SL(2, C)
 ł�
1
S =
2
 (t, )
t (t, )
t
t = 0
t > 0
"
" t
t
"
" " " " "
"x "
x
"x = {"x0, "x1, "x2, "x3} = {"x�}.
"x
x
(1, 2, 3)
(0, 1, 2, 3)
c = 1
"s2 = ("t)2 - (" )2 = g��"x�"x�,
�ł �ł
1 0 0 0
�ł �ł
0 -1 0 0
�ł �ł
g�� = .
�ł łł
0 0 -1 0
0 0 0 -1
0 3
g��
+ - --
-g�� - + ++
+- --
- + ++
- + ++
"s2
"s2
" " " " "
"x
c = 1
("t)2 = (" )2,
"s2 = 0.
("t)2 (" )2
{x�} U {x �}
U
x � = x �(x0, x1, x2, x3), � = 0, 1, 2, 3
"x � "x �
gą�dxądx� = g��dx �dx � = g�� dxądx�.
"xą "x�
dxądx�
"x � "x �
g�� = gą�.
"xą "x�
 g�� = -1
"x � 2
= 1.
"x�
+1
-1 ą = � = 0
"x 0 2 "x 1 2 "x 2 2 "x 3 2
= 1 + + + e" 1.
"x0 "x0 "x0 "x0
"x 0 "x 0
e" 1 d" -1
"x0 "x0
"
"x � "x 0
= 1; e" 1.
"x� "x0
Lę!+
"
"x � "x 0
= -1; d" -1,
"x� "x0
"
"x � "x 0
= -1; e" 1.
"x� "x0
"
"x � "x 0
= 1; d" -1.
"x� "x0
"t -"t " -"
 "t -"t " -"
Lę!+
z
" " " " "
{"x �} "x
U {"x�} U
"x "x
{"x�} "x U
{"x �} U
j
Dm m(ą, �, ł)
U "x = {"x �} "x = {"x�}
"x "x
U
U U
= {x , y }
U U
U ą z
U {x, y} {x , y } {x, y}
ą U {x, y} {x , y }
{x, y} {x , y } ą
" " " " "
" " " " "
U U
z
z
x , y , z
x , y , z
t
" " " " "
U U
z
z
z
"x = "x "y = "y .
("t)2 - ("z)2 = ("t )2 - ("z )2.
"x "x
"t = "t cosh � + "z sinh �, "z = "t sinh � + "z cosh �.
�
"x
U
U v U U
"z = 0
"z
tanh � = = v,
"t
U U
�
U U
 "x "x
" " " " "
"t = ł("t + v"z),
"z = ł("z + v"t),
1
ł = " .
1 - v2
v
-v
U z
U z a" 0
" " " " "
U t � = arctan v t
Ą
- 2� z
2
" " " " "
v > 1
= �
2
= R2 �R
R
� ą"
1
"t ą "z = ("t ą "z)eą�,
2
� " � -"
" " " " "
v1
v2 vsuma
z
"t = "t cosh � + "z sinh � ,
� "t "z
"t = "t cosh(� + �) + "z sinh(� + �),
�suma = � + � .
tanh � + tanh � v + v
vsuma = = .
1 + tanh � tanh � 1 + vv
vv c-2
� | tanh �| < 1 v < 1
dt
v = 1
z
1 + 1
z
{0, 0}
t
 t z
{0, 0}
x1
x2
U U v
U
t = ł(t + vz).
1.2 "
"s2 = t2 - z2 < 0,
|v| < 1 t > 0 t < 0
U
A B "s2
AB
U z z
� tan � = v v
z
z 2 > t 2
" " " " "
"s2 > 0.
A B
{0, 0}
t > |z|
y z t
3 + 1
t < -| |
A B
B A
A B
"s2 < 0 A
AB
B
 10
��
2, 2 � 10-8
c�� H" 660
U
U
U "z a" 0
U
"t = ł"t,
"t "t
��
ł
z
l0
U l
U
l l0
U
t
x1 x2
"z a" z2 - z1 = l0.
t1 t2
"z = ł(l0 + v"t); "t = ł("t + vl0).
U "z "t = 0
"t
l0
l = ł(l0 - v2l0) = .
ł
l l0 ł
l1 l2 < l1
"x
"x � = �� "x�,
�
�
z " "t
{a�}
a � = �� a�,
�
 {"x�}
{x�}
"s2 = g��"x�"x�.
a
a2 a" g��a�a�.
a
a b
ab a" g��a�b�
1
ab = [(a + b)2 - a2 - b2]
2
�
a� = g��a�.
a0 = a0; ai = -ai, i = 1, 2, 3.
a� = g��a�.
�
�
g
1.10
g�� = g��
a� = g��a�.
 g��
g��
�
g�ągą� = �� .
a� = b� a� = b�
ab = a�b� = a�b�
a � = ���a�.
a �b � = a������ bą = a�b�,
ą
a b 16 a�b�
a�bą
�
����� = �ą,
ą
��� = (�-1)�
�
a � = a�(�-1)� .
�
" " " " "
a�
a�b�
{a�}
{a�} {b�}
2 2
ds2 = dt2 - d = dt2(1 - )
"
d� a" ds2 = ł-1dt.
2
1.5 d" 1 ds2 e" 0
�
dx�
u� = = {ł, ł },
d�
u2 = 1
dt = ą|d | d� a" 0 �
t
az = | |
z
p� = mu�,
p0 p2 = m2
2
E2 = m2 + .
| | 1 m 0
mł
2
2
m2 E = m2 +
2
m2
E = mc2
E = mc2
2 2
( )2
E = m + - + . . . .
2m 8m3
"
"x�
"x �
dx � = dxą = �� dxą,
ą
"xą
"x �
= �� .
"xą ą
�
" "xą " ą "
= = � .
�
"x � "x � "xą "xą
� = �-1
ą "x � "xą "x �
�
�� � = = = �� .
ą �
"xą "x � "x �
" "
"� a" , "� a" ,
"x� "x�
= 1
Ć = -i".
pi -pi
"
p� = +i ,
Ć
"x�
"
$
= +i"t
"�
i = $
�,
"t
m1 E1 a" ELab
1
m2 = E2 = m2
2
Ecms
+ = .
1 2
2
s a" (p1 + p2)2 = Ecms.
s
s = p2 + 2p1p2 + p2 = m2 + 2m2ELab + m2.
1 2 1 2
"
s = Ecms = 2m2ELab + m2 + m2
1 2
m1 = m2 H" 1 Ecms = 100
ELab H" 5000
50
x = y = z = t = 0
z v ł
(z, x)
�0
�
x/z = tan � � �(�0, v, �)
u� = {1, 0, 0, 0}
pł
u�p� = �0.
ł
u� = {ł, 0, 0, łv}
p� = {�, � sin �, 0, � cos �}.
ł
�0 = ł� - łv� cos �
�
�0
� = .
ł(1 - v cos �)
cos � = ą1
"
1 - v2 1 ą v
� = �0 = �0 ,
1 " v 1 " v
�0 90ć% cos � = 0 � =
�0
ł
m n
m n
�l
...ąm
�1...�m ą1
k
T = �� T� ...�n �-1
�1...�n ąk 1 �l
k=1 l=1
g�� g��a�b�
g�� g��a�b�
ab
a ągą�b � a" a��ą gą��� b� = a�g��b�.
� �
a b
g�� = gą��ą �� .
� �
� �
�-1 �-1
� �
� �
g�� = g�� �-1 �-1 ,
� �
g��
g��
g��
1.7
3 � 3
" " " " "
ą�ł�
= 0
ą�ł�
ą, �, ł, � 0, 1, 2, 3
= 1 = -1
ą�ł� ą�ł�
a0 a1 a2 a3
b0 b1 b2 b3
D a" aąb�cłd� = ,
ą�ł�
c0 c1 c2 c3
d0 d1 d2 d3
a, b, c, d
ą�ł�
D
-1
a0 b0 c0 d0
-a1 -b1 -c1 -d1
D = - .
-a2 -b2 -c2 -d2
-a3 -b3 -c3 -d3
a2 ab ac ad
ba b2 bc bd
-D2 = ,
ca cb c2 cd
da db dc d2
D2
D
D
a, b, c, d
D = 0

