KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES SHMEIWSEIS … · KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES...

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Transcript of KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES SHMEIWSEIS … · KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES...

Page 1: KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES SHMEIWSEIS … · KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES SHMEIWSEIS-3 Kaj. Kuri koc Tamb khc PERIEQOMENA 1. Kat Tm mata Stajer Dunamik . 2. To

KBANTOMHQANIKH-I

SUNOPTIKES-PROQEIRES SHMEIWSEIS-3

Kaj. Kuri�koc Tamb�khc

PERIEQOMENA1. Kat� Tm mata Stajer� Dunamik�.2. To B ma DunamikoÔ.3. To Fr�gma DunamikoÔ (Fainìmeno S raggoc).4. To Tetragwnikì Phg�di (Dèsmiec Katast�seic).5. To Dunamikì Sun�rthshc Dèlta.

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1. Kat� Tm mata Stajer� Dunamik�.Mia tuqaÐa (omal ) sun�rthsh dunamikoÔ mporeÐ na proseggisjeÐ apì è-

na kat� tm mata stajerì dunamikì sthn epijumht  prosèggish (sq ma sthnepìmenh selÐda). H epÐlush thc anex�rththc tou qrìnou exÐswshc tou Sch-roedinger gia èna swmatÐdio enèrgeiac E se mia tètoia perioq  [a, b] sthnopoÐa h sun�rthsh dunamikoÔ èqei stajer  tim  V0 eÐnai apl  (deÔtero sq maepìmenhc selÐdac). Ac upojèsoume E > V0. H exÐsws  mac eÐnai

− h2

2m

d2ψEdx2

+ V0 ψE = E ψE , (a ≤ x ≤ b) .

Gr�fontac thn exÐswsh tou Schroedinger sthn morf  thc antÐstoiqhc exÐsw-shc gia to eleÔjero swmatÐdio,

− h2

2m

d2ψEdx2

= (E − V0) ψE ,

blèpoume ìti èqoume tic Ðdiec lÔseic epipèdwn kum�twn

eihpx, e−

ihpx ,

ìpou, ìmwc

p =

√2m

h2 (E − V0)

 

E = V0 +p2

2m,

ìpwc, ex�llou ja anamèname sthn perÐptwsh thc Klassik c Fusik c. epomè-nwc, sthn perioq  [a, b] h lÔsh ja eÐnai

ψE(x) = Aeihpx + B e−

ihpx

me sunoriakèc sunj kec

ψE(a) = Aeihpa + B e−

ihpa, ψE(b) = Ae

ihpb + B e−

ihpb

ψ′E(a) =i

hp(Ae

ihpa −B e−

ihpa), ψ′E(b) =

i

hp(Ae

ihpb −B e−

ihpb),

ìpou ψE(a), ψ′E(a), ψE(b), ψ′E(b) eÐnai oi timèc thc kumatosun�rthshc kai thcparag¸gou thc (gia tic perioqèc x ≤ a kai x ≥ b) sta sunoriak� shmeÐa.H sunèqeia thc parag¸gou sta shmeÐa a, b proupojètei ìti to dunamikì denapeirÐzetai sta shmeÐa aut�. Shmei¸ste ìmwc ìti asunèqeia sto dunamikìepitrèpetai.

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V(x)

a b

V0

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2. To B ma DunamikoÔ.Mia polÔ apl  perÐptwsh dun�mewn eÐnai dun�meic pou exaskoÔntai mìno se

èna shmeÐo. EpÐ paradeÐgmati, e�n h perioq  sthn opoÐa kineÐtai èna swmatÐdioqwrÐzetai se dÔo tm mata me stajerì all� diaforetikì dunamikì, to swmatÐdioden ja dèqetai dun�meic stic perioqèc autèc all� mìno sto shmeÐo sun�nths ctouc ìpou kai to dunamikì ja parousi�zei asunèqeia. To aploÔstero tètoiopar�deigma eÐnai to B ma DunamikoÔ

V (x) =

0 −∞ < x < 0

V0 0 < x < ∞

(Sq ma mejepìmenhc selÐdac) Oi epitrepìmenec timèc thc enèrgeiac eÐnai 0 ≤E ≤ V0 kai E ≥ V0. H perÐptwsh E < 0 den odhgeÐ se fusik� apodektèclÔseic. Sqetik� me to autì isqÔei to akìloujo {Je¸rhma}: Den up�rqounfusik� apodektèc lÔseic thc exÐswshc tou Schroedinger, oi opoÐec na antistoi-qoÔn se enèrgeia mikrìterh apì to el�qisto tou dunamikoÔ.

