!
"
#
$%
$
"
"
# &
%
#
'
(
)
*
+
,
(
)
,
$
,
-
µ
#
"
(
!
.
,
/
0
"
%
#
1
$
# &
2
#
,
#
3
"
+
(
$
(
4
5
6
78
9
:
;
<
= > ? @
A
=
B @
?
C
B
DB
> B
:
B
DB
E F
;G
B
'
"
$
,
,
#
"
/ +
H
IJ
# &
,
!
&
(
%
#
'
(
)
*
K
+
−
1
r
2
d
dr
r
2
d
dr
+
2µ
¯h
2
(V (r) − E) +
l(l + 1)
r
2
u(r) = 0
L
M
N
4O
P
"
(
!
2
V (r) = −
Ze
2
r
L
M
N
4
M
P
-
!
H
"
,
(
!
&
(
+ /
(
0
$
% &
H
IJ
#
"
3
(
$,
'
(
)
Q
4
R
"
!
0
H
"
2
1
,
$
(
!
1
#
"
-
J
ψ
n
lm
(r, θ, ϕ) = u
n
l
(r)Y
m
l
(θ, ϕ),
L
M
N
4
N
P
+
1
,
$
(
!
u
H
!
#
l
(
/
$
#
#
!
n
#
!
S
TU
V
T
WX
Y
4
n = ν + 1 + l,
+
ν = 0, 1, . . . l = 0, 1, . . .
L
M
N
4
Z
P
[
-
(
# %
+
3
,
#
"
/ +
"
-
J
\
E
n
= −
µe
4
Z
2
2¯h
2
n
2
= −
e
2
Z
2
2a
0
n
2
,
a
0
=
¯h
2
µe
2
,
L
M
N
4
]
P
+
-
2
a
0
= 5, 29 × 10
−
9
(
!
-
# &
^
"
/
,
#
,
L
"
_
`
)
P
4
a
,
#
H
0
H
n
-
3
1, 2
∞
4
b
$
H
+
n
H
#
-
&
!
$
%H
#
^
(
ν
l
0
(
!
U c
d
c
X
c
S
T e
f
g
#
,
#
"
/ +
4
R
^
I
"
$
2
$
$,
"
#
-
(
)
"
%
#
n = ν + l + 1 ν l
+
(
!
2l + 1
nazwa
+
(
!
(
2
$
#
1
0 0
1
1s
1
2
0 1
3
2p
1 0
1
2s
4
3
0 2
5
3d
1 1
3
3p
2 0
1
3s
9
L
M
N
4
h
P
i
#
"
$
J
-
3
/
"
^
(
)
,
$
0
H
(
2
$
#
+
(
!
#
-
n
2
:
n−1
X
l=0
(2l + 1) = 2
n(n − 1)
2
+ n = n
2
.
L
M
N
4
j
P
O
N M
[
#
#
$,
,
#
-
+
,
u(r)
#
u
n
l
-
&
#
3
V
c
W
V
T
X
T
d
c
SS
c
T
4
,
$
(
!
-
,
"
-
#
#
+
"
-
J
ψ
n=1
l=0,m=0
=
1
√
4π
Z
a
0
3/2
2 exp
−
Zr
a
0
.
L
M
N
4
P
R
#
-
-
#
/
,
"
-
#
!
-
(
1
,
$
(
!
1
#
ψ
n=2
l=0,m=0
=
1
2a
0
3/2
2 −
r
a
0
exp
−
r
2a
0
Y
0
0
(θ, ϕ),
ψ
n=2
l=1,m=0,±1
=
1
2a
0
3/2
1
√
3
r
a
0
exp
−
r
2a
0
Y
0,±1
1
(θ, ϕ).
L
M
N
4
P
'
&
1
,
$
(
!
4
5
6
7
5
A
C
<
B
9
=
A
>
<
A
[
/
(
#
"
2
#
-
2
+
"
$
(
+
/
,
"
,
#
,
4
"
"
$,
!
+
"
-
$
#
+
%
#
+
-
z,
)
/
,
!
&
(
"
!
,
!
"
-
J
ˆ
H
0
= eEz = eEr cos θ = eErn
z
,
L
M
N
4O
P
+
e > 0
L
2
,
$
$
,
#
-
−e
P
4
"
-
#
-
$,
-
!
,$
#
(
)
%
#
(
#
(
)
#
$
#
&
K
(
+
~n
#
0
H
+
n
z
$
!
&
0
(
# &
!
-
1
$
0
H
cos θ
!