D
ą�ł�
a, b, c, d
" " " " "
d4x
d4p
" " " " "
D
D
ą�ł�
ą�ł
" " " " "
d3p = dpxdpydpz
dpx, dpy, dpz
d3p
z pz = mT sinh y
E( ) = mT cosh y mT = m2 + p2 + p2
x y
d3p = dpxdpymT cosh ydy = E( )dpxdpydy
y
z dpx
dpy 1.4
dy
d3p
d&! a"
E( )
z z
d3p
d3p f( ) = E( )f( )
E( )
d3p f( )
d3
E( )f( )
E( )
�(p2 - m2)d4p
p0 2
p0
" " " " "
xł
"2x � "x � "x � "2x �
g�� + g�� = 0.
"xą"xł "x� "xą "x�"xł
ą, �, ł ą ł
ł � � ą
"2x � "x � "x � "2x �
g�� + g�� = 0.
"xł"x� "xą "xł "xą"x�
g
"2x � "x � "x � "2x �
g�� - g�� = 0.
"xą"xł "x� "xł "xą"x�
ą ł
"x � "2x � "2x � "x �
g�� - g�� = 0,
"xą "x�"xł "xą"xł "x�
"2x � "x �
g�� = 0.
"xą"xł "x�
 ą ł
"2x �
� = 0, 1, 2, 3 � = 0, 1, 2, 3
"xą"xł
g = 0

"2x �
= 0.
"xą"xł
 A�
"
= " � , = -"A0 - ,
"t
(x) a" A0
��
F = "�A� - "�A�,
A�(x) A �(x) = A�(x) + "��(x),
�(x)
x � = 1, 2, 3 "�
"
-
"x�
(x) = (x) - "�(x).
�ł �ł
0 -Ex -Ey -Ez
�ł
Ex 0 -Bz By �ł ��
��
�ł �ł
F = = -F .
�ł
Ey Bz 0 -Bx łł
Ez -By Bx 0
v z
Ex = ł(Ex + vBy), Ey = ł(Ey - vBx), Ez = Ez,
Bx = ł(Bx - vEy), By = ł(By + vEx), Bz = Bz.
" " " " "
( , )
�(x)
d = dŁ " � = ŚŁ( )
C Ł
C
Ł
ŚŁ( )
Ł
��
"�F = 4ĄJ�,
J = {�, }
"�
��
"�"�F = 0.
"�"�
��
F
"�
"�J� a" + " � = 0.
"t
"
" � = 0, " � + = 0.
"t
" " " " " "
dp�
��
= qF u�,
d�
� q u�
d
= q
dt
=
, =

" " " " "
C+ C-
C+ C-
Cą
C+ C-
=
=
 A �
A�
= "� =
=
�(x)
(x) = -"�(x).
(x) a"
(x)
�(x)
a"
d = 0 C
Ł = Ł C
(x)
Ł Ł
Ł
Ł �(x)
(x) a"
C C- C+
C
C
=