2.1. PerÐptwsh E > V0.H exÐswsh Schroedinger eÐnai stic dÔo perioqèc

− h2

2md2ψEdx2 = E ψE (x < 0)

− h2

2md2ψEdx2 + V0 ψE = E ψE (x > 0)

  ψ′′E = −p2

h ψE (x < 0)

ψ′′ = − q2

h ψE (x > 0)

ìpou

E =p2

2m, E − V0 =

q2

2m.

Oi lÔseic eÐnai

ψE(x) =

Ae

ihxp + B e−

ihxp (x ≤ 0)

C eihxq + De−

ihxq (x ≥ 0)

H ermhneÐa k�je ìrou eÐnai h akìloujh:

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O ìroc Aeihxp antistoiqeÐ se èna {kÔma prospÐpton apì arister�}.

O ìroc B e−ihxp antistoiqeÐ se èna {anakl¸meno kÔma}.

O ìroc C eihxq antistoiqeÐ se èna {dierqìmeno kÔma}.

Tèloc, o ìroc De−ihxq antistoiqeÐ se èna {kÔma pou prospÐptei apì dexi�}.

QwrÐc bl�bh thc genikìthtac, mporoÔme na aplopoi soume thn an�lus mac periorizìmenoi se swmatÐdia, ta opoÐa prospÐptoun mìno apì arister�.Autì isodunameÐ me thn epilog 

D = 0 .

Tìte, h lÔsh mac eÐnai

ψE(x) =

Ae

ihxp + B e−

ihxp (x ≤ 0)

C eihxq (x ≥ 0)

H sunèqeia thc kumatosun�rthshc kai thc parag¸gou thc sto shmeÐo a-sunèqeiac tou dunamikoÔ x = 0 dÐnei

A + B = C

pA − pB = q C .

EpilÔontac autì to sÔsthma paÐrnoume

B

A=

p− qp+ q

,C

A=

2p

p+ q.

Telik�, h kumatosun�rthsh eÐnai

ψE(x) =

A(eihpx +

(p−qp+q

)e−

ihxp)

(x ≤ 0)

A(

2pp+q

)eihqx (x ≥ 0)

Katarq n parathroÔme ìti up�rqei merik  an�klash parìlo pou h enèr-geia eÐnai megalÔterh apì to fr�gma dunamikoÔ. Autì den ja mporoÔse nasumbaÐnei gia èna {klassikì} swmatÐdio. Antijètwc, eÐnai èna anamenìmenofainìmeno gia kÔmata.

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V(x)

0

V0

V(x)

E<V0

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Ac upologÐsoume thn puknìthta reÔmatoc pou antistoiqeÐ se k�je komm�tithc kumatosun�rthshc. To prospÐpton reÔma, ofeilìmeno ston ìro Aeipx/h,eÐnai

Jπ = |A|2 p

m.

To anakl¸meno reÔma, ofeilìmeno ston ìro (...) e−ipx/h, eÐnai

Jα = − p

m|B|2 = − p

m|A|2

(p− qp+ q

)2

.

Tèloc, to dierqìmeno reÔma sthn perioq  x ≥ 0 eÐnai

Jδ =q

m|C|2 =

q

m|A|2

(2p

p+ q

)2

.

'Opwc kai sthn Optik , ìpou to an�logo mègejoc proc thn puknìthtareÔmatoc eÐnai h èntash, ètsi kai ed¸ mporoÔme na orÐsoume ton Suntelest An�klashc wc

R ≡ |Jα|Jπ

=|B|2

|A|2=

(p− qp+ q

)2

.

