KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES SHMEIWSEIS ?· KBANTOMHQANIKH-I SUNOPTIKES-PROQEIRES SHMEIWSEIS-3…

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KBANTOMHQANIKH-I

SUNOPTIKES-PROQEIRES SHMEIWSEIS-3

Kaj. Kurikoc Tambkhc

PERIEQOMENA1. Kat Tmmata Stajer Dunamik.2. To Bma Dunamiko.3. To Frgma Dunamiko (Fainmeno Sraggoc).4. To Tetragwnik Phgdi (Dsmiec Katastseic).5. To Dunamik Sunrthshc Dlta.

1

1. Kat Tmmata Stajer Dunamik.Mia tuqaa (omal) sunrthsh dunamiko mpore na proseggisje ap -

na kat tmmata stajer dunamik sthn epijumht prosggish (sqma sthnepmenh selda). H eplush thc anexrththc tou qrnou exswshc tou Sch-roedinger gia na swmatdio enrgeiac E se mia ttoia perioq [a, b] sthnopoa h sunrthsh dunamiko qei stajer tim V0 enai apl (detero sqmaepmenhc seldac). Ac upojsoume E > V0. H exsws mac enai

h2

2m

d2Edx2

+ V0 E = E E , (a x b) .

Grfontac thn exswsh tou Schroedinger sthn morf thc antstoiqhc exsw-shc gia to elejero swmatdio,

h2

2m

d2Edx2

= (E V0) E ,

blpoume ti qoume tic diec lseic epipdwn kumtwn

eihpx, e

ihpx ,

pou, mwc

p =

2m

h2(E V0)

E = V0 +p2

2m,

pwc, exllou ja anamname sthn perptwsh thc Klassikc Fusikc. epom-nwc, sthn perioq [a, b] h lsh ja enai

E(x) = Aeihpx + B e

ihpx

me sunoriakc sunjkec

E(a) = Aeihpa + B e

ihpa, E(b) = Ae

ihpb + B e

ihpb

E(a) =i

hp(Ae

ihpa B e

ihpa), E(b) =

i

hp(Ae

ihpb B e

ihpb),

pou E(a), E(a), E(b),

E(b) enai oi timc thc kumatosunrthshc kai thc

paraggou thc (gia tic perioqc x a kai x b) sta sunoriak shmea.H sunqeia thc paraggou sta shmea a, b proupojtei ti to dunamik denapeirzetai sta shmea aut. Shmeiste mwc ti asunqeia sto dunamikepitrpetai.

2

V(x)

a b

V0

3

2. To Bma Dunamiko.Mia pol apl perptwsh dunmewn enai dunmeic pou exaskontai mno se

na shmeo. Ep paradegmati, en h perioq sthn opoa kinetai na swmatdioqwrzetai se do tmmata me stajer all diaforetik dunamik, to swmatdioden ja dqetai dunmeic stic perioqc autc all mno sto shmeo sunnthsctouc pou kai to dunamik ja parousizei asunqeia. To aplostero ttoiopardeigma enai to Bma Dunamiko

V (x) =

0 < x < 0

V0 0 < x <

(Sqma mejepmenhc seldac) Oi epitrepmenec timc thc enrgeiac enai 0 E V0 kai E V0. H perptwsh E < 0 den odhge se fusik apodektclseic. Sqetik me to aut isqei to akloujo {Jerhma}: Den uprqounfusik apodektc lseic thc exswshc tou Schroedinger, oi opoec na antistoi-qon se enrgeia mikrterh ap to elqisto tou dunamiko.

2.1. Perptwsh E > V0.H exswsh Schroedinger enai stic do perioqc

h22md2Edx2

= E E (x < 0)

h22md2Edx2

+ V0 E = E E (x > 0)

E =

p2

h E (x < 0)

= q2

h E (x > 0)

pou

E =p2

2m, E V0 =

q2

2m.

Oi lseic enai

E(x) =

Ae

ihxp + B e

ihxp (x 0)

C eihxq + De

ihxq (x 0)

H ermhnea kje rou enai h akloujh:

4

O roc Aeihxp antistoiqe se na {kma prosppton ap arister}.

