INTERNET
 :
  |    

/

 /

,

 

: .

:

..

3- :

1997 .
() .

, , , .

:

() x, :

-2/(2m)׶2/x2 + U0 , x < -a

٠

H = -2/(2m02/x2 , -a < x < a

-2/(2m)׶2/x2 + U0 , x > a

m - I , III ;

m0 - II.

:

2YI/x2 + 2m/2(E - U0)YI = 0 , x £ -a

2YII/x2 + 2m0/2EYI = 0 , -a £ x £ a

2YIII/x2 + 2m/2(E - U0)YI = 0 , x ³ a

I :

1- :

YI(x) = Aexp(nx) + Bexp(-nx).

, B = 0. ,

YI(x) = Aexp(nx).

:

YII(x) = Cexp(ikx) + Dexp(-ikx).

:

YIII(x) = Fexp(-nx).

k = (2m0E/2)1/2

n = (2m(U0-E)/2)1/2.

:

¨    4 , .

¨    4 A,C,D F. .

¨    , . , , .

, .. :

YI(x=-a) = YII(x=-a)

YII(x=a) = YIII(x=a)

YI¢(x=-a)/m = YII¢(x=-a)/m0

YII¢(x=a)/m0 = YIII¢(x=a)/m

:

Aexp(-na) = Cexp(-ika) + Dexp(ika)

m-1A nexp(-na) = ik/m0(Cexp(-ika) - Dexp(ika))

Cexp(ika) + Dexp(-ika) = Fexp(-na)

ik/m0(Cexp(ika) - Dexp(-ika)) = - n/mFexp(-na).


:

|exp(-na) -exp(-ika) -exp(ika) 0 |

|m-1nexp(-na) -1/m0ikexp(-ika) 1/m0ikexp(ika) 0 |

|0 exp(ika) exp(-ika) -exp(-na) |

|0 1/m0ikexp(ika) -1/m0ikexp(-ika) 1/mnexp(-na)|

, :

((n/m)2 - (k/m0)2)Sin(2ka) + 2kn/(mm0)Cos(2ka) = 0.

, , .

A, C, D, F . , .

C = Fexp(-na){exp(ika) + exp(-3ika) ( ik/m0 - n/m)/(n/m + ik/m0)}

D = Cexp(-2ika)( ik/m0 - n/m)/(n/m + ik/m0)

A = exp(na)(Cexp(-ika) + Dexp(ika)) .

A, C D F, :

A = RAF

C = RCF

D = RDF.

RA, RC, RD - .

, - F, A, C, D. .

:

YI(x) = FRAexp(nx)

YII(x) = F( RCexp(ikx) + RDexp(-ikx)).

YIII(x) = Fexp(-nx).

I1 + I2 + I3 = 1


I1 = |F|2|RA|2Qexp(2nx)dx = |F|2|RA|2(2n)-1exp(2nx) =

= |F|2|RA|2(2n)-1exp(-2na)

I2 = |F|2{ L|RC|2dx + L|RD|2dx + RCRD*Lexp(2ikx)dx +

+ RC*RDLexp(-2ikx)dx } = |F|2{ 2a(|RC|2 + |RD|2) +

((exp(2ika) - exp(-2ika))RCRD*/(2ik) +

+ i((exp(-2ika) - exp(2ika))RC*RD/(2k) }

I3 = |F|2Wexp(-2nx)dx = |F|2(2n)-1exp(-2na)

|F|2 = { |RA|2(2n)-1exp(-2na) + 2a(|RC|2 + |RD|2) +

((exp(2ika) - exp(-2ika))RCRD*/(2ik) +

+ i((exp(-2ika) - exp(2ika))RC*RD/(2k) + (2n)-1exp(-2na) }-1.

, F, A, C, D, , .

, , , . .

, . , , .

:

U(x)=U(x+2a) (1)

(1) , .

, , I, III, :

2Y/x2 + 2m/2(E - U0)Y = 0

, .

:

r = exp(i 2ak)

Y(x+2ma) = Y(x)rm , m=0, 1, 2,... (2)

, ( E<U0) I, II, III. , (2), .

I:

:

2YI/x2 + 2m2/2(E - U0)YI = 0 , 0 > x > -a

:

YI(x) = Aexp(nx) + Bexp(-nx).

n = (2m2 (U0-E) /2)1/2

II:

:

2YII/x2 + 2m1/2E YII = 0 , a ³ x ³ 0

:

YII(x) = Cexp(ipx) + Dexp(-ipx).

p = (2m1E/2)1/2

III:

2YIII/x2 + 2m2/2(E - U0)YIII = 0 , 2a > x > a

:

YIII(x) = r (Aexp(nx) + Bexp(-nx)).

:

YI(x=0) = YII(x=0)

YII(x=a) = YIII(x=a)

YI¢(x=0)/m = YII¢(x=0)/m0

YII¢(x=a)/m0 = YIII¢(x=a)/m

, A, B, C, D:

A+B=C+D

C exp(i p a)+D exp(-i p a) = exp(i 2 a k) (A exp(n a)+B exp(-n a))

(A-B) n/m2 = (C-D) i p / m1

(C exp(i p a)-D exp(-i p a)) i p / m1 = exp(i 2 a k) n/m2 (A exp(n a)-B exp(-n a))

, :

|1 1 -1 -1 |

|exp(ik2a+na) exp(ik2a-na) -exp(ipa) -exp(-ipa) |

|n/m2 -n/m2 -ip/m1 ip/m1 |

|n/m2exp(ik2a+na) -n/m2exp(ik2a-na) - ip/m1exp(ipa) ip/m1exp(-ipa) |

.

.

.

a=10; U=10; m1=4; m2=1

0.1135703312666857

0.6186359585387896

0.2019199605676639

0.3155348518478819

0.05047267055441365

1.263391478912778

0.4544326758658974

2.137353840637548

0.808172718170137

2.479933076698526

0.4544326758658974

6.168062551132728

5.611693924351967

1.820461802850339

1.529165865668653

1.023077302091622

a=10 U=10 m1=2 m2=1

0.1032788024178655

0.2324238959628721

0.41331603936642

0.6460490460448886

0.930750939555283

1.26759057783714

1.656787195799296

2.098624192369327

2.593469359607937

3.141805331837109

3.744277072860902

5.887485640841992

a=10 U=10 m1=1 m2=1

0.05408120469105441

0.2163802958297131

0.4870681554965061

0.86644533469418

1.354969224117534

1.953300729714778

2.662383817919513

4.418966218448088

7.961581805911094

a=10 U=10 m1=0.5 m2=1

0.118992095909544

4.249561710930034

1.068004282376146

0.4754473139332004

5.78216724725356

2.955345679469631

1.895012565781256

a=10 U=10 m1=.25 m2=1

0.2898665804439349

4.30026851446248

2.479039415645616

1.132264393019809


Copyright © 2005—2007 «RefStore.Ru»