O Suntelest c An�klashc ekfr�zei to posostì twn anaklwmènwn swmatidÐ-wn. O Suntelest c Dièleushc T orÐzetai wc

T ≡ JδJπ

=q

p

|C|2

|A|2=

(2p

p+ q

)2

.

O Suntelest c Dièleushc ekfr�zei to posostì twn dierqomènwn swmatidÐwn.EÐnai anamenìmeno ta anakl¸mena swmatÐdia kai ta dierqìmena prostijè-

mena na mac dÐnoun ton sunolikì arijmì twn swmatidÐwn, opìte ja prèpei naisqÔei h sqèsh

R + T = 1 .

Antikajist¸ntac tic ekfr�seic twn R kai T , blèpoume ìti h sqèsh aut  ika-nopoieÐtai tautotik�. Up�rqei kai ènac �lloc trìpoc na katano soume aut thn sqèsh. H exÐswsh sunèqeiac gia tic kumatosunart seic ψE(x) paÐrnei thnmorf  diat rhshc tou reÔmatoc

dJdx

= 0 =⇒ J = const.

 

J (x < 0) = J (x > 0) =⇒ Jπ + Jα = Jδ =⇒ Jπ − |Jα| = Jδ

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1 − |Jα|Jπ

=JδJπ

=⇒ 1 −R = T =⇒ R + T = 1 .

2.2. PerÐptwsh E < V0.Sthn perÐptwsh pou èna {klasikì} swmatÐdio enèrgeiac E prospÐptei se

èna fr�gma dunamikoÔ Ôyouc V0 > E p�ntote anakl�tai. Ac doÔme poiaeÐnai h problepìmenh kumatosun�rthsh sthn perÐptwsh aut . H exÐswsh touSchroedinger eÐnai

− h2

2md2ψEdx2 = E ψE (x < 0)

− h2

2md2ψEdx2 + V0 ψE = E ψE (x > 0)

  ψ′′E = −p2

h ψE (x < 0)

ψ′′ = s2

h ψE (x > 0)

ìpou

E =p2

2m, V0 − E =

s2

2m.

Shmei¸ste thn allag  pros mou sthn exÐswsh tou Schroedinger gia thn pe-rioq  x > 0. Oi lÔseic eÐnai

ψE(x) =

Ae

ihxp + B e−

ihxp (x ≤ 0)

C e−xs/h + Dexs/h (x ≥ 0)

H ermhneÐa k�je ìrou eÐnai h akìloujh:

O ìroc Aeihxp antistoiqeÐ se èna {kÔma prospÐpton apì arister�}.

O ìroc B e−ihxp antistoiqeÐ se èna {anakl¸meno kÔma}.

Oi ìroi C e−xs/h kai Dexs/h den èqoun thn ermhneÐa diadidomènwn kum�-twn.

M�lista, o ìrocDexs/h aux�netai ekjetik� kai apoklÐnei sto �peiro, ¸stena eÐnai fusik� apar�dektoc. Sunep¸c, mìno h epilog 

D = 0

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eÐnai fusik¸c apodekt .Epib�lontac thn sunèqeia thc kumatosun�rthshc kai thc parag¸gou thc

sto shmeÐo x = 0, paÐrnoume

A + B = C

ip (A −B) = −sC

 B

A=

p+ is

p− is,

C

A=

2p

p− is.

Telik�, h kumatosun�rthsh eÐnai

ψE(x) =

A(eihpx +

(p+isp−is

)e−

ihpx)

(x ≤ 0)

A(

2pp−is

)e−sx/h (x ≥ 0)

ParathroÔme ìti h kumatosun�rthsh eÐnai mh-mhdenik  sthn {klassika apa-goreumènh perioq }. MporeÐ �rage na parathrhjeÐ to swmatÐdio sthn perioq aut ? Ac shmeiwjeÐ ìti lìgw thc ekjetik c pt¸shc thc kumatosun�rthshc hpijanìthta parousÐac tou swmatidÐou eÐnai upologÐsimh mìno gia thn perioq 

δx ∼ h

s.

