Документ взят из кэша поисковой машины. Адрес оригинального документа : http://genphys.phys.msu.su/rus/edu/kvant/II_2/ShredingerEq.pdf Дата изменения: Sun Mar 22 09:51:03 2015 Дата индексирования: Sun Apr 10 01:08:32 2016 Кодировка: Поисковые слова: m 2

m
U ( x)

x
( x, t )
2 2 2



x (t )
i

dp dt

dU dx

t

2m x

U

^ H

( x, t

t)

x (0) p ( 0)

1^ ( x, t ) H ( x, t ) t i

x (t )

( x , 0)


( x, t )
dP
*

, - :

( x, t ) ( x, t )dx



U U ( x)
( )

i

( x, t ) t
2

2

2

2m
2

( x, t ) U ( x ) ( x, t ) 2 x

( x, t )

( x)T (t )



T (t ) i t T (t )

2m

( x) U ( x) ( x) 2 x ( x)

E

const

i
2

T (t ) t
2 2

ET (t )

T (t ) e

E it

2m x
( x, t ) ( x )e

( x) U ( x) ( x)
*

E ( x)
*

E it

( x, t ) ( x, t )

( x) ( x)





i ( x, t ) t
( x, t )
^ H ( x)

^ H ( x, t )
E it

=const.

( x )e



E ( x)



( )

*

( x, t ) ( x, t )

*

( x ) ( x)




0
2 2 2



x



2m x

U ( x)

E

2

x
n

2

k

2

0 k

2

2mE
2

(0)

(a)

0

( x)

2 n sin x a a

2

2 2

En

2ma

n

2



n

( x, 0)

n

( x)

2 n sin xe a a

i

En

t

|

n

|

2

2

2 2

En

2ma

n

2

3

2 3 sin x a a
2 2 sin x a a

n=3

E3

9E1

n=2 n=1

E2
E1

4E1
2 2 2

2

0

a

1

x

2 sin x a a

0



a a

x

2ma




2

2 2 sin( x)e a a 2 sin( x)e a a
i

i

E2

t

E2

4E1
2 2 2

E1

t

1

E1

2ma



a



: c

( x, t ) C1 1 ( x, t ) C2
1

2

( x, t )

( x, t )

1 2
2

( x) exp(

i

E1t )

1 2

2

( x, t ) exp(

i

E2t )

( x, t )

2

1 2

1

( x)

1 2

2

( x)

2 1

( x) 2 ( x) cos( t ) E2 E1


a /2

x
a /2 a /2 * 1 a /2

1 x 2

1

( x)

2

1 x 2

2

( x)

2 1

( x) x 2 ( x) cos( t ) dx x12 - x

( x) x 2 ( x) cos( t )dx

1

x

2

(d=e ) :



Ae
i

Ae

it

Ae
1

i

A1e
1

i

1

A2 e
2

i

2

t
i
2 1

2

t
i
2

A

2

AA
2 1 2 1

* 2 2

( A1e

i

1

A2e )( A1e
2

i

A2e
1

)

A A

A2 A2
1

A1 A2 (e
( )t

i(

2

1

)

e

i(

)

)

2 A1 A2 cos(
2 1

2

1

)

2




C



Re (x,t)

Re (x,t)




C





C 15- 16-







( x, t ) C1 1 ( x, t ) C2

2

( x, t )
i En t

: , , , H.

