量子力学概论第三章

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1、1nÙ³

2、¯•³³²•³^•³²•f421nÙ³

3、¯Ù§·&?éØ5âf3«³

4、V(x)¥¯"du·ÏA½Uþ½)§·òżê¤ψ(x,t)=ψ(x)e−iEt/¯h,(244)d?ψ(x)÷vع(½)Schr¨odinger§µd2ψ2m=[V(x)−E]ψ(245)dx2¯h2ó§kÔn¿Â)ψ(x)7µ1)2k"ÄAÇÝ

5、ψ

6、¬"432)ëY"ÄAÇ6¬"•³³²ÄþmUþEâf3e¡³¥$Ä:(0for0≤x≤aV(x)=.(246)∞otherwise22Xdψ/dx(±9ψ)
±k§(245)3³?£x≤0§

7、x≥a¤§ψ=0"T¯durâfÛ3ÝǑaÝfp"ψ30.^Ǒψ(0)=ψ(a)=0.(247)?§â(245)§ψ3T«÷vd2ψ=−k2ψ(248)dx2d?22mEk=.(249)¯h2ùp§·b½E>0"´y§vkE<0)U÷v>.^(247)"½>.^(247)§§(248))Ǒψn(x)=Ansin(knx),(250)d?An´?¿£¢¤~ê§n=1,2,3,···knπkn=,(251)ay3§l(249)Ú(251)wѧUþEUlѵn2π2¯h2En=(252)2ma2å)£

8、x

9、→∞§

10、ψ

11、→0¤AƵUþÆ´l

12、Ñ"½Ƽêψn(x)A÷v8^µZaψn(x)ψm(x)dx=δnm.(253)0qXkAn=2/a§y²(¢Xd"Ïd§éun=1,2,3,···§s2xψn(x)=sinnπ(254)aaFig.anThethevPassideandergiesbblinesdashservwprobabilitthe***³²¥å"£þ¤Å¼ê¶£e¤AÇÝ"44Top:Timedevelopmentofawavepacketmovinginaninfinitelydeepsquare***£þ¤³²SÅmüz"£e¤²;mAÇÝmüz"45§

13、¹)±¤½)5UµX∞c−iEnt/¯hψ(x,t)=nψn(x)e,(255)n=1d?Zacn=ψn(x)ψ(x,0)dx.(256)0•³^ÄþmUþE>0âf3³^¥$Ä(V0for0≤x≤aV(x)=,(257)0otherwised?V0>0"463³^üý§ψ(x)÷vd2ψ=−k2ψ,(258)dx2d?kd(249)Ñ"³^ý(x<0))Xe/ªψ(x)=eikx+Re−ikx.(259)1²¡Å)ºǑmŧ1)ºǑÅ"

14、R

15、2´AÇ"«x<0¥AÇ6(115)Xe/ªj2left=v(1−

16、R

17、),(260)d?v=p

18、/m=¯hk/mǑ²;âfÝ"³^mý(x>a))Xe/ªψ(x)=Teikx.(261)ù)ºǑBß³^ßÅ"

19、T

20、2´ßXê"«x>a¥AÇ6Xe/ªj2right=v

21、T

22、.(262)2â(131)§½(∂

23、ψ

24、/∂t=0)¥§AÇ6´m~ê(∂j/∂x=0)"Ïd§7kjleft=jright§½ö

25、R

26、2+

27、T

28、2=1.(263)ó§ÚßAÇoÚǑ1"³^SÜ(0≤x≤a)§ψ(x)÷vd2ψ=−q2ψ,(264)dx2d?22m(E−V0)q=.(265)¯h2ÄkÄE>V0/"³^Sܯ(264))/ªǑψ(x)=Aeiqx+

29、Be−iqx,(266)q2d?q=2m(E−V0)/¯h.y3,³^>.(x=0Úx=a)þ>.^ǑψÚdψ/dxëY"ψÚdψ/dx3>.(x=0)ëY51+R=A+B,(267)k(1−R)=q(A−B).(268)aq§ψÚdψ/dx3m>.(x=a)ëY5ÑAeiqa+Be−iqa=Teika,(269)qAeiqa−Be−iqa=kTeika.(270)²$§þão§Ñ2(k2−q2)2sin2(qa)

30、R

31、=,(271)4k2q2+(k2−q2)2sin2(qa)±94k2q2

32、T

33、2=.(272)4k2q2+(k2−q2)2sin2(qa)

34、5¿±þüª÷vå^(263)"***³^ߣ¢¤AÇ£¤"³^°Ýa=1.25λ"—(λ=2π/k´gdmdeBroglieÅ")47***³^ߣ¢¤AÇ£¤"³^pÝV0=0.75E"48â(271)§éuXeAϳ^°Ý§AÇǑ""qa=nπ,(273)d?n=1,2,3,···"y3ÄE.(x=0)ëY51+R=A+B,(275)ik(1−R)=