-
"
-
&
1
,
$
(
!
&
θ
(
$,
0, π
4
#
"
#
-
3
(
)
,
$,
K
/
,
_
1
$
-
2
+
"
$
(
+
$
-
,
"
-
#
#
+
4
a
+
H
#
!
-
"
#
,
+
3
(
)
,
$,
/
,
_
1
$
-
#
3
TU
S
T
c
d
!
#
-
$
'
$
4
b
-
%
#
#
/
,
(
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(
+
#
(
)
,
$
/
,
K
_
1
$
-
(
"
"
%
#
+
(
(
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"
!
#
-
3
!
,H
#
"
#
-
3
(
)
,
$,
/
,
_
L
#
4
W
V
X
V
c
!
#
-
$
'
$
P
4
(
)
$
"
$
2
,
"
"
#
-
#
/
,
-
,
#
,
n =
2
0
+
#
-
3
"
,
!
"
(
#
%
+
(
!
#
+ 3
,
l
m
4
[
(
$
!
/
,
-
"
#
!
&
(
+
E
(0)
2
= −
me
2
8¯h
2
L
M
N
4OO
P
!
$
ψ
1
= ψ
n=2
l=0,m=0
,
ψ
2
= ψ
n=2
l=1,m=−1
,
ψ
3
= ψ
n=2
l=1,m=0
ψ
4
= ψ
n=2
l=1,m=1
.
R
#
H
#
$
#
&
(
n
z
#
(
#
$
3
-
%H
(
)
l
0
ˆ
H
0
H
J
#
$
3
ψ
1
ψ
2,3,4
4
b
$
#
0
O
NN
"
#
H
n
z
!
-
-
$
2
#
&
,$
#
+
"
-
#
+
m = 0
0
!
#
ˆ
H
0
H
0
13
= (ψ
1
| ˆ
H
0
|ψ
3
)
H
0
31
= H
0∗
13
4
,
#
"
2
^
#
,H
(
^J
#
+
(
$
4
R
"
!
0
H
"
#
-
&
"
"
# $
3
+
#
+
#
(
)
,
$,
/
,
K
_
,
!
#
(
#
(
$
0
$
%
(
(
)
+
#
(
)
"
K
%
#
"
!
,
!
"
-
J
\
−E
(1)
2
0
H
0
13
0
0
−E
(1)
2
0
0
H
0∗
13
0
−E
(1)
2
0
0
0
0
−E
(1)
2
= 0.
L
M
N
4O
M
P
b
$
#
0
"
#
H
m = 0
1
,
$
(
!
$,
-
-
&
(
#
-
0
(
)
H
0
13
= H
0∗
13
≡ h
%
#
L
M
N
4O
M
P
"
!
,
!
"
-
J
E
(1)2
2
E
(1)2
2
− h
2
= 0
L
M
N
4O
N
P
(
!
(
# &
E
(1)
2
= 0, 0, −h, h.
L
M
N
4O
Z
P
a
!
I
#
$
#
2
-
#
^
(
#
2
-
!
E
(1)
2
= 0
\
0 0 h 0
0 0 0 0
h 0 0 0
0 0 0 0
a
b
c
d
= h
c
0
a
0
= 0.
L
M
N
4O
]
P
# &
+
%
#
!
-
c = a = 0
4
#
"
-
!
&
b
d
0
(
(
0
H
#
$
#
2
-
#
^
(
#
2
-
!
0
-
&
$
/
(
!
#
1
,
$
(
!
ψ
2
ψ
4
4
-
(
"
,
,
+
!
&
-
l = 1
m = ±1
4
$
#
2
-
E
(1)
2
= −h
!
,
!
# &
,
!
&
(
%
#
h 0 h 0
0 h 0 0
h 0 h 0
0 0 0 h
a
b
c
d
= h
a + c
b
a + c
d
= 0.
L
M
N
4O
h
P
# &
+
%
#
!
-
c = −a
0
b = d = 0
4
-
+
E
(0)
2
− h
!
-
$
/
(
!
&
#
&
a (ψ
1
− ψ
3
) = a ψ
n=2
l=0,m=0
− ψ
n=2
l=1,m=0
,
L
M
N
4O
j
P
+
a
#
(
#
,
$,
,
#
a = 1/
√
2
4
,
$
(
!
3
1
#
&
-
,
+
E
(0)
2
+ h
H
#
(
J
%
#
+
(
+
L
M
N
4O
h
P
,
/
#
,
$,
+
K
^
(
-
,
L
M
N
4O
j
P
\
1
√
2
(ψ
1
+ ψ
3
) =
1
√
2
ψ
n=2
l=0,m=0
+ ψ
n=2
l=1,m=0
.