'Exw apì aut  thn perioq  h pijanìthta fjÐnei t�qista. Apì thn Arq  thcAbebaiìthtac ìmwc h abebaiìthta sthn orm  ja eÐnai

δp ≥ h

δx∼ s .

H antÐstoiqh abebaiìthta sthn enèrgeia ja eÐnai

δE ∼ (δp)2

2m≥ s2

2m= V0 − E =⇒ E + δE ≥ V0 .

Autì ìmwc shmaÐnei ìti h abebaiìthta sthn enèrgeia aneb�zei to swmatÐdioèxw apo thn (klassik� apagoreumènh) energeiak  z¸nh 0 < E < V0. Parì-lo pou sto sugkekrimèno sÔsthma, ìpou h {klassik� apagoreumènh perioq }eÐnai �peirh, o mh-mhdenismìc thc kumatosun�rthshc ekeÐ den èqei sunèpeiec,sto sÔsthma tou tetragwnikoÔ fr�gmatoc dunamikoÔ pou ja exet�soume pa-rak�tw, to gegonìc autì jètei tic proupojèseic gia to perÐfhmo FainìmenoS raggoc.

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Mia epal jeush tou gegonìtoc ìti den up�rqei di�dosh gia x ≥ 0 parè-qetai apì ton mhdenismì tou reÔmatoc pijanìthtac sthn perioq  aut 

J (x > 0) = 0 .

Apì thn diat rhsh reÔmatoc paÐrnoume ìti

Jπ + Jα = 0 =⇒ −JαJπ

= 1 =⇒ R = 1 .

To apotèlesma autì shmaÐnei ìti èqoume olik  an�klash kai sunodeÔetai apìton mhdenismì tou suntelest  dièleushc T = 0 .

3. To Fr�gma DunamikoÔ (Fainìmeno S raggoc).Ac jewr soume t¸ra èna swmatÐdio enèrgeiac E > 0, to opoÐo kineÐtai sto

dunamikì

V (x) =

0 x < −a

V0 −a < x < a

0 x > a

Ja exet�soume thn perÐptwsh E < V0. H exÐswsh tou Schroedinger eÐnai

− h2

2md2ψEdx2 = E ψE (x < −a)

− h2

2md2ψEdx2 + V0ψE = E ψE (−a < x < a)

− h2

2md2ψEdx2 = E ψE (x > a)

 

ψ′′E = −p2

h ψE (x < −a)

ψ′′E = s2

h ψE (−a < x < a)

ψ′′E = − p2

2mψE (x > a)

ìpou

E =p2

2m, V0 − E =

s2

2m.

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Oi lÔseic eÐnai

ψE(x) =

Aeihxp + B e−

ihxp (x ≤ −a)

C e−xs/h + Dexs/h (−a ≤ x ≤ a)

F eihxp + Ge−

ihxp (x ≥ a)

Sto shmeÐo autì ac k�noume tic ex c dÔo parathr seic:1) Sthn peperasmènh perioq  [−a, a] den suntrèqei lìgoc anexèlegkthc

aÔxhshc thc kumatosun�rthshc kai, epomènwc, kai oi dÔo ìroi C e−xs/h kaiDexs/h eÐnai fusik¸c apodektoÐ.

2) O ìroc Aeihxp antistoiqeÐ se swmatÐdio pou prospÐptei apì arister� en¸

o ìroc Ge−ihxp antistoiqeÐ se swmatÐdio pou prospÐptei apì dexi�. QwrÐc bl�-

bhn thc genikìthtoc mporoÔme na epilèxoume G = 0, pr�gma pou antistoiqeÐse prìsptwsh mìno apì arister�.

Apì thn sunèqeia thc kumatosun�rthshc kai thc parag¸gou thc sta sh-meÐa −a kai a, paÐrnoume

Ae−ipa/h + B eipa/h = C eas/h + De−sa/h

ip(Ae−ipa/h −B eipa/h

)= s

(−C eas/h + De−sa/h

)C e−as/h + Deas/h = F eipa/h

s(−C e−as/h + Deas/h

)= ip F eipa/h .

