CX
�64M|�
E =
mc2√
1 + Z
2α2
(n′+
√
(j+1/2)2−Z2α2)2
,
14
6�:� n = n′ + |k| = n′ + j + 1/2 �O���
E = mc2
[
1− 1
2
(Zα)2
n2
− 1
2
(Zα)4
n3
(
1
j + 1/2
− 3
4n
)
+ · · ·
]
.
Ra;I�� Balmer ��;d�����℄��bd�
6[V
7 α
�*4I*1hp�
15
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