L
M
N
4O
P
O
N
Z
[
0
$
%
(
)
# ,
-
%
#
L
M
N
4O
j
0
M
N
4O
P
H
-
&
+
3
(
,
!
$
h
4
b
(
)
(
-
#
#
H
"
#
^
$
#
"
(
,
#
2
-
^
(
-
#
K
+
(
$
4
R
-
!
!
-
(
#
^J
(
/
#
h
4
(
,
#
"
-
!
#
&
"
-
J
1
,
$
(
!
1
#
(
)
\
ψ
1
= ψ
n=2
l=0,m=0
=
1
2a
0
3/2
2 −
r
a
0
exp
−
r
2a
0
Y
0
0
(θ, ϕ),
ψ
3
= ψ
n=2
l=1,m=0
=
1
2a
0
3/2
1
√
3
r
a
0
exp
−
r
2a
0
Y
0
1
(θ, ϕ).
L
M
N
4O
P
'
,$
(
#
#
H
-
3
#
h = eE
1
2a
0
3
1
√
3
∞
Z
0
drr
3
r
a
0
2 −
r
a
0
exp
−
r
a
0
h1, 0| n
z
|0, 0i .
L
M
N
4
M
P
.
3^J
$
&
#
&
#
+
(
$
h1, 0| n
z
|0, 0i =
r 1
3
.
.
2
$
3
&
#
(
"
-
#
!
&
(
ρ = r/a
0
\
h = eE
1
2a
0
3
1
3
a
4
0
∞
Z
0
dρ ρ
4
(2 − ρ) exp (−ρ)
= eE
a
0
24
(2 − 5) 4! = −3eEa
0
.
L
M
N
4
M
O
P
/
"
-
#
J
!
-
$
0
#
I
"
E
#
n×
(
0
"
3
!
&
(
0
H
O
O
N
√
(
+
/
-
\
h = −3 × 4, 8 × 10
−
10
(
√
(
+
-
n
300
√
(
+
-
(
5, 29 × 10
−
9
(
= n × 0, 25 × 10
−
19
+
= n × 1, 56 × 10
−
8
= 12, 48 × 10
−
6
1
(
,
L
M
N
4
MM
P
+
O
(
!
-
"
#
&
!
-
$
&
#
&
+
(1
= 8 × 10
3
O
(
)
4
a
"
%
3
,
10
5
(
1
$
'
$
!
-
(
"
3
,
10
−
4
0
#
3
(
(
3
#
$
^
(
!
H
+
2
^J
3
-
"
-
#
#
#
"
n =
2
4
5
6
7
6
?
=
<
C
:
B?
;
;
<
= > B @
A
G
F
;
=
F
@
b
(
)
(
-
%
#
^
^J
-
(
+
%
2
#
2
#
(
&
-
$
2
#
!
#
3
K
"
$
(
4
/
"
-
J
$H
,
(
)
#
"
,
+
(
0
,
-
-
$
#
K
#
J
"
#
)
$
-
(
H =
1
2m
~p −
q
c
~
A(~r, t)
2
+ qV (~r, t),
L
M
N
4
M
N
P
O
N ]
+
q
!
-
2
,
$
(
&
-
$
0
^
c
"
3
$
^
(
&
^
#
2
4
R
"
!
0
H
"
$
K
(
+
(
#
H
!
&
-
3
"
"
(
!
2
#
-
3
"
,
!
&
(
-
"
-
%
/
\
~
E = −
1
c
∂ ~
A
∂t
− ~
∇V,
~
B = ~
∇ × ~
A.
L
M
N
4
M
Z
P
[
(
3^J
$
(
"
-
#
"
/
%
#
0
(
3^J
#
!
&
(
"
(
!
2
#
$
K
#
~
A
!
-
3
"
-
-
$
#
#
J
0
+
H
"
(
!
2
~
A
!
-
1
,
$
(
!
&
~r
0
~r
$
,
,
!
"
"
3
,
4
a
"
(
&
"
(
)
,
"
-
,
0
/
$
#
#
"
K
/
2
)
#
-
$
4
%
#
(
-
~p −
q
c
~
A(~r, t)
2
→ ˆp
2
+
q
2
c
2
ˆ
A
2
−
q
c
ˆ
p · ˆ
A + ˆ
A · ˆp
= −¯h
2
~
∇
2
+
q
2
c
2
~
A
2
+ i
q¯h
c
~
∇ · ~
A
+ 2 ~
A · ~
∇
.
