Atom wodoru w zewntrznych polach

(1)

!" # $% $ " " # & %# '( ) * + ,( ) , $ , -

µ

# " ( ! . , / 0 " %# 1 $ # & 2# , # 3 " + ( $ ( 4 5678 9 :; < = > ? @ A=B @? CBDB> B : BDB E F ;G B ' " $ , ,# " / + H IJ # &, ! &( %# '( ) * K +

−

_r

1 ₂

_dr

d

r

2 d

dr

+

2µ

¯h

2 (V (r) − E) +

l(l + 1)

r

2 u(r) = 0

L MN 4O P " ( ! 2

V (r) = −

Ze

2 r

L MN 4 MP - ! H " ,( ! &( +/ ( 0 $ % & H IJ # " 3 ( $, '( ) Q 4 R " ! 0H " 2 1 , $( ! 1 # " - J

ψ

n

_lm

(r, θ, ϕ) = u

n

_l

(r)Y

_l

m

(θ, ϕ),

L MN 4 NP + 1 , $( !

u

H ! #

l

( / $# # !

n

# ! S TU V T WXY 4

n = ν + 1 + l,

+

ν = 0, 1, . . . l = 0, 1, . . .

L MN 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 MN 4 ] P + - 2

a

0 = 5, 29 × 10

−

9

( ! - # & ^ " / ,#, L " _ ` ) P 4 a,#H0 H

n

- 3

1, 2

∞

4 b $H +

n

H # -& ! $ %H # ^ (

ν

l

0 ( ! U cdc X c S T e fg # , # " / + 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 MN 4 hP i # " $ J - 3 / " ^ ( ) , $ 0H (2$# + ( ! # -

n

2 _:

n−1

X

l=0

(2l + 1) = 2

n(n − 1)

2 + n = n

2 _.

L MN 4 j P N M

(2)

[ # # $, ,# - + ,

u(r)

#

u

n

l

- & # 3 V c W V T X Tdc SS cT4 , $( ! - , " - ## + " - J

ψ

n=1

_l=0,m=0

=

√

1 4π

Z

a

0 3/2

2 exp

−

Zr

_a

0 .

L MN 4 P R #- - # / , " -# ! - ( 1 , $( ! 1 #

ψ

n=2

_l=0,m=0

=

1 2a

0 3/2

2 −

_a

r

0 exp

−

_2a

r

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

−

_2a

r

0 Y

₁

0,±1

(θ, ϕ).

L MN 4 P '& 1 , $( ! 4 5675 AC <B 9=A><A [/ ( # " 2# - 2 + " $ ( + / , " ,#,4 " " $, ! + " -$ # + %# + -

z,

) / , ! &( " ! , !" - J

ˆ

H

0 = eEz = eEr cos θ = eErn

z

,

L MN 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 cd ! # -$ ' $4 b - %# # / , ( ) ( + # ( ) , $ / , K _ 1 $ -( " " %# + ( ( )" ! # - 3! ,H # " #- 3 ( ) , $, / , _ L #4 WVXV c ! # -$ ' $ P 4 ( ) $ " $2, " " #- # / , - ,#,

n =

2

0 + #-3" , ! " (#% + ( ! # +3 ,

l

m

4 [ ( $ ! / , - " # ! &( +

E

₂

(0)

_{= −}

me

2 8¯h

2

L MN 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 NN

(3)

" #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

2 (1)

0 H

0

13

0

0 _−E

2 (1)

0

0 H

0∗

13

0 −E

(1)

2

0

0 _−E

₂

(1)

= 0.

L MN 4O M P b $#0 " #H

m = 0

1 , $( ! $, - -& (# - 0 ( )

H

0

13 = H

0∗

13 ≡ h

%# L MN 4O MP " ! , ! " - J

E

₂

(1)2

E

₂

(1)2

_{− h}

2 = 0

L MN 4O NP ( ! ( # &

E

₂

(1)

_{= 0, 0, −h, h.}

L MN 4O Z P a ! I # $ # ^ ( !

E

₂

(1)

= 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 MN 4O ] P # & + %# ! -

c = a = 0

4 # " - ! &

b

d

0 ( ( 0 H # $ # ^ ( !

0

-& $ / ( ! # 1 , $( !

ψ

₂

ψ

₄

4 -( " , , + ! & -

l = 1

m = ±1

4 $ #2-

E

₂

(1)

_{= −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 MN 4O hP # & + %# ! -

c = −a

0

b = d = 0

4 - +

E

2 (0)

− h

! - $ / ( ! & #&

a (ψ

₁

_{− ψ}

₃

) = a ψ

n=2

_l=0,m=0

_{− ψ}

n=2

_l=1,m=0

,

L MN 4O jP +

a

# ( #, $, , #

a = 1/

√

2

4 , $( !3 1 #& - , +

E

₂

(0)

+ h

H # ( J %# + ( + L MN 4O h P , / #, $, + K ^ ( - , L MN 4O jP\

1 √

2 (ψ

1 + ψ

3 ) =

1 √

2 ψ

n=2

l=0,m=0

+ ψ

n=2

l=1,m=0

.

L MN 4O P NZ

(4)

[ 0$ % ( ) # , - %# L MN 4O j 0 MN 4O 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 −

_a

r

0 exp

−

_2a

r

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

−

_2a

r

0 Y

₁

0 (θ, ϕ).

L MN 4O P ',$ ( ## H - 3 #

h = eE

1 2a

0

3

1 √

3 ∞

Z

0 drr

3 r

a

0 2 −

_a

r

0 exp

−

_a

r

0 h1, 0| n

z

|0, 0i .