Met� apì (k�poiec) pr�xeic mporoÔme na upologÐsoume ton lìgo

F

A=

2ips e−2 ihpa

[ (p2 − s2) sinh(2sa/h) + 2ips cosh(2sa/h) ].

Apì ton lìgo autìn mporoÔme na prosdiorÐsoume ton suntelest  dièleushc

T =JδJπ

=|F |2

|A|2=

1[1 + (p2+s2)2

4p2s2sinh2(2sa/h)

]

=

1 +V0

4E(1− E

V0

) sinh2

2

√2ma2

h2 (V0 − E)

−1

.

H basik  parat rhsh eÐnai ìti parìlo pou h enèrgeia tou swmatidÐou eÐnai mi-krìterh apì to Ôyoc tou fr�gmatoc, èqoume di�dosh pèran tou fr�gmatoc kai

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mh-mhdenikì suntelest  dièleushc. Autì eÐnai to {Fainìmeno S raggoc}.Sto fainìmeno s raggoc ofeÐlontai pollèc sÔgqronec teqnologikèc efarmo-gèc (p.q. hmiagwgoÐ) all� kai shmantik� fusik� fainìmena ìpwc h radienèrgeiatÔpou α.

EÔkola analÔetai h perÐptwsh enìc eurèwc (2a >>) kai uyhloÔ (V0 >>E) fr�gmatoc. Tìte, to ìrisma tou uperbolikoÔ hmitìnou pou emfanÐzetaisthn èkfrash tou suntelest  dièleushc ja eÐnai meg�lo. Eis�gontac thnpar�metro

γ ≡√

2m

h2 (V0 − E)

ja èqoume(2a) γ >> 1

kai mporoÔme na k�noume thn prosèggish

sinh(2aγ) ≈ 1

2e2γ a ,

opìte, o suntelest c dièleushc ja eÐnai

T (E) ≈ 16E

V0

(1− E

V0

)e−4γa .

Shmei¸noume ìti h sun�rthsh pou pollaplasi�zei to ekjetikì metab�letaibradèwc me thn enèrgeia en¸, antÐjeta, to ekjetikì exart�tai isqur� apì au-t n. H ekjetik  euaisjhsÐa tou suntelest  dièleushc apì thn enèrgeia para-threÐtai sthn meg�lh diakÔmansh tou qrìnou zw c twn radienerg¸n pur nwnpou diasp¸ntai mèsw thc radienèrgeiac α.

To Fainìmeno twn Suntonism¸n. 'Ena endiafèron fainìmeno parousi�ze-tai sto fr�gma dunamikoÔ sthn perÐptwsh enèrgeiac E > V0. MporoÔme nap�roume amèswc ton lìgo F/A apì thn an�logh èkfrash pou èqoume sthnperÐptwsh E < V0 me thn tupik  allag 

s → −is, s =√

2m(E − V0) .

'Eqoume

F

A=

2pse−2i pah

[ 2ps cos(2sa/h) − i(p2 + s2) sin(2sa/h) ].

O suntelest c dièleushc eÐnai

T (E) =4p2s2[

4p2s2 cos2(2sa/h) + (p2 + s2)2 sin2(2sa/h)] .

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ParathroÔme ìti gia tic eidikèc timèc

s = nhπ

2a, (n = 1, 2, . . .) =⇒ T = 1

to fainìmeno thc merik c an�klashc exafanÐzetai kai èqoume pl rh dièleush.To fainìmeno autì onom�zetai Fainìmeno twn Suntonism¸n kai oi timèc thcenèrgeiac gia tic opoÐec emfanÐzetai eÐnai

En = V0 +n2π2h2

8ma2.

4. To Tetragwnikì Phg�di (Dèsmiec Katast�seic).Sto ed�fio autì ja melet soume thn kÐnhsh enìc swmatidÐou upì thn epÐ-

drash tou dunamikoÔ

V (x) =

0 (x < −a)

−V0 (−a < x < a)

0 (x > a)

H enèrgeia to swmatidÐou mporeÐ na eÐnai eÐte E > 0 eÐte −V0 < E < 0. Jamelet soume thn deÔterh perÐptwsh (arnhtik  enèrgeia) giatÐ parousi�zei toendiafèron fainìmeno twn Dèsmiwn Katast�sewn.