O roc B eihxp antistoiqe se na {anaklmeno kma}.

O roc C eihxq antistoiqe se na {dierqmeno kma}.

Tloc, o roc Deihxq antistoiqe se na {kma pou prospptei ap dexi}.

Qwrc blbh thc genikthtac, mporome na aplopoisoume thn anlusmac periorizmenoi se swmatdia, ta opoa prospptoun mno ap arister.Aut isoduname me thn epilog

D = 0 .

Tte, h lsh mac enai

E(x) =

Ae

ihxp + B e

ihxp (x 0)

C eihxq (x 0)

H sunqeia thc kumatosunrthshc kai thc paraggou thc sto shmeo a-sunqeiac tou dunamiko x = 0 dnei

A + B = C

pA pB = q C .

Epilontac aut to ssthma parnoume

B

A=

p qp+ q

,C

A=

2p

p+ q.

Telik, h kumatosunrthsh enai

E(x) =

A(eihpx +

(pqp+q

)e

ihxp)

(x 0)

A(

2pp+q

)eihqx (x 0)

Katarqn parathrome ti uprqei merik anklash parlo pou h enr-geia enai megalterh ap to frgma dunamiko. Aut den ja mporose nasumbanei gia na {klassik} swmatdio. Antijtwc, enai na anamenmenofainmeno gia kmata.

5

V(x)

0

V0

V(x)

E

Ac upologsoume thn puknthta rematoc pou antistoiqe se kje kommtithc kumatosunrthshc. To prosppton rema, ofeilmeno ston ro Aeipx/h,enai

J = |A|2p

m.

To anaklmeno rema, ofeilmeno ston ro (...) eipx/h, enai

J = p

m|B|2 = p

m|A|2

(p qp+ q

)2.

Tloc, to dierqmeno rema sthn perioq x 0 enai

J =q

m|C|2 = q

m|A|2

(2p

p+ q

)2.

'Opwc kai sthn Optik, pou to anlogo mgejoc proc thn puknthtarematoc enai h ntash, tsi kai ed mporome na orsoume ton SuntelestAnklashc wc

R |J|J

=|B|2

|A|2=

(p qp+ q

)2.

O Suntelestc Anklashc ekfrzei to posost twn anaklwmnwn swmatid-wn. O Suntelestc Dileushc T orzetai wc

T JJ

=q

p

|C|2

|A|2=

(2p

p+ q

)2.

O Suntelestc Dileushc ekfrzei to posost twn dierqomnwn swmatidwn.Enai anamenmeno ta anaklmena swmatdia kai ta dierqmena prostij-

mena na mac dnoun ton sunolik arijm twn swmatidwn, opte ja prpei naisqei h sqsh

R + T = 1 .

Antikajistntac tic ekfrseic twn R kai T , blpoume ti h sqsh aut ika-nopoietai tautotik. Uprqei kai nac lloc trpoc na katanosoume autthn sqsh. H exswsh sunqeiac gia tic kumatosunartseic E(x) parnei thnmorf diatrhshc tou rematoc

dJdx

= 0 = J = const.

J (x < 0) = J (x > 0) = J + J = J = J |J| = J

7

1 |J|J

=JJ

= 1 R = T = R + T = 1 .

2.2. Perptwsh E < V0.Sthn perptwsh pou na {klasik} swmatdio enrgeiac E prospptei se

na frgma dunamiko youc V0 > E pntote anakltai. Ac dome poiaenai h problepmenh kumatosunrthsh sthn perptwsh aut. H exswsh touSchroedinger enai

h22md2Edx2

= E E (x < 0)

h22md2Edx2

+ V0 E = E E (x > 0)

E =

p2

h E (x < 0)

= s2

h E (x > 0)

pou

E =p2

2m, V0 E =

s2

2m.

Shmeiste thn allag prosmou sthn exswsh tou Schroedinger gia thn pe-rioq x > 0. Oi lseic enai

E(x) =

Ae

ihxp + B e

ihxp (x 0)

C exs/h + Dexs/h (x 0)

H ermhnea kje rou enai h akloujh:

O roc Aeihxp antistoiqe se na {kma prosppton ap arister}.