L
M
N
4
M]
P
'
&
ˆ
H = −
¯h
2
2m
~
∇
2
+ i
q¯h
2mc
~
∇ · ~
A
+ i
q¯h
mc
~
A · ~
∇ +
q
2
2mc
2
~
A
2
+ qV.
L
M
N
4
M
h
P
R
,
#
,
2
,
-
2
+
"
+
(
+
"
#
,
!
K
+
-
%
#
# &
(
)
4
'
"
%
/
,
!
/
(
J
3
3
#
"
#
-
3
(
)
,
$,
/
,
_
4
(
,
!
$
/
,
"
!
$
#
&
#
ˆ
A
(
3^J
L
M
N
4
M
h
P
\
ˆ
H
0
= −i
e¯h
2mc
h ~
∇ · ~
A
+ 2 ~
A · ~
∇
i
,
L
M
N
4
M
j
P
+
2
,
$
q
"
!
3
^
2
,
$
$
,
q = −e, e > 0.
'
2
"
+
(
~
B
"
-
#
!
-
"
(
!
2
~
A =
1
2
~
B × ~r.
L
M
N
4
M
P
(
#
^
(
B
i
=
~
∇ × ~
A
i
=
1
2
ijk
∂
j
klm
B
l
r
m
=
1
2
B
l
(δ
il
δ
jm
− δ
im
δ
jl
) δ
jm
= B
i
.
R
#
H
~
∇ · ~
A =
1
2
klm
B
l
∂
k
r
m
= 0,
#
#
L
M
N
4
M
j
P
"
-
!
$
,
+
(
2
\
2 ~
A · ~
∇ =
i
¯h
~
B × ~r
· ˆp =
i
¯h
~
B · (~r × ˆp) =
i
¯h
~
B · ˆ
L.
O
Nh
/
,
"
!
,
!
"
-
J
ˆ
H
0
=
e¯h
2mc
1
¯h
~
B · ˆ
L.
L
M
N
4
M
P
$
^J
µ
B
=
e¯h
2mc
= 9, 27 × 10
−
21
+
-
= 9.2740154 × 10
−
24
= 5.78838263 × 10
−
5
L
M
N
4
N
P
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T
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# $
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v/ 2πr
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%
#
0
I =
ev
2πr
.
L
M
N
4
N
O
P
r
v
L
r
I
−
e
→
→
→
µ = −−
evr
−
2
c
→
n
z
-
,
$
O
\
+
(
$
,
"
,
-
!
&
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+
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R
3
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S
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0
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(
)
K
+
2
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#
+
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+
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#
~µ
~µ =
I
c
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S,
L
M
N
4
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P
+
~
S
!
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-
$
#
"
#
(
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K
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#
(
S
c
d
T
S
S
T
S
g
X
c
f
4
"
#
"
-
"
"
$,
S = πr
2
µ =
evr
2c
=
e
2mc
mvr =
e
2mc
L,
L
M
N
4
NN
P
O
N
j
+
L
!
-
#
^
(
&
,
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3
,
$
,
4
R
#
H
$
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,
-
-
3
#
$
,
$,
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(
#
H
"
&
0
# &
$
L
M
N
4
NN
P
H
"
-
J
#
$
#
~µ = −
e
2mc
~
L.
L
M
N
4
N
Z
P
a
# %
J
,
#
+3
0
H
−e
!
-
2
,
$
$
,
4
a
# %
L
M
N
4
M
P
H
"
"
-
J
+
%
ˆ
H
0
= − ~
B · ~µ,
L
M
N
4
N ]
P
+
~µ
!
-
c
S
T
S
c
,
+
(
+
(
&
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$
0
ˆ
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0
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(
!
3
c
S
T
S
T
+
2
#
,
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#
+
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+/
(
~
B
4
b
"
~
B
-
$
#
+
#
2
,H
-
z
(
#
3
-
ψ
nlm
!
-
/
"
-
/
(
hn, l, m| ˆ
H
0
|n, l, mi = mµ
B
B ,
L
M
N
4
Nh
P
+
m µ
B
!
-
+
(
,
#
-
|n, l, mi .
0
H
"
+
(
,
#
,
,
+
!
&
-
(
"
,
#
-
"
,
"
"
(
!
B
m
4
a
!
#
-
$
a
$
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$,
!
-
3
#
"
"
$,
,
#
+ 3
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,
$
,
4
O
N
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