(x) a" Cą
(x)
Ł
Ł a"
(x) a" C
" " " " "
C+ = (x) =
C+
C+
C-
(x) = C-
Ł
C
 C+
Cą
" " " " "
�(x) q
�(x)
� (x) = e-iq�(x)�(x).
�(x) � (x)
e-iq�(x) x
�(x) =
e-iq�(x)
1�1 U(1)
U(1)
�(x)
px
�(x) = - �(x) = e-ipx � (x) = 1
q
(p��(x)) = p �(x)� (x),
Ć Ć
p��(x)
Ć
p��(x) = iD�(x)�(x).
Ć
D�(x) a" "� + iqA�(x),
A�
i q i
iD�
q
A�
(D��(x)) = D �� (x),
e-iq�(x)("� + iqA�(x))�(x) = ("� + iqA �(x))e-iq�(x)�(x).
A �(x) = A�(x) + "��(x).
t = 0 {�(x) = ei , A = 0}
1
{�(x) = 1, = , A 0 = 0}
q
A
"
"��"(x) D��"(x)
{x, y}
1
r d"
2
1
z r > = 0 {x, y}
2
1
Ś(B) = ĄB.
4
{x, y}
Br Br 1
A = {0, - sin Ć, cos Ć, 0}, r d" ,
2 2 2
B B 1
A = {0, - sin Ć, cos Ć, 0}, r e" .
8r 8r 2
{r = 1, Ć = Ą}
{r = 1, Ć = 0} C+ 0 < Ć < Ą C-
Ą < Ć < 2Ą r a" 1
d = {0, - sin ĆdĆ, cos ĆdĆ, 0},
C+ (dĆ < 0) C-
ĄB 1
d = " = " Ś(B).
8 2
Cą
Aą
Cą
1
(x)d = 0, ( (x) - (x))d = ą Ś( ).
ą ą
2
Cą Cą
"0
ą
Cą
Aą "0
ą
Cą Aą
ą
�ą
= - "�ą.
ą
( (x) - (x))d = "�ąd = "�ą
ą
Cą Cą
"�ą �ą
C+ C-
1
"�ą = ą Ś( )
2
"�ą
-q"�ą
Cą "ą
1
"ą = "0 " qŚ( ),
ą
2
 Aą
A
" a" "+ - "- = "0 - "0 - qŚ(B).
+ -
2Ą
2Ąn
"Ś(B) = ,
q
n
Cą
Cą
= 0
" " " " "
" = "0 - "0
+ -
" " " " "
Ą
Ś(B) = n,
e
(-e) n
-2e
" " " " "
V (x)
1
2
iD0�(x) = - �(x) + V (x)�(x).
2m
V (x)
1
2
iD 0� (x) = - � (x) + V (x)� (x),
2m
2
D 0� (x) = D0�(x) � (x) = ( �(x)) =
2
( �(x))
 e-iq�
=
"�(x) 1
i = - "2�(x) + V (x) + qA0(x) �(x),
"t 2m
V (x)+qA0(x)
E
i"t iD0
e-iEt
q c c = q
A�(x)
V (x)
�
HB = -� .
z = {0, 0, B}
yB xB
A = {0, - , , 0}.
2 2
(" - iq )2 "2 iq q2 2
- = - + " + .
2m 2m m 2m
2
HB
iqB " " qB q
HB = (-y + x ) = (ypx - xpy) = - .
2m "x "y 2m 2m
z BLz =
e
�B = H" 5, 8 � 10-5eV/T,
2m
(-e) m
HB = �B .
� = -�B .
e
�N = H" 3.15 � 10-8eV/T,
2mp
mp
HB = g�B , g = 2,
g
Jz = Lz + Sz
" = "
 Lz + 2Sz
g = 2
g - 2 ą
= + O(ą2).
2 2Ą
1
ą H"
137
2.7 � 10-13
ą
-1, 91�N
2.79�N
(p�p� - m2)�(x) = 0.
Ć Ć
p2 - m2 = 0
2
= E p� = i"�
2m
"�"� + m2 �(x) = 0.
a" "�"�
( + m2)�(x) = 0.
Ć(a)
T x x = x + a
Ć(a)x Ć-1(a)x = x - a.
T = x = x + a, x = T
�(x)
x|� |�
Ć(a)f(x).
f (x) = T
f (x ) = f(x).
x
x x
Ć-1(a)x)
f (x) = f(T
f(x) = x
" " " " "
f
x f
a x
f
z 90ć% y
x
y ( = )
T (x, y)
z
z Ć
" " " " "
� f(x)
�
f (x) = f(�-1x),
�
f
f(x) x
�-1 = �,
�
f (x) = f(�x),
f( ) = cos Ć
z
" " " " "
 x
x + a
Ć(a)( Ć(a)�(x).
0 = T + m2)�(x) = ( + m2)T
Ć(a)
T
+ m2
Ć(a)
T
�p(x) = N(p)e-ipx,
N(p) x p
Ć(a)N(p)e-ipx = N(p)e-ip(x-a) = eipaN(p)e-ipx,
T
p� = i"� p�
Ć
p� N(p)
Ć(a)
T
a
" " " " "
+m2
0
�p0(x) = e-ip t�p0(0, ).
�p(x)
2
(p0)2 - = m2,
2
p0 = ą + m2.
p0 > 0 E = p0
p0 < 0
 t = 0
p0 < 0
E = -p0 > 0 - E2 - 2
Sz z
S
 -ip0"t
-ip0(-"t) = -i(-p0)"t -p0 = E > 0
E > 0
t = t0 t0
t < t0
t > t0
e+e-
" " " " "
 J "�J� = 0
J0 = |�(x)|2
J
j�
j0
" " " " "
�"(x) �(x)
�"(x)( + m2)�(x) = 0; �(x)( + m2)�"(x) = 0.
�"(x) �(x) - �(x) �"(x) = 0.
0 = �"(x)"�"��(x) - �(x)"�"��"(x) = "� [�"(x)"��(x) - �(x)"��"(x)]
"�J� = 0,
J� = N (�""�� - �"��") ,
N m V (x)
"�� D�� "��"
"
D��"
"
J� = N �"D�� - �D��" ,
D� = "� + iqA� N
" " " " "
0
�(x) = e-ip t�(0, )
J0(x) = -2iNp0|�(0, )|2.
N J� = qu� u q
" " " " "
i
p0 > 0 N =
2m
p0
J0(x) = |�(0, )|2.
m
p0
1
m
p0 < 0 -iN
J0(x) d" 0
N
p0
" " " " "
E
J0(x) = q |�(0, )|2,
m
q E = |p0|
q = 0 J�(x) a" 0
E
ł
ł E
 A
��(x) = 0,
�(x)
�A(x) a" ��(x).
���(x) = 0 = ���(x).
�(x)
A
[�, �] = 0,
� �
Ć(a)
3.1 � = + m2 � = T
Lę!+
= "�"� �(x)
Ć
( + m2)Lę!+�(x) = 0.
Ć
Lę!+�(x) �(x)
�(x)
Ć
P
Ćx Ć2x
P = {t, - }, P = x.
Ć2
P = 1
2Ą
Ć2
P = 1
 Ć(x) Ć
�
Ć
ĆĆ(x) Ćx), Ć2Ć(x) 2
P = Ć(P P = Ć(x)
2
Ć2
P = 1 = 1
= ą1.
Ą
S = (-1)S
= (-1)S+1 S = 0
S = 1
Ćx) L
Ć(P = Ć(x)
= ą1
L
ĆĆ(x) Ćx) L
P = Ć(P = Ć(x).
Ć(x)
L
Ć
P
L
= (-1)L
L
" " " " "
" " " " "
" " " " "
Ć
"�"� P
ĆP (x)
Ć(x)
Ą
" " " " "
x, y z 180ć%
Ć
z Lę!+


ĆC(x) = Ć"(x).
C
C
Ć(x)
C
Ć(x)

2Ć(x) = | |2Ć(x),
C

2 = 1
C
q
("� + iqA�(x))("� + iqA�(x))Ć(x) + m2Ć(x) = 0,
C
("� - iqA�(x))("� - iqA�(x))ĆC(x) + m2ĆC(x) = 0.
-q 



0 0
e-ip t e+ip t
p0 < 0
qA (-q)(-A) =
qA
(Ć"(x))" = Ć(x)
Ć(x) ĆC(x)
Ą0
n Ą+ n Ą-
n0 Ą0
" " " " "
C
+1
 Ą
Ć
T
Ćx
T = {-t, }.
Ć2
T = 1
ĆĆ(x) T Ćx).
T = Ć"(T
T
Će-iEt
T = (e-iE(-t))" = e-iEt,
t
("�"� +m2)
ĆT
Ć