H exÐswsh tou Schroedinger eÐnai

− h2

2md2ψEdx2 = E ψE (x < −a)

− h2

2md2ψEdx2 − V0ψE = EψE (−a < x < a)

− h2

2md2ψEdx2 = E ψE (x > a)

 

ψ′′E =

s2

h2 ψE (x < −a)

ψ′′E = − q2

h2ψE (−a < x < a)

ψ′′E = s2

h2 ψE (x > a)

ìpou

E = − s2

2m, E + V0 =

q2

2m.

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Oi apodektèc lÔseic sthn exwterik  perioq  eÐnai ta fjÐnonta ekjetik�esx/h, gia x < −a, kai e−sx/h, gia x > a. Sthn eswterik  perioq  oi lÔ-

seic eÐnai ta kÔmata e±ihxq. IsodÔnama, mporoÔme na epilèxoume wc lÔseic

sunart seic kajorismènhc parity, dhlad  hmÐtona kai sunhmÐtona sin(qx/h),cos(qx/h). H kumatosun�rthsh ja eÐnai

ψE =

Aesxh (−a ≤ x)

C cos(qx/h) + D sin(qx/h) (−a ≤ x ≤ a)

B e−sxh (x ≥ a)

To gegonìc ìti to dunamikì (kai, epomènwc, kai o telest c Hamilton) eÐnaikatoptrik� summetrikì mac epitrèpei na dialèxoume tic lÔseic ψE na eÐnai �rtieckai perittèc. H �rtia lÔsh antistoiqeÐ se D = 0 kai B = A kai eÐnai

ψ(+)E (x) = ψ

(+)E (−x) =

Aesxh (−a ≤ x)

C cos(qx/h) (−a ≤ x ≤ a)

Ae−sxh (x ≥ a)

=

Ae−

sh|x| |x| ≥ a

C cos(qx/h) |x| ≤ a

H peritt  lÔsh antistoiqeÐ se C = 0 kai B = −A kai eÐnai1

ψ(−)E (x) = −ψ(−)

E (−x) =

Aesxh (−a ≤ x)

D sin(qx/h) (−a ≤ x ≤ a)

−Ae−sxh (x ≥ a)

=

Asign(x) e−

sh|x| |x| ≥ a

D sin(qx/h) |x| ≤ a

'Artia LÔsh.H sunèqeia thc �rtiac lÔshc dÐnei (arkeÐ h sunèqeia sto shmeÐo a.)

Ae−sha = C cos(qa/h)

−sA e−sha = −C q sin(qa/h) .

Apì autèc tic sqèseic paÐrnoume

C

A=

e−sha

cos(qa/h)=

s

q

e−sha

sin(qa/h),

1Me sign(x) èqoume sumbolÐsei to prìshmo tou x.

14

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ìpou, ektìc apì ton prosdiorismì tou lìgou C/A, paÐrnoume kai mia sunj khsthn enèrgeia

e−sha

cos(qa/h)=

s

q

e−sha

sin(qa/h)=⇒ tan(qa/h) =

s

q.

H sunj kh aut  kajorÐzei tic epitrepìmenec idiotimèc thc enèrgeiac. Ektìc apìtic paramètrouc tou sust matocm, a, V0, h exÐswsh aut  perièqei mìno thn e-nèrgeia. EÐnai, epomènwc, h exÐswsh idiotim¸n thc enèrgeiac. Mìno lÔseic thcexÐswshc idiotim¸n sthn perioq  −V0 < E < 0 eÐnai apodektèc. Antijètwc,sthn perÐptwsh jetik c enèrgeiac olìklhrh h hmieujeÐa 0 < E < ∞ epi-trèpetai. Apì majhmatik c pleur�c, h exÐswsh idiotim¸n eÐnai mia uperbatik exÐswsh giatÐ sundèei trigwnometrikèc kai �rrhtec sunart seic (rizik�).

