Мартыненко Т.П. и др. Практический курс физики. Квантовая физика. Элементы физики твёрдого тела и ядерной физики
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61
) ,
r
1
;
) r > 2r
1
.
2.70.
:
(
)
(
)
1
r/rexpAr
−
⋅
=
Ψ
, –
,
2
2
0
1
me
4
=r – . :
)
r ;
) r ;
)
( )
2
rr −=σ .
2.71.
:
( )
−⋅
π
=Ψ
1
3
1
r
r
exp
r
1
r
, r
1
– .
:
)
2
r ;
) 1/r
;
) 1/r
2
,
;
) r>r
1
;
) r<10
-15
(
).
2.72. ,
(
)
(
)
(
)
arexpr1Ar
−
⋅
β
+
=
Ψ
.
.
?
2.73.
:
( )
−⋅−⋅=Ψ
11
r2
r
exp
r2
r
1Ar , r
1
-
. :
.
2.74. m
L (0<x<L)
.
62
( )
π
⋅=Ψ
L
x
sin
L3
8
x
2
.
<T>.
2.75. m
( )
2
kx
xU
2
= .
:
(
)
(
)
2
xexpAx β−=Ψ
, –
.
.
2.76. m
( )
2
kx
xU
2
= :
(
)
(
)
2
xexpAx β−=Ψ , A – ;
2
m
ω
=β ,
m
k
=ω . :
) x ;
) x ;
)
(
)
2
2
xx −=σ .
2.77. m
( )
2
kx
xU
2
=
(
)
(
)
2
xexpAx β−=Ψ , A – ;
2
m
ω
=β ,
m
k
=ω . U
T .
2.78. m
( )
2
kx
xU
2
=
(
)
(
)
22
xaexpxBx −⋅⋅=Ψ
( – ).
.
2.79. E = 50 ,
x,
U
0
=20 . :
) ;
) .
63
2.80. ,
n
R.
2.81. n
R = 0,5.
2.82.
R
U
0
<< E.
2.83.
R = 2,5·10
-5
.
,
.
2.84. ,
,
D R.
2.85. ,
D
n .
2.86. n
, D = 0,8.
2.87.
U
0
= 99,75
= 100 .
2.88. m E
X
U
0
> . :
) ;
)
( > 0), - ,
x < 0 .
2.89. m
( . .).
.
. : ) E<<U
0
, )
E>>U
0
.
2.90. m,
,
x
U(x)
ll U
0
E l
0
x
U
E
-
U
0
64
U
0
> .
> 0, . .
,
.
, 1EU
0
=
−
.
2.91. = 5
,
U
0
= 10 l = 0,1 .
:
) D.
) ,
U
0
, .
:
( )
−⋅
⋅
− EUm2
2
expD
0
l
.
2.92.
( )
[ ]
−⋅−
2
1
x
x
dxExUm2
2
expD
,
1
2
– , U > ,
m E
, ,
) );
) );
(
)
(
)
22
0
L/x1UxU −= .
U, E
0 L x
)
E
U
0
E
U
0
-L 0 L x
)
U, E
65
3. .
.
,
+Ze. (Z = 1)
Z,
, .
, .
:
r4
Ze
U
0
2
πε
−= (3.1)
r – . (3.1)
(2.13) :
0
r4
Ze
E
m2
0
2
2
=Ψ
πε
++∆Ψ . (3.2)
, , -
, (3.2)
(r, , ) ( . 3.1).
:
ϕθ
∆⋅+
∂
∂
∂
∂
⋅=∆
,
2
2
2
r
1
r
r
r
r
1
. (3.3)
ϕθ
∆
,
r, (3.2)
:
(
)
(
)
(
)
ϕ
θ
⋅
=
ϕ
θ
Ψ
,YrR,,r . (3.4)
R(r) ,
Y( , ) – . (3.4) (3.2)
R(r) Y( , ).