P
" Lę!+
"
"
CP T
CP T
CP T
Ć

P
CP
CP T
CP
 SL(2, C)
Lę!+
SU(2)
2Ą
2Ą
Lę!+ SL(2, C)
L
SU(2)
SL(2, C)
SL(2, C) Lę!+
� SL(2, C) A
� A = AA �B.
B
{1, 2}
�
Lę!+
 U
U-1
U� = UAU-1U�.
A �
�
� = UAU-1 � = U�.
A �
SL(2, C)
1
j = m = ą1
2 2
�1�2 - �2�1
Lę!+
(�1�2-�2�1) = (A1 �1+A1 �2)(A2 �1+A2 �2)-(A2 �1+A2 �2)(A1 �1+A1 �2).
1 2 1 2 1 2 1 2
(�1�2 - �2�1) = (�1�2 - �2�1) det A = �1�2 - �2�1,
det A A
det A = 1.
Lę!+
A = 1 + c + ( + i )�.
c �i i =
x, y, z 2 � 2
|c|
|ai| |bi|
c = 0.
|c|, |ai|, |bi|
det A = 1 c = 0
" " " " "
=
 1 + i �
" " " " "
�A�A a" �1�1 + �2�2 = �1�2 - �2�1.
�1 = �2; �2 = -�1.
�A = �B.
AB
0 1 -1
T -1 T
= = - = - = = i�y.
-1 0
e1 e2
� = �AeA.
eA
�A �
1.10
�A�B = �A�A
AB
AB
(�A�B + �B�A) = 0
AB
�A�A = -�A�A.
�A�A = 0.
A
B
AB BA BA
�A = - �B = �B = �B ,
AB
= ,
AB
AB
= AB
" " " " "
A
DC
�A = � B = AB �C = AB �D a" �AB�B.
AB AB AB
C C
�
T
� = A = �yA�y = (A-1)T .
�A = �B(A-1)B .
A
�A�A
" " " " "
a b d -c
A = , � = .
c d -b a
" " " " "
� A" = AA "�B".
B
A" SL(2C)
�
�
A
A"
A A"
U A
A" = UAU-1.
A = 1 + ( + i )�,
A" = 1 + ( - i )�".
( - i )�" = ( + i )U�U-1.
=
�" = U�U-1.
=
�" = -U�U-1.
A"
A � �"
A"
Ł Ł
� A = A"A �.

A" �" x = y"
y = x"
a
� �"
b
 �, � A
A
A
Ł
�A" = �"A.
�
I
Ł
�I, �I, �I �I
Ł
=
�"U = -U�.
U �y
�x �z �y
0 1
U = i�y = = .
-1 0
Ł
�A <" �A, �A <" -�A.
Ł
a <" b a b
Lę!+ A A"
" " " " "
" " " " "
Lę!+ A A"
1.10 m
n
2m+n
1 1 m 1 n 1
ś A ,...,Am,1,...,n = AA . . . AA A" . . . A" śC ,...,Cm, 1,..., n,
C1 Cm 1 n
A1 Am 1 n
1
śA ,...,Am,1,...,n <" �1 . . . �m �m+1 . . . �m+n,
<"
A �
a b
śAB = ,
c d
d -c
�
śAB = = śAB = (ś-1)BA det ś,
-b a
ś śAB
" " " " "
ś
Lę!+
1
śA ,...,Am,1,...,n,
A1A2
Lę!+
ś A1 "! A2
ś
m n
Ł
1 1
m + 1 n + 1
d(m, n) = (m + 1)(n + 1).
SL(2C)
(m + 1)(n + 1) = 4
m = n
śA
p
śA
śA
Lę!+
1 j
Tm j = 0, 1
2
ą1 ą2 ą
j
� Tm
p
1 1
1
T1 = " (-px - ipy) <" ą1ą1 = - " ś11,
2 2
1 1
1
"
T0 = pz <" (ą1ą2 + ą2ą1) = - (ś12 + ś21),
2
2
1 1
1
"
T-1 = (px - ipy) <" ą2ą2 = - " ś22,
2 2
1 1
0
"
T0 = p0 <" (ą1ą2 - ą2ą1) = - (ś12 - ś21),
2
2
"
śAB = - 2ąAąB.
0
T0
0 1
T0 Tm
  
śAB
śAB = śBA
śA
Ł
ś11 = ś2 <" ś21
1
Ł
ś12 = ś2 <" ś22
2
Ł
ś21 = -ś1 <" -ś11
1
Ł
ś22 = -ś1 <" -ś12.
2
<"
Ł
�21 = px + ipy,
Ł
�22 = -p0 - pz,
Ł
-�11 = p0 - pz,
Ł
-�12 = -px + ipy,
 �A
p�
śA
�A
 śA

�A
Ł -p0 + pz px - ipy Ł
�IJ = = (-p0 + �)IJ.
px + ipy -p0 - pz
śA
ś A = AA A" śC  = (AśA )A.
C

det A = 1
Ł Ł
det ś IJ = det śIJ.
śIJŁ
�A
Ł
det �IJ = 2(p0)2 - 2
2 = 1
 = ą1

 = -1.
�A  = -1 śA
Lę!+ śA
p
2 = 1 �A�A
" " " " "
 = -1
śA = (p���)A,
�
�� = {1, -�}.
�
śIJŁ �I �JŁ
Ł
1.10 ��IJ
�
T
(p0 + �T ) = �y(p0 + �T )�y = p���,
�
�A = {1, �}A
Lę!+
z z
x, y, z
z
z
A = 1 + � = A .
=
A
v
ś
ś = ś + ( �)ś + ś( �).
ś
śA
ś - ś
� ś
ś - ś = ( �)(p0 + �) + (p0 + �)( �) = 2p0( �) + 2 ,
�j�k + �k�j = 2�jk
ś - ś = p 0 - p0 + ( - )�.
v z
p 0 - p0 = vpz, p x - px = 0, p y - py = 0, p z - pz = vp0,
ax = ay = 0, vpz = 2azpz, vp0 = 2p0az.
1
az = v
2
z v
v
A = 1 + �z.
2
A
z Ć
Ć
N
N
v
y = v
y y
N
y
y y�z
�z
2
A(y) = AN ( ) = (1 + )N = e .
N 2N
1 0
u+ = u- = A
0 1
ey/2 e-y/2
y z
y
2
e 0
A(y) = A"(y) = y .
0 e- 2
" " " " "
uą
z ą1
2
cą
y y
2 2
A(y)(c+u+ + c-u-) = e c+u+ + e- c-u-.
z y
" c+ = 0

z
y -"
" " " " "
A
uą
Ć
P
z
Ćx Ć2x ĆSzP
Ć-1
P = {t, - }, P = x, P = Sz
Ć2
P
Ć2
P
2Ą
Ć2�(x)
P = �(x) 2Ą
Ć2�(x)
P = -�(x).
Ć2
P
z
Ć�A(x) P Ćx) Ć�A(x) Ł Ćx).
P = �A(P P = i�A(P
z A
P
P
" " " " "
Ć
x = 0 P
P
ĆL(v)�A(0) = P L(v)�A(0) P
ĆL(v)�A(0) = L(-v) P �A(0),
P
Ć
x = 0 Lx = x
" " " " "
� ś
Sz
" " " " "
Ć�A(x) = i�A(P P Ł = i�A(P
Ćx), Ć�A(x) Ćx),
P
Ł
Ł Ł
Ć�A(x) Ćx), Ć�A(x) Ćx).
P = -i�A(P P = -i�A(P
�I
Ł
Ć
P
Ć
-iP


