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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.
1
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.
2
V(x)
a b
V0
3
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:
4
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.
5
V(x)
0
V0
V(x)
E<V0
6
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
7
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
8
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.
9
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.
10
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
11
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)] .
12
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.
13
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
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.
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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
mλ
A =
√−mλh3 .
Telik�,
ψE =
√−mλh3 e
λmh3 |x| =
(−2mE
h2
)1/4
e−√− 2mE
h2 |x|.
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