ϕθ
∆
,
2
L
ˆ
22
,
/L
ˆ
−=∆
ϕθ
. (3.5)
Y
L
Y
L
ˆ
22
⋅
=
(
)
1L
22
+⋅⋅= ll , l = 0, 1, 2, … (3.6)
Z
Y
X ϕ . 3.1
θ
r
66
2
L
ˆ
–
Y
l,m
( , ), l m,
m m = 0, ±1, ±2, … ±l.
(3.4) (3.2), (3.5) (3.6)
R(r) :
0R
r
)1(
r4
Ze
E
m2
dr
dR
r
dr
d
r
1
2
0
2
2
2
2
=⋅
+
−
πε
++⋅
ll
(3.7)
,
, > 0,
( ). < 0
, :
22
0
2
42
n
nh8
meZ
E
ε
−= , (3.8)
n = 1, 2, … n l + 1. (3.9)
(3.7)
n, l R
n,l
(r). ,
(
)
(
)
(
)
ϕθ⋅=ϕθΨ ,YrR,,r
m,,nm,,n lll
n, l, m.
n (n = 1, 2, 3…).
(3.8) n . l
;
(3.6) (3.9) l = 0, 1, 2, …(n - 1)
)1(L +⋅= ll (3.10)
l:
l
0
1
2
3
…
s
p
d
f
…
m (m = 0, ±1,
±2, ±3, …±l )
( Z).
mL
z
⋅
=
(3.11)
.
, ,
,
, .
)1s(sS +⋅⋅= , (3.12)
s = 1/2.
67
Z:
zz
mS
⋅
=
, (3.13)
m
s
= +1/2, -1/2.
J
L
S
.
(
)
1jjJ +⋅⋅= , (3.14)
j
s,sj −+= ll (3.15)
J
z
:
jz
mJ ⋅= ,
m
j
= j, (j - 1), (j - 2)…- j.
: n ( ), l
( ), m ( ) m
s
( ).
, (
L
,
S
J
)
( , Z).
. ,
.
–
.
.
L
S
( . 3.2).
.
(3.8).
.3.3
.
,
l, m,
L
S
S
J
L
j=
l
-s
j=l+s
J
. 3.2
1
2
∞=n
n=3
n=2
n=1
0
E
3
E
2
E
1
. 3.3
68
m
s
. .
. (n = 1) –
.
,
.
“ ”.
, –
.
.3.3 1 ,
. n = 1 n =
( 2 .3.3).
“ ” .
, .
, , .
.
:
EEh
−
=
ν
. (3.16)
(3.8) (3.16), ,
−
ε
=ν
222
0
3
42
n
1
n
1
h8
meZ
(3.17)
n – , n –
.
:
−
ε
=
λ
222
0
3
42
n
1
n
1
ch8
meZ1
. (3.18)
17
2
0
3
4
10097,1
ch8
me
R
−
⋅=
ε
= .
:
2
2
n
n
RhcZ
E −= (3.19)
, ,
−=
λ
22
2
n
1
n
1
RZ
1
. (3.20)
( ). .3.3
.
( n = 1 )
69
. ( n = 2)
, –
. (
, . .) .
,
, . ,
( l = ±1).
.
,
.
, :
n, l, m m
s
.
n, l,
n l. ,
n, n ,
l.
.
,
. ,
n, l, m, m
s
n l 2(2l + 1)
; n – 2n
2
.
n .
:
n 1 2 3 4 …
K L M N …
K – S - (l = 0),
L – s - (l = 0) p - (l = 1)
. . s -
; - 6;
K – ; L –
8 . .
1
.
.
70
1
-
L-
-
S S p S p d
H
He
Li
Be
B
1
2
2
2
2
1
2
2 1
1S
1S
2
1S
2
2S
1S
2
2S
2
1S
2
2S
2
2p
.
,
L - S .
L
M , –
s
M .
L
M
s
M
J
M .
:
(
)
1LLMM
LL
+== ,
(
)
1SSMM
SS
+== ,
(
)
1JJMM
JJ
+== ,
L – , S –
, J – . (
, ,
).
, ,
L = 0, S = 0 , , J = 0.
.
,
. .
. , ,
.
, L S
J.
.
– , ,
,
. ,