Ł
"

�A(x) = �"A(x), 
�A(x) = �A(x),
C Ł C
Ł
"

�A(x) = �A(x), 
�A(x) = �"A(x),
C C
Ł
C

2 = 1.
Ł

2�A(x) = 
�"A(x) = | |2�A(x).
C C
Ł
�I(x) �I(x)
" " " " "


1
0
Ł Ł
I I I I
a 1 1 a"

= 
a = a" = C .
C
0 0 0 0
Ł
I I
a a

= C ,
0 0
1
0
C
"
C C
" " " " "
CP
Ł
"
�A(x) = -i C�A(P 
P Ł = -i C�"A(P
Ćx), Ć�A(x) Ćx),

P
Ć�A(x) C Ćx), 
P Ł = i C�A(P
Ć�A(x) " Ćx).

P = i �"A(P
Ł
i
Ć
P


" " " " "
Ć

P
Ć Ć
.

P = P
Ć2
i P = -1
Ć
T
Ćx Ć2x ĆSzT
Ć-1
T = {-t, }, T = x, T = -Sz
Ć�A(x) T A Ćx) Ć�A(x) T Ł Ćx),
T = �" (T T = �"A(T
T
Ć
� T
Ć
x = 0 T
L(v)�A L(v) z
Ł Ł
ĆL(v)�A(0) = T L(v)�"A(0), T
ĆL(v)�A(0) = T L(-v)�"A(0),
T
�
" " " " "
Ć�A(x) T A Ćx), Ć�A(x) T Ł Ćx),
T = �" (T T = - �"A(T
Ł
Ł
Ć�A(x) T Ćx), Ć�A(x) T Ł Ćx).
T = - �"A(T T = �" (T
A
Ć2
T = -1.
"
Ć2�A(x) Ć Ćx) T
T = T �A(T = -| |2�A(x).
T
Ł Ł Ł Ł
1 1
Ć2śA ...AmAm+1...An = (-1)nśA ...AmAm+1...An.
T
n
1
n n
2
�(x)
$
�(x) = E�(x),
Ć$
Ć Ć�(x)
T = $
T T
E
�(x)
T �(x)
Ć2�(x) Ć Ćx)
T = T �(T = | |2�(x),
 �(x)
" " " " "
ĆT ĆT Ł
�A(x) = i C " �A(-x), 
P �A(x) = -i C " �A(-x),

P
Ł
T T
Ł
ĆT = i C " �A(-x), 
P �A(x) = -i C " �A(-x),
Ć�A(x) ĆT Ł

P
T T
ĆT
Ćx
P = -x
"
= 1
C
T
" " " " "
Lę!+
Ć Ć
Lę!+ P
�I �I
Ł
�I(x)
�(x) = .
�I(x)
Ł
�(x) � (x) = U�(x) U
4 � 4
� (x) �I(x)
�I(x)
Ł
A 0
Ć Ć
L�(x) = S�(L-1x), S = .
0 A-1
Lę!+
 4 � 4 ł0
0 1
ł0 = .
1 0
4 � 4 2 � 2
2�2 ł0
a b d c
ł0 ł0 = ,
c d b a
a, b, c, d 2 � 2
ł0
a b
(ł0)2 = 1
c d
ł0 a = d b = c a = -d b = -c
" " " " "
a b
ł0 = .
b a
ł0
ł0S ł0 = S-1.
Ć, Ć
� = P 
, T
�I, �I ԾI, ԷI
Ł Ł
�(x) ��(x) = 0
���-1��(x) = 0
���(x) = 0 �
��(x) �
��(x)
�(x)
�
UO
ԾI(x)
�O(x) = UO .
ԷI(x)
Ł
 (iDx + px)�(x) = 0
" " " " "
" " " " "
-
�I(x) -�I(P
Ćx)
P
�P (x) = ił0
�I(x) = -�I(P ,
Ćx)
P
Ł Ł
0 - 
�I(x) �"I(x)
�C(x) = = ,
"
0 �I (x)

�I(x)
Ł Ł
Ćx)
�I(x) -�"I(T
0 T
�T (x) =
"
Ćx)
�I(x) = -�I (T ,
0
T
Ł
Ł
UP = ił0,
0 -
UC = ,
0
0
UT = .
0
- 0
UCP = i ,
0
UCP T = ił5,
1 0
ł5 = .
0 -1
Ć Ć
Ć Ć


P = P 
T = -T
ĆP ĆT
Ć Ć
T = -P
CP T
" " " " "
Ć Ć
P 
T �(x)
� (x) = U�(x)
� 0
�O(x) = UO �(x)
0 �
U U-1U = 1
�(x)
� 0
� O(x) = UUO U-1� (x) a" h
O�� (x).
0 �
Ć
� = P U-1
h
P = UUP U-1,
h
P UP � =
Ć

� = T �
U-1
h
O = UUOU"-1.
U
U
h
O = UUOUT
" " " " "
ł5
S ł0
ł5�(x) = ��(x), � = ą1.
�
� = +1 � = -1
1 0 �I(x) �I(x)
=
0 -1 �I(x) -�I(x)
Ł Ł
�I(x) a" 0
Ł
�I(x) �I(x) a" 0
Ć
�I(x) P 