( x, t )
n

C En
* n n

n

n

( x) e

^ H C
n

n

( x) ( x)dx

C
n * n

2 n

1 ( x)dx
Cn
nm
2

m

n- P(n)=ICnI2

P ( n)


U=const
2 2 2

2m x
2

U ( x)
2

E 2m( E U )
2
x

x

2

k

2

0 k



( x) ei
2




k

2 1, 2

k

2

( x)

Ae

i 1x

Be

i

2

x

k2 > 0, ­ (t=0)

( x) ( x) Ae
x

Ae
i,

ikx

Be
0

ikx

k2 < 0, ( ) ­ :

,

k

( " ")



:

q t

jqdS
s

j
j dS
s

q

-

P : t
P

*

j -


dV

:
i

i j ( 2m
*

*

*

)

( x, t )
m
q

Ae
m
q
*
*

( Et px )

j

p m

p2 A m

vA

2

: :

jm m j
jq q j




P : t q : t jq dS
s

jq -
j dS
s
2

q

t

divjq

j -
2


*

t

divj

q

j

q

i i

t
*

2m
2 2

U ( x)
*

t i i t j
*

2m
*

U ( x)

*



2

*

2

2

*

t
* 2

t
*

2m
*

t divj 2mi (
*

2m

*

)



j 2mi (
* *

)

:
i

( x, t )
:
m

Ae
m
q
*

( Et px )

j

p m

*

p2 A m

vA

2

jm m j
jq q j

:

*

q





EU
0



j

p m

*

p2 A m

k2 A m

kA2



EU
0



U E 1
: ,

l

10

10






C 1 /2 2



U0 E < U0
2 1

( x) ( x)

A1e

ikx

B1e
x

ikx

k 2m(U
0 2

2mE
2

A2e

E)


3

a
A3e
2
ikx

1

( )

( x)

T
T0

A3 A1

T0 exp( 2a
T0
a 10
10

2m(U
1


0 2

E)

)

16 E (U 0 E ) U 02

U E 1


T
T

0, 3
10
8

a 10 9

T T

2

a

2%

a

0.001


(STM)



T

exp( 2 a )

2m(U

0 2

E)



T T

2

a
a 0.001

a

2%


http://www.almaden.ibm.com/vis/stm/stm.html



STM image, 7 nm x 7 nm, of a single zig-zag chain of Cs atoms (red) on the (110) surface of a GaAs crystal (blue)
L.J. W hitman, J.A. Stroscio, R.A. Dragoset, and R.J. Celotta, "Geometric and Electronic Properties of Cs Structures on III-V (110) Surfaces: From 1-D and 2-D Insulators to 3-D Metals"


-



T
T ln T

exp( 2a
T1 T2 T3 T4 ln T1 ln T2

2m(U

0 2

E)

)

ln T3 ln T4

ln Tn

2
x2

2m(U ( xn ) E )
2

dxn

ln T

2 dx
x1
x2

2m(U ( x) E )
2

)

T

exp

2 dx
x1

2m(U ( x ) E )
2



dN Ndt N (t ) N 0e
t






N( ) N0 / 2

dN Ndt

(1896) 1903 . « , ».

1

ln 2

-
-

- ( )

-


-
10-6 1010 - 8 4


-
E

R r0 A1/

3

r0 10 15

1 2e 2 ( Z 2) U ( r1 ) 40 r1 Ra
226

88

R 7.3 10 15

E

4.78 r1 50 10 15
p v v 2R m T exp( exp( 2 4 4
r1

nT n T 2m R t1/
2

mR dr 2m (U (r ) E ))
R

2

2m R 2 ln 2 B E

qq
0

2m ) E

ln t1/

2

A


-
ln t1/
2

A

B E


-

-

88Ra

226

- 86Rn222

92U

238


-
41 -->
4,03188
4

4,0026

p + p ІH + e+ + e + 0,4
1
T exp 2
x2

dx 2m(U ( x) E )
x1

~1









U

1 kx 2

2

2 2 2m x

2

12 kx 2

E

En

n

1 2 k m

;n
1/ 2

0,1, 2, ...

=




0

( x) exp( ( x) x
0

m2 x) 2 x exp(

2

m m2 x) 2

1

( x)








dt dP T x A cos x A si
A A

dx xT t nt 1 2 A
2

dP dx

( x)

x

2

A

A





( x, t )
n

Cn

n

( x) e

i

En

t

() . . , - , . , .