L MN 4 MP . 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 MN 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 MN 4 MMP + 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 5676 ? =< 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 MN 4 MN P N ]

(5)

+

q

! - 2, $ ( &- $0 ^

c

" 3 $ ^ ( & ^ # 24 R " ! 0 H " $ K ( + ( # H ! &- 3 " " ( ! 2 # -3" , ! &( -" -% /\

~

E = −

1 _c

∂ ~

_∂t

A

− ~

∇V,

B = ~

~

_{∇ × ~}

A.

L MN 4 MZP [ ( 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 A

ˆ

2 −

q

_c

_{p · ˆ}

ˆ

A + ˆ

_{A · ˆp}

= −¯h

2 ∇

~

2 +

q

2 c

2 A

~

2 _{+ i}

q¯h

c

~

∇ · ~

A

+ 2 ~

_{A · ~}

_∇

.

L MN 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 MN 4 MhP R , #, 2 , - 2 + " + ( + " #, ! K + - %# # & ( ) 4 ' " % / , ! / ( J 3 3 # " #- 3 ( ) , $, / , _ 4 ( , ! $ / , " ! $ #& #

ˆ

A

( 3^J L MN 4 Mh P\

ˆ

H

0 _{= −i}

e¯h

2mc

h ~

∇ · ~

A

+ 2 ~

_{A · ~}

_∇

i

,

L MN 4 Mj P + 2, $

q

" ! 3 ^ 2, $ $ ,

q = −e, e > 0.

' 2 " + (

~

B

" -# ! - " ( ! 2

~

A =

1

2 B × ~r.

~

L MN 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 MN 4 MjP " - ! $, + (2 \

2 ~

_{A · ~}

_{∇ =}

i

¯h

~

B × ~r

· ˆp =

i

¯h

B · (~r × ˆp) =

~

i

¯h

B · ˆ

~

L.

Nh

(6)

, , -

ˆ

H

0 =

e¯h

2mc

1 ¯h

B · ˆ

~

L.

L MN 4 M P $ ^J

µ

_B

=

e¯h

2mc

= 9, 27 × 10

−

₂₁

+ -

= 9.2740154 × 10

−

₂₄

= 5.78838263 × 10

−

₅

L MN 4 N P # Td X c X c S T4 [ " # H ! ) ^ (-2 # & $ $ -( ,( ) $ , 2, $,

−e

" 3 $ ^ ( &

v

" / ( $2# ! " ,

r

#$%2 ( 3 H$ + ! & 4 $H ! ( ) # ( -# ! ,$2 " K $ # & " ! $H , ^ " ( - " # + ,$2, ! - %# ,4 - , ^ " + ( " ( ) &( g WV SY U c f c W c SX ! - %H L " 3 ! 0 H " & ! $ / $ H ! " 2 #$ , $, " (# ,( ) , $ , P 4 3 H " &, + # " ,( ) $ , ^J 2, $,0 $ % " " 2# " " , $ / # ( & + , - $, 4 $ # ( & + , - $, # $ , !

v/ 2πr

/ %#0

I =

ev

2πr

.

L MN 4 N O P

r

v

L

r

I

−

e

→

µ = −−

evr

₂

−

c

→

n

_z

-, $ O \ + ( $ , " ,- ! &( + - " / ( $2# ! 4 R 3 " & " # ( )

S

+ , ! " + (

~

B

0$ % ,H( ) K + 2 ^ ( # + & ! $ " " + ( + ( " #

~µ

~µ =

I

c

S,

~

L MN 4 N M P +

~

S

! - -$ # " # ( ) 0 " ( !+ $ , $ # # +3 K $ , $, " &, # ( S cd T S S T SgX c f 4 " # " -" " $,

S = πr

2 µ =

evr

2c

=

e

2mc

mvr =

e

2mc

L,

L MN 4 NN P Nj

(7)

+

L

! - # ^ ( & , "3 , $ ,4 R #H $ " ,- - 3 #$ , $, " (# H " &0# & $ L MN 4 NNP H " - J # $ #

~µ = −

_2mc

e

L. ~

L MN 4 NZ P a# % J ,# +3 0H

−e

! - 2, $ $ ,4 a # % L MN 4 MP H " " - J + %

ˆ

H

0 = − ~

B · ~µ,

L MN 4 N ] P +

~µ

! - c S T S c , + ( + ( &- $0

ˆ

H

0

" ( ! 3 c S T S T + 2# , " # + " +/ (

~

B

4 b "

~

B

-$ # + # 2,H -

z

( # 3 -

ψ

nlm

! - / " - / (

hn, l, m| ˆ

H

0 |n, l, mi = mµ

B

B ,

L MN 4 NhP +

m µ

B

! - + ( , # -

|n, l, mi .

0H " + ( , #, , + ! & -( " , # - " , " " ( !

B

m

4 a ! # -$ a $ " $, ! - 3 # " " $, ,# +3 -" , $ ,4 N

Atom wodoru w zewntrznych polach

µ



−

r

1

2

dr

d



r

2

d

dr



+

2µ

¯h

2

(V (r) − E) +

l(l + 1)

r

2



u(r) = 0

V (r) = −

Ze

2

r

ψ

n

lm

(r, θ, ϕ) = u

n

l

(r)Y

l

m

(θ, ϕ),

u

l

n

n = ν + 1 + l,

ν = 0, 1, . . . l = 0, 1, . . .

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

,

a

0

= 5, 29 × 10

−

9

n

1, 2

_r

₂

_dr

_lm

_l

_l

_Z

_Z

_:

_.

_l=0,m=0

Z

_a

_l=0,m=0