Ac eisag�goume thn metablht 

ξ ≡ qa

h=

√2ma2

h2 (E + V0)

kai thn par�metro

β2 ≡ 2mV0a2

h2 .

Sunart sei aut¸n èqoume

sa

h=√β2 − ξ2

kai mporoÔme, telik�, na gr�youme thn exÐswsh idiotim¸n wc

tan ξ =

√β2

ξ2− 1 .

H exÐswsh aut  epilÔetai grafik� kai èqei arijm simo kai peperasmèno pl jocapì lÔseic

ξ1, ξ2, . . . =⇒ V0 < E1, E2, . . . . < 0 .

Oi energeiakèc autèc st�jmec antistoiqoÔn se katast�seic stic opoÐec to sw-matÐdio eÐnai entopismèno sto phg�di tou dunamikoÔ kai onom�zontai DèsmiecKatast�seic. To pl joc touc exart�tai apì thn par�metro β, dhlad  apì toeÔroc kai to b�joc tou dunamikoÔ. 'Oso megalÔtero eÐnai to β tìso perissì-terec dèsmiec katast�seic up�rqoun. Up�rqei ìmwc p�ntote mÐa toul�qistondèsmia kat�stash (basik  st�jmh), ìso rhqì kai an eÐnai to phg�di.

15

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Peritt  LÔsh.Saut  thn perÐptwsh h sunèqeia dÐnei

D sin(qa/h) = −Ae−sa/h

q D cos(qa/h) = sA e−sa/h .

Autèc oi sqèseic afenìc prosdiorÐzoun ton lìgo D/A kai afetèrou dÐnoun miasunj kh idiotim¸n thc enèrgeiac, ìpwc kai sthn perÐptwsh thc �rtiac lÔshc

D

A= − e−sa/h

sin(qa/h)=

s

q

e−sa/h

cos(qa/h)=⇒ tan(qa/h) = −q

s.

Qrhsimopoi¸ntac touc orismoÔc tou ξ kai tou β pou eisag�game gia thn �rtialÔsh, paÐrnoume

tan ξ = − 1√β2

ξ2 − 1.

H exÐswsh aut  lÔnetai grafik� kai odhgeÐ se arijm simo pl joc lÔsewn

ξ′1, ξ′2, . . . =⇒ V0 < E′1, E

′2, . . . . < 0 .

Se antÐjesh me thn �rtia lÔsh, den up�rqei p�ntote peritt  lÔsh par� mìnoe�n β ≥ π/2.

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5. To Dunamikì Sun�rthshc Dèlta.Mia oriak  perÐptwsh tetragwnikoÔ phgadioÔ   tetragwnikoÔ fr�gmatoc

dunamikoÔ eÐnai èna phg�di   fr�gma me eÔroc 2a → 0 kai b�joc   Ôyoc±V0 = λ

2a → −∞. 'Ena tètoio dunamikì parist�netai me thn sun�rthshDèlta tou Dirac

V (x) = λ δ(x) .

H exÐswsh Schroedinger eÐnai

− h2

2m

d2

dx2ψE(x) + λ δ(x)ψE(x) = E ψE(x) .

5.1. H perÐptwsh Fr�gmatoc (λ > 0). Sthn perÐptwsh aut èqoume lÔseic mìno me E > 0. Qrhsimopoi¸ntac thn idiìthta thc sun�rthshcdèlta δ(x)ψE(x) = δ(x)ψE(0), h exÐswsh Schroedinger gr�fetai

ψ′′E(x) − 2mλ

h2 ψE(0) δ(x) = −2mE

h2 ψE(x) .

Ac shmeiwjeÐ ìti sto shmeÐ x = 0 to dunamikì den up�rqei kai, epomènwc,kai h deÔterh par�gwgoc ψ′′E(0) den ja up�rqei, Autì shmaÐnei ìti h pr¸thpar�gwgoc den ja eÐnai suneq c. Gia na broÔme thn asunèqeia thc pr¸thcparag¸gou mporoÔme na oloklhr¸soume thn exÐswsh Schroedinger se miamikr  perioq  [−ε, ε] me ε→ 0 gÔrw apì to shmeÐo x = 0. 'Eqoume∫ +ε

−εdxψ′′E(x) − 2mλ

h2 ψE(0)

∫ +ε

−εdx δ(x) = −2mE

h2

∫ +ε

−εdxψE(x)

 

ψ′(ε) − ψ′(−ε) − 2mλ

h2 ψE(0) = −2mE

h2 (2ε)ψE(0)

 

ψ′(ε) − ψ′(−ε) =2mλ

h2 ψE(0) .

