( x) ( px), , , .
:
1. t , ℇ .
2. t , ℇ - . .
.
1.4 ..
1.4.1. -
.. -.
ψ(r,t) , .
ψ(r,t) :
dw = ψ ψ* d 3 r,
d 3 r = dx·dy·dz.
:
= d 3 r = 1 (1.4)
(1.4) , .
1.4.2. .
( ψ ).
ψ1 ψ2 - , () ,
Ψ = 1 ψ1 + 2 ψ2,
. │ 1 │2 │ 12 │2
.
Ψ =
, ( , ).
,
Ψ1,2(, ) = ψ1( ψ22(,
│Ψ1,2(, ) │2= │ψ1( 1)│2 ·│ψ2( 2)│2
.
1.4.3. .
. , , , ψ , , . , .. │ ψ │2.
. f - . , , . f .
ψ = f ψ.
:
f = ) d 3 r
|
|
ψn . f fn ,
ψn = fn ψn.
. .
f = ψn d 3 r = fn
( ) . .
1.4.4. .
1.
() φ()
: , -, .
2.
px x = - jħ , ψp (x) = A ) .
x ψp (x) = px ψp (x)
:
= - jħ = - jħ ( x + y + z).
3.
: ℇ = = + +
.. 2 ψ = - ħ2 ψ = ψ
: = - [ + + ] = - =
+ (r) = = - + (r)
ψ = ℇ ψ
. 2
2.1
2.1.1.
, ψ to, : ψ(t)
-:
j ħ =- + ψ(x,t) (2.1, )
j ħ =- + ψ( ,t) (2.1)
. (, t) = 0, .
ψ( ,t) .
2.1.2.
. - (), ℇ = const
ψ(r,t) = ψ()· (*)
( ℇ )
(2.1) , :
- +(ℇ - ) ψ( ,t) = 0 (2.2)
, .
:
1). (*) ( = ω),
2). (│ψ│2 = const)
3). (2.2) . .
2.2
2.2.1. , -
(!). , .
:
1. ψ() .
2. ψ() - . │ψ│2
3. ψ() . -.
.
(2.2) (): , ( ). .
():
|
|
: ψI(l)│ l-0 = ψII(l)│ l+0
()I│ l-0 =()II│ l+0
, ψ, r .
, │ψ│2.
d 3 r = 1
ψ.
2.3
:
ψp (x) = A )
ℇ ( ), │ψ│2 = const
.. . .
() =0 :
+ ℇ ψ = 0 (2.3)
+ k2 ψ =0 (2.3, a)
ψ = e jkx + e jkx (2.4)
k2 = ℇ Þ ℇ = =
,
ψ = e jpx / ℏ + e - jpx / ℏ (2.4, a)
, = 0, . ℇ . ℇ min. ℇ 0 .
2.4
(x) ( .), , . . F = - . ℇ x=xo ( ).
F .
2.4.1
() =0 I () = II.
ℇ
. . , :
I (x ), , :
ΨI = e jkx + e jkx
. - . .
II (x
+ (ℇ - ) ψ =0 (2.5)
:
δ2 ψ = 0, δ =
:
ΨII = e - δ x + De δ x (2.6)
D = 0.
ψ :
A = , B = 1, C =
,
1) ψ ℇ;
2) , ΨI .
ΨI = G cos (kx + φ), G φ .
, x .
.
R = = │ │2,
α1 = │ΨI,│2 α2 = │ΨII,│2
ΨII → 0 II.
ℇ
ΨII = e j ℜ x ℜ =
R = ()2 1
.
2.4.2.. .
(. .)
I (=0)
: ΨI = e jkx + Be jkx,
, k = .
II ( )
ΨII De δ x , δ = (2.7)
.
III (=0)
ΨIII De δ a e jkx
. :
= = e 2δ a = exp (-2 a ), (2.8)
|
|
. . 1. .
2.5
( )
. . I III → , II . . =0.
. . . II .
0
ψ () = e jkx + e jkx
x =0 : + = 0 Þ = ,
ψ () = sin kx. k = .
x =a sin k= 0, ak =n π, n= 1, 2, 3
k
Ψn () = sin x (2.9)
ℇ n = = . , n .
=
Ψn () .
.
. 3 .
.
3.1
.
(x) = = m ,
ω o = .
. :
+ (ℇ - m ) ψ =0
ℇ > () . ψ(,t) , .
xn = ψn .
, :
ℇ n = (n+ ) ħ ω, n = 0, 1, 2,
n = 0 . Ÿ . ( - ).
3.2
3.2.1. ..
= [ ]
, , .
.
. , .
- .
Lx ( ). - Lz │ │. . :
Lz = ρ p φ; Lx = y p z = ρ sin φ· p z; Ly = - xp z = ρ cos φ· p z
L = ρ
3.2.2.
() . ψ(r,t) .
z ψ = Lz ψ (3.1)
2 ψ = L2 ψ (3.2)
ψ
= const.
z = ρ φ = - jℏ ρ = - jℏ
ψ(r, θ) = (r, θ) e(Lz·φ/ ℏ ) = A exp(jmL φ)
, p φ = ℏ k φ
|
|
-
ψ (φ + 2 π) = ψ (φ),
-
Lz = m ℓ ℏ, m ℓ mL = 0, 1, 2, 3 -
│ m ℓ │ l
: - ℓ, - ℓ +1, - ℓ +2 -1, 0, +1, . ℓ -1, ℓ
2 ℓ +1 .