O roc B eihxp antistoiqe se na {anaklmeno kma}.

Oi roi C exs/h kai Dexs/h den qoun thn ermhnea diadidomnwn kum-twn.

Mlista, o rocDexs/h auxnetai ekjetik kai apoklnei sto peiro, stena enai fusik apardektoc. Sunepc, mno h epilog

D = 0

8

enai fusikc apodekt.Epiblontac thn sunqeia thc kumatosunrthshc kai thc paraggou thc

sto shmeo x = 0, parnoume

A + B = C

ip (A B) = sC

B

A=

p+ is

p is,

C

A=

2p

p is.

Telik, h kumatosunrthsh enai

E(x) =

A(eihpx +

(p+ispis

)e

ihpx)

(x 0)

A(

2ppis

)esx/h (x 0)

Parathrome ti h kumatosunrthsh enai mh-mhdenik sthn {klassika apa-goreumnh perioq}. Mpore rage na parathrhje to swmatdio sthn perioqaut? Ac shmeiwje ti lgw thc ekjetikc ptshc thc kumatosunrthshc hpijanthta parousac tou swmatidou enai upologsimh mno gia thn perioq

x hs.

'Exw ap aut thn perioq h pijanthta fjnei tqista. Ap thn Arq thcAbebaithtac mwc h abebaithta sthn orm ja enai

p hx s .

H antstoiqh abebaithta sthn enrgeia ja enai

E (p)2

2m s

2

2m= V0 E = E + E V0 .

Aut mwc shmanei ti h abebaithta sthn enrgeia anebzei to swmatdioxw apo thn (klassik apagoreumnh) energeiak znh 0 < E < V0. Par-lo pou sto sugkekrimno ssthma, pou h {klassik apagoreumnh perioq}enai peirh, o mh-mhdenismc thc kumatosunrthshc eke den qei sunpeiec,sto ssthma tou tetragwniko frgmatoc dunamiko pou ja exetsoume pa-raktw, to gegonc aut jtei tic proupojseic gia to perfhmo FainmenoSraggoc.

9

Mia epaljeush tou gegontoc ti den uprqei didosh gia x 0 par-qetai ap ton mhdenism tou rematoc pijanthtac sthn perioq aut

J (x > 0) = 0 .

Ap thn diatrhsh rematoc parnoume ti

J + J = 0 = JJ

= 1 = R = 1 .

To apotlesma aut shmanei ti qoume olik anklash kai sunodeetai apton mhdenism tou suntelest dileushc T = 0 .

3. To Frgma Dunamiko (Fainmeno Sraggoc).Ac jewrsoume tra na swmatdio enrgeiac E > 0, to opoo kinetai sto

dunamik

V (x) =

0 x < a

V0 a < x < a

0 x > a

Ja exetsoume thn perptwsh E < V0. H exswsh tou Schroedinger enai

h22md2Edx2

= E E (x < a)

h22md2Edx2

+ V0E = E E (a < x < a)

h22md2Edx2

= E E (x > a)

E = p2

h E (x < a)

E =s2

h E (a < x < a)

E = p2

2mE (x > a)

pou

E =p2

2m, V0 E =

s2

2m.

10

Oi lseic enai

E(x) =

Aeihxp + B e

ihxp (x a)

C exs/h + Dexs/h (a x a)

F eihxp + Ge

ihxp (x a)

Sto shmeo aut ac knoume tic exc do parathrseic:1) Sthn peperasmnh perioq [a, a] den suntrqei lgoc anexlegkthc

axhshc thc kumatosunrthshc kai, epomnwc, kai oi do roi C exs/h kaiDexs/h enai fusikc apodekto.

2) O roc Aeihxp antistoiqe se swmatdio pou prospptei ap arister en

o roc Geihxp antistoiqe se swmatdio pou prospptei ap dexi. Qwrc bl-

bhn thc genikthtoc mporome na epilxo