Ac gr�youme t¸ra thn lÔsh thc exÐswshc Schroedinger. Stic perioqèc

x < 0 kai x > 0 den up�rqei dunamikì kai, epomènwc, h lÔseic eÐnai e±ihpx.

Epilègontac prìsptwsh mìno apì arister� èqoume

ψE(x) =

Ae

ihpx + B e−

ihpx (x < 0)

C eihpx (x > 0)

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Epib�lontac tic sunj kec sunèqeiac/asunèqeiac sto x = 0, paÐrnoume

A + B = C

i

hpC − i

hp (A −B) =

2mλ

h2 C .

Apì autèc prosdiorÐzoume touc lìgouc

B

A=−iλmhp

1 + iλmhp,

C

A=

1

1 + iλmhp.

O suntelest c an�klashc eÐnai

R =|B|2

|A|2=

λ2m2

λ2m2 + h2p2.

H kumatosun�rthsh eÐnai

ψE(x) =

A(eihpx −

(iλm

hp+iλm

)e−

ihpx)

(x > 0)

A(

hphp+iλm

)eihpx (x > 0)

=

A(eihpx −

(iλm

hp+iλm

)e−

ihpx)

(x > 0)

A(

1 −(

iλmhp+iλm

) )eihpx (x > 0)

kai gr�fetai apl� wc

ψE(x) = A

(eihpx −

(iλm

hp+ iλm

)eihp|x|

).

To komm�ti thc lÔshc eihpx apoteleÐ to prospÐpton kÔma en¸ to kamm�ti e

ihp|x|

apoteleÐ to skedazìmeno kÔma. O suntelest c

−(

iλm

hp+ iλm

)onom�zetai pl�toc skèdashc.

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5.2. H perÐptwsh PhgadioÔ (λ < 0). Sthn perÐptwsh aut  mpo-roÔme na èqoume E > 0 kai E < 0.

Skèdash (E > 0). Ta b mata eÐnai panomoiìtupa me thn an�lush pou proh-g jhke. Apl� qrei�zetai na èqoume upìyh mac ìti λ < 0. h kumatosÔnarthsheÐnai p�li

ψE(x) = A

(eihpx −

(iλm

hp+ iλm

)eihp|x|

).

Dèsmiec Katast�seic (E < 0). H exÐswsh tou Schroedinger eÐnai tupik�h Ðdia all� oi lÔseic thc eÐnai eqx/h gia x < 0 kai e−qx/h gia x > 0. 'Eqoume

eisag�gei thn par�metro q mèsw thc sqèshcE = − q2

2m . Sugkekrimèna, èqoume

ψE(x) =

Aeqx/h (x < 0)

C e−qx/h (x > 0)

H sunèqeia/asunèqeia dÐnei

A = C, −q C − q A =2mλ

h2 A .

Apì tic sqèseic autèc paÐrnoume thn sunj kh idiotim¸n

q = −mλh2 =⇒ E = −mλ

2

2h4 .

Epomènwc, èqoume mia dèsmia kat�stash me thn enèrgeia aut . H antÐstoiqhkumatosun�rthsh eÐnai (λ < 0)

ψE(x) = Aeλmh3 |x| .

H stajer� A prosdiorÐzetai apì thn kanonikopoÐhsh

1 = |A|2∫ +∞

−∞dx e2λm

h3 |x| = 2|A|2∫ ∞

0dx e2λm

h3 x = −|A|2h3

 

A =

√−mλh3 .

Telik�,

ψE =

√−mλh3 e

λmh3 |x| =

(−2mE

h2

)1/4

e−√− 2mE

h2 |x|.

19