Ł
Ć Ć Ć ĆT
Ć
Lę!+ T 
P 
P
1
P+ = (1 + ł5),
2
1
P- = (1 - ł5),
2
Pą
Ć"(x)�j(x), i = 1, 2, 3, 4, j = 1, 2, 3, 4.
i
x
� Ć
Ć
Lę!+
Ł
�I
ąI
Ć" = , � = .
�I
�I Ł
Ć ą �
Ć" �
Ł Ł
ąA�B, ąA�, �A�B, �A�,
Ł
ąA�B
�A�
�A�
��
F
��
F
Ć�i�, Ć = Ć ł0,
 �i 4 � 4
Ć
Ć
LĆ = (SĆ) ł0 = Ć S ł0 = Ć ł0ł0S ł0 = ĆS-1,
ĆĆ
P = (ił0Ć) ł0 = -iĆ ł0ł0 = -iĆł0.
Ć
LĆ�i� = ĆS-1�iS�,
ĆĆ�i�
P = -iĆł0�iił0� = Ćł0�ił0�.
ł0�ił0 = P �i, P = ą1,
ĆĆ�i�
P = P �i�
ĆĆł5�i� = Ćł0ł5�ił0� = -Ćł5ł0�ił0� = -P Ćł5�i�,
P
�i ł5
Ć
Lę!+
S-1�iS = aij�j,
j
aij �i
cj�j = 0
j
ci
Ć
LĆ�i� = ĆS-1�iS� = aijĆ�j�
j
Ć
LĆł5�i� = ĆS-1ł5�iS� = Ćł5S-1�iS� = aijĆł5�j�.
j
�S = 1
Ć
LĆ� = ĆS-1S� = Ć�,
ĆĆ�
P = Ćł0ł0� = Ć�,
 �P = ł5
��
V
�
a b
�� = ,
V
c d
Ł Ł
Ć�� � = �Aa�A�B + �Ab�A� + ąAc� �B + ąAd��.
V B Ł Ł
AB A
�
a = d = 0.
Ł
�� �Ab�A� ąAc� �B
V Ł
AB
b�A c�
Ł
AB
b c
b� = ei� {1, -�}, c� = ei� {1, �},
�b �c [0, 2Ą)
Ć
Lę!+ Ć�� �
V
0 c�
ł0�� ł0 = ,
V
b� 0
�b = �c
�b = �c = 0
0 ��
�
�� a" ł� = ,
V
�� 0
ł�
�� �
A
�� = ł5ł�,
A
i
��� = ��� a" (ł�ł� - ł�ł�) .
T
2
���
�i 0 �k 0
�0i = -i , �ij = ijk ,
0 �i 0 �k
ijk
����
T
Ƴ��
ĆS-1ł�S� = �� Ćł�� = Ć�� ł��.
� �
Ć �
S-1ł�S = �� ł�.
�
���
S-1 S
i
� �� = (S-1ł�SS-1ł�S - S-1ł�SS-1ł�S).
2
���
��� ł5
" " " " "
16
�S = 1, �P = ł5, �� = ł�, �� = ł5ł�, ��� = ���,
V A T
�i�
ił0
Ć2
ł0 P = 1
" " " " "
Ć(x) �(x)
�
x �-1x
�i ��
� i = U�iU-1
 ł�
ł�
0 1 0 -�
ł0 = , ł = .
1 0 � 0
ł�
ł�ł� + ł�ł� = 2g��.
ł�łł� = -2ł
ł�łł�ł� = 4g�
" " " " "
ł�
ł �
U
ł � = Uł�U-1, � = 0, 1, 2, 3.
ł
U = ł5 U = ł0
" " " " "
ł5 ł�
ł5 = ił0łxłyłz.
ł5 ł�
ł�ł5 + ł5ł� = 0, � = 0, 1, 2, 3.
�i
ł�
ł� ł5
ł� = ł�.
U
ł � a" (Uł�U-1) = Uł� U = Uł�U = ł�.
a = a�ł�,
/
a
a/ + /a = 2ab
/b b/
" " " " "
UC = iły, UT = łxłz, UCP = ł0ły.
ł � = Uł�U-1
U
1
S =
2
1
S =
2
p�
p0 - pz -px + ipy
pA = = (p0 - �)A
-px - ipy p0 + pz
p0 + pz px - ipy
pA = = (p0 + �)A.
px + ipy p0 - pz
p� = i"�
Ć
1957
1957
1997
1997
Ł
pBA�A(x) = 0.
Ć
Ł
pA�A(x) = 0.
Ć
ś+ = �I, ś- = �I,
Ł
(p0 ą Ć�)śą(x) = 0.
Ć
śą
� = "1
(p0 " �)
Ć
( Ć�)2 = Ć2
p�p�śą = 0,
Ć Ć
śą(x) = uą(p)e-ipx.
i"�e-ipx =
p�e-ipx
2
p0 = ą = ą| |.
śą
(p0 ą �)uą(p) = 0.
1
S =
2
�
 = .
2| |
p0 = | | > 0
| |
(1 ą 2)uą = 0.
 �I = ś-
1
( = + ) �I = ś+ ( =
Ł
2
1
- )
2

-
 = ą1  = 0
p0 = -| |
(1 " 2)uą = 0.
ś+
(1 - ł5)
0 0 �I 0
(1 - ł5)� = 2 = 2 .
0 1 �I �I
Ł Ł
(1 + ł5)
� �śą(x)
śą(x)
ś+ ś-
T CP CP T
Ć�I(x) T " Ćx) T Ćx), Ć�I(x) T Ćx) T " Ćx).
T = �I (T = �"I(T T = - �"0
(T = + �I (T
Ł
Ł
Ćśą(x) T " Ćx),
T = śą(T
"
(-p0 " px�x ą py�y " pz�z)śą(x) = 0,
Ć Ć Ć Ć
p�
Ć
�y t -t p0
Ć
Ćx
T x
"
Ćx)
(p0 " px�x ą py�y " pz�z)śą(T = 0.
Ć Ć Ć Ć
= i�y
T T
"
Ćx)) Ć Ćśą(x)
(p0 ą px�x ą py�y ą pz�z)(ą śą(T = (p0 ą �)T = 0.
Ć Ć Ć Ć
T
Ć�I(x) " Ćx) C Ćx), Ć�I(x) Ł Ćx) C " Ćx).

P = -i�I (P = -i �"I(P 
P = -i�"I(P = i �I (P
Ł
Ł
-
"
Ćx)
(-p0 ą px�x " py�y ą pz�z)śą(P = 0.
Ć Ć Ć Ć
"i
c
"
Ćx) Ć Ćśą(x)
(-p0 " px�x " py�y " pz�z)("i )śą(P = -(p0 ą �)
P = 0,
Ć Ć Ć Ć
c
CP
CP T
ĆT
Ćśą(x) C "