3.3 .. (3.3.1.)
3.3.1. ..
(3.2) . .
L2 Lz , ℓ. .. Lz 2max. = ℏ 2 ℓ 2, , , ℓ
L2 = ℏ 2 ℓ(ℓ +1) (3.3),
ℓ = 0, 1, 2, . .
ℓ . - , Lz.
: L2 = L2 L x2 = L y2 = L z2
3.3.2 .
, () .. i. -
L =ℏ L z =ℏM z
L ℓ 1 , ℓ 2 , ℓ 3 , L L max L min.
Lmax = . Lmax = ℓ 1+ ℓ 2 Lmin = │ ℓ 1- ℓ 2 │.
L :
ℓ 1+ ℓ 2, ℓ 1+ ℓ 2-1, ℓ 1+ ℓ 2- 2, ., │ ℓ 1- ℓ 2 │
2 ℓ m +1 , lm ℓ 1 ℓ 2 .
3.4 (3.4.1). (3.4.2)
3.4.1.
.
m = g = , g .. .
:
mz = - L z = - m ℓ ℏ = μ m ℓ
μ ,
μ = |
m ℓ .
3.4.2
.
. .
:
│ │ = ℏ , Sz = ℏ m s
s .
- s s
s = : , ,
s = 1, s = 0.
. z .
C :
μz = - .
, :
(μz / Sz): (pm / L z) =2
. (- ), .
ψ( ,s,t).
, .
: , ( ).
. (, ) .
ψ(ξ1, ξ 2) = + ψ (ξ2, ξ 1) -
ψ (ξ1, ξ 2) = - ψ(ξ2, ξ 1) -
.
.4
.
4.1 .
.. = + .
1. , .
2. , .. . .
3. ()
() = - k , k =
4. .
4.2
4.2.1 -
ψ = ℇ ψ
|
|
- ψ - k ψ = ℇ ψ
ψ() = ψ(r, θ, φ)
. . ,
ℇ = ℇ + ℇ = +
:
[ℇ + + (r)] ψ = ℇ ψ
.. L . (3.2)
2 ψ = ℏ 2 ℓ(ℓ +1) ψ
, , .
Ψ = ℜ (r) Y(θ, φ)
- ( ) + [ - k ]ℜ = 0 (4.1)
( ).
ℇ 0,
ℇn =- = - ,
ℇn = - , (4.2)
- ( ), n , nr = 0,1, 2, 3 ( ).
n nr ℓ , . m ℓ .
n ℓ, , . n
ℓ = 0, 1, 2, , n 1
m ℓ
m ℓ = - ℓ, - ℓ+1, - ℓ +2, -1, 0, +1, +2,, ℓ-1, ℓ,
n 2. , , : 2 n 2 n, 1 .
4.2.2
Z =1 , ,
= 13,6 (. .).
, , ℓ:
ℓ = 0, s -
ℓ = 1, p-
ℓ = 2, d -
ℓ = 3, f -
ℓ = 3, g
........
m ℓ m s.
4.3
4.3.1
(4.1) n =1 ℓ = 0 :
- [ + ] (ℇ1 + k ) = 0, (4.3)
ℜ10
, r →0 .
= A1 exp[- ( r/ ao)]
. (4.3) , :
a o = - .. ,
ℇ1= - = - - .
→0 r →0 .
. .
4.3.2 ()
n =2, ℓ =0
= A2 (1- r/2 ao) exp[- ( r/2 ao)]
.
ℓ 0 .
, n = 2 ℓ = 1, m ℓ = 0 e :
= A2,1 ·(r/ ao) exp[- ( r/2 ao)]· cos θ
n = 2 ℓ = 1, m ℓ = :
= A2,1,1 ·(r/ ao) exp[- ( r/2 ao)]· sin θ ·
, .
dw = │ψ│2 4 r2 ·dr
. . .
4.4
4.4.1
.
, :
) , ;
) - ; .
. Zeff . Z
(Z =1 )
1s, 13,6 , . ( ).
(Z =2 )
, .. ,
ℇ = - =13,6 · = 54 ,
.
Z* Zeff = Z - σ,
σ = , n = 2 .
.
(Z =3 )
σ = 1,25. , . 2s. n ℓ . p - . , 2(2ℓ +1)= 6 . n =2
2 +6 = 8.
. , .
4.3.2
, (), , . .
,
ℓ = 1
L = 1 S =0. J : J = 1, J = 0 . , .
ψ(r,t) f fn x ()I│ l-0 =()II│ l+0 │ψ│2
d 3 r = 1 L L L m ℏ 2 ℓ(ℓ +1)
υ2 = υ = φ = (2 m +1) π φ = 2π m ħω ħω hν ħ ħ
ψn ψ*n
ψ(x,t) j ħ -
ψ → ω → = υ2
δ = ωo2 = cos Σ
Þ ^ > < α φ q φo β ω ω 2 π ψ (x,t) x k
ℓ δ λ φ ε θ α π υ ν ω τ μ ψ ρ ∙ ΄ z γ σ ℓ
→ ∶ π tg φ ψ → ω → t μ μ εoε ∠
δ = ωo2 = cos(ωt + φo) sin (ωt + φo) sin2
- δt ω = εm Cambria Math ( .)
ħ const