P = "i śą(-x).
T
"
" x -x
C
T
ĆT
Ćśą(x)
(p0 ą �)
P = 0.
Ć
Ć Ć
T 
P
1
m
2
e-ipx
2
p0 = ą + m2.
�I(P
Ćx)
�I(x) �I(x) = -iP
Ł
(p0 + � Ć)�(x) = m1�(x),
Ć
m1
(p0 - � Ć) (� Ć)2 = Ć2
Ć
p�p��(x) = m1(p0 - � Ć)�(x)
Ć Ć Ć
(p0 - � Ć)�(x) = m2�(x),
Ć
m2
m2 = .
m1
� �
1 1
m0 = (m1 + m2), �m = (m1 - m2),
2 2
p�ł��(x) = (m0 + �mł5)�(x).
Ć
Ć(x)
Ć(x)p�ł��(x) = m0Ć(x)�(x) + �mĆ(x)ł5�(x).
Ć
�m = 0,
m1 = m2 m0 = m
(p - m)�(x) = 0.
/
Ć
(p - m)
/
Ć
ĆAPC�C = m2�A
Ć
P
1
Ć
� = PC�C
m
" " " " "
(p - qA - m)�(x) = 0,
/ /
q E
m H" 0.5
" " " " "
�(x)
ł0 ł�
� (x)(-p0ł0 - ł - qA0ł0 - q ł - m) = 0
Ć
4 � 4
i -i
p = pj � (x)
Ćj Ć
-ł0
�!
�(x)( / +qA + m) = 0.
p /
"
-i
"x
1 � 1
"
px = -i"x
Ć
" " " " "
�(x)
�(x)
�!
�(x) p �(x) + �(x)p = i"�[�(x)ł��(x)] = 0.
/ /�(x)
J� = N(p)�(x)ł��(x),
N(p)
�(x)ł0�(x) = � (x)�(x) e" 0.
J0
 J0
"�(x)
i = $
�(x).
"t
"�(x)
ił0 = (ł Ć + m)�(x).
"t
ł0
$
= ą Ć + �m,
ą = ł0ł, � = ł0.
ąi
0 1 0 -�i �i 0
ąi = = .
1 0 �i 0 0 -�i
p0 > m p0 < -m
2m
1021
p0 <
-m p0 > m
1
m
4 � 10-11
m-1
e
 10-8
10-13
O = P, C, T
CP CP T O
O �O(x)
�(x) �O(x)
-
�
Ć
P
-1
UP (p0ł0 + Ćł - m)UP �P (x) = 0
Ć
UP = ił0 ł�
(p - m)�P (x) = 0.
/
Ć
P
�O(x)
�I
�0
UO
�
" " " " "
z


-1
UC(-p0ł0 + pxłx - pyły + pzłz - m)UC �C(x) = 0.
Ć Ć Ć Ć
p� = i"�
Ć
ły UC = iły
ł�
(p - m)�C(x) = 0.
/
(p - qA - m)�(x) = 0,
/ /
(p + qA - m)�C(x) = 0,
/ /
qA A�(x)
/
p�
Ć
-q
A
" " " " "
4
"
J0 = N� � = N �i �i,
i=1
4
0 "C C
JC = N�C �C = N �i �i ,
i=1
�C = iły�" �C =
-i�T ły = +i�T ły
0 "
JC = Ni�T łyiły�" = N�T �" = N �i�i .
i
� �
N NC = -N
J0
p0 > 0
e-ipx
p0ł0u( ) = mu( ).
p0
u
p0 = m
�
u( ) = .
�
= 0
Ću(
P ) = ił0u( ) = iu( ).
uC( )
Ć
P
Ć(u(
P )uC( )) = iu( )iuC( ) = -(u( )uC( )).
Ć2
P = +1
1
s =
2
t -t
-1
UT (p0ł0 + pxłx - pyły + pzłz - m)UT �T (x) = 0.
Ć Ć Ć Ć
UT = łxłz ł�
(p - m)�T (x) = 0.
/
C P
C, P T
CP CP T
CP T
 CP
C, P, T
�O(x)
�(p)
�(x) = u(p)e-ipx, u(p) = ,
�(p)
p
(p0 + � )� = m�, (p0 - � )� = m�.
�
�
" " " " "
�
� � = 1.
� �
�
u(p) = Ną p0-� ,
�
m
N+ p0 > 0 N- p0 < 0
�
1
� � = 2| |�,  = ą .
2
�
1
uą(p) = Ną p0-2| | ,  = ą , ąp0 = E > 0.
� 2
m
p0 = ąm cosh y, | | = m sinh y,
�
uą(p) = Ną .
ąe"2y�
p0 > 0 y 1
1
 = +  = -1
2 2
|y| 1
m2
2p2
0
" " " " "
�� = u(p)u(p) = ą2|Ną|2e"2y.
p0
Ną = eąy,
�(x)
eąy�
�ą(x) = e-ipx.
ąe"y�
�
u+(p) =
�
" " " " "
p0 > 0
= y = 0
�
u(p) = , J0 = u u = 2,
�
=
J�
{2, }
p�
J� = 2 .
m
 J�
" " " " "
p0 < 0
0 -
C
C
� (x) a" uC(pC)e+ipx = u"(p)e-ip x,
 
0
pC = -p
pC
"
0 - ey �
uC(pC) = u"(p) = .
  "
0 e-y �
" "
�i = i�y�i = -�i , i = x, y, z.
-
" "
� � = -2| | �,
" "
� � = 2| | �.
C C
"
� 
C
" C
� = � .
C
ey�
C
C
� (x) = e-ip x,
C
e-y�
p0 > 0
p pC � �C
�C(x)
�
łD
�
łS
� �
łD = UłSU ,
1 1
1 1
0 5
"
U = U = " = (łS + łS).
1 -1
2 2
1 0
0 5
łD = = łS,
0 -1
0 �
łD = = -łS,
-� 0
0 1
5 0
łD = = łS.
1 0
Ć(p)
�(x) = N e-ipx.
�(p)
Ć �
Ć �
(p0 - m)Ć = � �, (p0 + m)� = � Ć.
p0 = E > 0.
Ć
� Ćs
�zĆs = 2sĆs, Ć Ćs = 1.
s
1
s = ą
2
Ćs z �s
Ćs
Ćs 1
�+,s(x) = Ns � e-ipx, p0 > 0, s = ą .
Ćs
2
E+m
| |
| | = 1
E
�s � �s 1
s
�+,s(x)
�+,s�+,s = 2.
� Ćs
1 0
2
2 = �+,s�+,s = Ns Ć , Ć
�
s s
0 -1 Ćs
E + m
E+m
2
(� )2 = = E2 - m2 = (E + m)(E - m).
Ns
E + m
Ns = .
m
E + m Ćs 1
�+,s(x) = e-ipx, p0 > 0, s = ą .
�
Ćs
m 2
E+m
p0 = -E < 0.
�s
�z�s = 2s�s, � �s = 1.
s
s
z
Ćs
�
1
- �s
�-,s = Ns E+m e-ipx, p0 < 0, s = ą ,
�s 2
Ns
2 2m
�-,s�-,s = -Ns .
E + m
-2
E + m
Ns = Ns = ,
m
�
E + m - �s
1
E+m
�-,s = e-ipx, p0 < 0, s = ą .
m �s 2
U
�s
Ćs
" " " " "
UC
0
y
UC = iłD = .
- 0
�"
E + m
0
-E+m �"
C
s
�-,s(x) = e+ipx
- 0
m
�"
s
"
�i =
-�i
E + m + �"
C s
�-,s(x) = e+ipx.
�
m - �"
s
E+m
�z �" = -2s �".
s s
�" = Ć-s.
s
�C
p� = -p�
C
E + m +Ć-s
C
C
�s (x) = e-ip x,
� c
m Ć-s
E+m
pC0 = -p0
= - z -s
C
1
S =
2
p0 > 0
�A( )
�( , t) = e-iEt, E = p0 > 0.
�B( )
(E - qA0 - m)�A( ) - �( Ć - q )�B( ) = 0,
�( Ć - q )�A( ) - (E - qA0 + m)�B( ) = 0,
A
m q
q = -e
m
| | |qA�| E m
ENR = E - m
2
m
2m
�B( )
1
m
�B( )
1
(E - qA0 - m)�A( ) - �( Ć - q ) �( Ć - q )�A( ) = 0.
E - qA0 + m
 (E-qA0+m)-1 �( Ć-
q )
1
$
�A(x) = ENR�A(x), $
= �( Ć - q ) �( Ć - q ) + qA0.
ENR - qA0 + 2m
q A0
1 1
= + O(m-2)
ENR - qA0 + 2m 2m
[�( Ć - q )]2
$
= + qA0.
2m
�j Ć = Ć -q
(� Ć)2 = Ć2
(� Ć)2 = Ć2 + i�( Ć � Ć).
$
= $
0 + $
1,
$
0 $
1
Ć
[( Ć - q ) � ( Ć - q )] = iq(" � + � "),
" � = - � ",
= (" � )
1
= �
2
q
$
1 = - .
m
2.4
$
1
104
 ą5.8 � 10-5
10
�
Emagn = -� ,
q
� = g , g = 2.
2mc
g
g = 1
( = )
3 � 10-26
=
$
2 = $
2,1 + $
2,2 + $
2,3,
( Ć2)2
$
2,1 = - ,
8m3
q
$
2,2 = - �( � Ć),
4m2
q
$
2,3 = - (" � ).
8m2
q = -e
1
ą = e2 H" .
137
e2
ą = ą = 10-7e2
4Ą
�(x)
$
2
E2 = dx �"(x)$
2�(x).
$
2,i
2
m2
2 2
( )2
2
E( ) = + m2 = m + - + � � � .
2m 8m3
$
2,1
| |
ą
m
$
2,1 mą4
$
2,2
"V (r)
= -"V (r) = - .
r "r
$
2,2
q "V (r) q "V (r)
Ć.
$
2,2 = �( � Ć) =
4m2r "r 2m2r "r
 = +
Ć2 Ć2
2 2 2 1 1 3
= (1 + ) =
2 2 4
1 3 1 3
= J(J + 1) - L(L + 1) - = [(J - L)(J + L + 1) - ].
2 4 2 4
1
J = L + J =
2
1
L-
2
L = 0
qV (r)
ą
-
r
ą
$
2,2 =
2m2r3
r | |
1
H2,1 r H"
mą
mą4
$
2,3
"�
4Ą
" � = 4Ą|e|�3( ).
$
2,3
Ąą
$
2,3 = �3( ).
2m2
Ąą
dx �"( )$
2,3�( ) = |�(0)|2,
2m2
|�( )|2
1
r
mą
|�(0)|2 H" m3ą3.
mą4
L = 0 �( ) =

S
L = 0
1
m
e+e- 2m
1
m
1
m
$
2,3
�ij
�xi = 0, �xi�xj =
3m2
�ij �
(� )2 = (�x1)2 + (�x2)2 + (�x3)2
�
3
"V ( ) 1 "2V ( )
V ( + � ) = V ( ) + �xi + �xj�xk ,
"xi 2 "xj"xk
i=1 j,k
1 q
V ( + � ) - V ( ) = "2V ( ) = - " � .
6m2 6m2
$
2,i, i = 1, 2, 3
mą2 mą4 3 n
ENR(n, J) = - + - ,
2n2 n4 8 2J + 1
1
n = 1, 2, . . . J = L ą
2
ą
ą4
0, 511
mą2 H" 27, 2 , mą4 H" 0, 00145 ,
 S L = 0
L = 0

J
2P n = 2, L = 1
"E H" 4, 53 � 10-5
 H" 2, 74
S
1S 1 5.9 � 10-6
2
21
21
n
J 2S 1 2P 1
2 2
2P 1 4 � 10-6
2
29 1040
A0
qA0 = V (z) V (z) = 0
z < 0 V (z) = V0 > 2m z > 0
z = 0
z = 0
E
m < E < V0 - m.
 pz > 0
1
z +
2
z > 0
5.6
0
e-ip t
z -"
"
1
�in+(z) pz = + E2 - m2 Sz = +
2
�Rą(x) -pz z
1 1
+ -
2 2
z < 0
z z z
�(z) = uin+(pz)eip z + R+uR+(-pz)e-ip z + R-uR-(-pz)e-ip z, z < 0,
Rą
uin+
uRą
Rą
z < 0
�ł �ł �ł �ł �ł �ł
1 1 0
�ł �ł �ł �ł �ł �ł
0 0 1
z z z
�ł �ł
�(z) = eip z+R+ �ł -pz �ł e-ip z+R- �ł 0 �ł e-ip z, z < 0.
pz
�ł łł �ł łł �ł łł
E+m E+m
pz
0 0
E+m
z > 0
+z E z
ą1
2
V0
E E - V0 pz
p z = (E - V0)2 - m2.
p z(E, V0, m)
�ł �ł �ł �ł
1 0
�ł �ł �ł �ł
0 1
z z
�(z) = T+ �ł p z �ł eip z + T- �ł 0 �ł eip z, z > 0,
�ł łł �ł łł
E-V0+m
-p z
0
E-V0+m
Tą
-" < z < "
Rą Tą �(z) z = 0
p z E + m
r = .
pz E - V0 + m
r
pz p z
(E + m)(V0 - E + m)
r = - .
(E - m)(V0 - E - m)
m2 - E2 + V0E + V0m
m2 - E2 + V0E - V0m
r < -1.
z = 0
1 + R+ = T+
R- = T-
1 - R+ = rT+
R- = -rT-,
pz
E+M
r < -1
R- = T- = 0.
2 1 - r
T+ = , R+ = .
1 + r 1 + r
z
Jin JR JT
2pz 2pz 2p z
z z 2 z 2
Jin = , JR = -R+ , JT = T+ .
E + m E + m E - V0 + m
2
z z z z
JR 1 - r JT 4r -JR + JT
= - , = , = 1.
z z z
Jin 1 + r Jin (1 + r)2 Jin
z
JR
e+e-
p2 = 0

e+e-
1
m
2m
p2 = m2
c3
TH = ,
4kBMG
kB M G
M
TH H" 3.9 � 10-7 K,
M
M 2 � 1030
106
1023
4 � 1026
1012
�
Z Z + 1
Z + 2
76
Ge
�
�
�
 �
�(x)
�C(x)
�(x)
�(x)
1 1
" (�(x) + �C(x)), " - �C(x))
(�(x)
2 i 2
" " " " "
