Panepistmio Patrn Tmma Majhmatik Antnhc Strklac

Eisagwg

sthn

Kbantomhqanik Ptra 5/2/2008

J E

M A T

A

Jma

1.

(mon.1)

Na

gryete

kai

na

analsete

ta

aximata

thc

Kban-

tomhqanikc.

Na apodexete ti to .

2

enai anexrthto ap ton qrno

(i h t

2.

2

= 0)

Jma

(mon.2)

Na

orsete poic

ton

qro o

tou

Qlmpert. telestc

Ti (

onomzoume ). . Na Na

ermhtian

telest

kai

enai

monadiaoc

gryete nan ermhtian kai nan monadiao pnaka diastsewc exetsete an o diaforikc telestc

unitary 22

d/dx

enai ermhtianc.

Jma 3. megjh

(mon.3) An kai

T1

kai

T2

enai oi telestc pou antistoiqon se do

m1

m2

tte na apodeixete ti isqei h akloujh genikeumnh

sqsh abebaithtac

(m1 )(m2 ) Pte isqei h isthta. jshc

1 [T1 , T2 ] 2i1 2

Na efarmsete thn sqsh gia touc telestc thc

x

kai thc

kinhtikc enrgeiac

p2 x

.

Jma 4. peira

(mon.2) 'Ena swmatdio enai dsmio se na phgdi dunamiko me

toiqmata

V (x) = 0

gia

a < x < a

kai

V (a) = V (a) =

.

Na brejon

oi idiosunartseic kai oi

idiotimc thc energeac tou.

Jma

5.

(mon.2) .

'Ena An

swmatdio

kinetai

se

mia

distash

sto

disthma

a/2 x a/2

h kumatosunrths tou enai

(x, t) = N eit cos (3kx) sin3 (kx)na brejon

) ) )

Oi dunatc

timc tou

k

.

Ta

endeqmena gia

apotelsmata

thc

ormc,

thc

energeac

tou

kai

oi pijanthtec

kje endeqmeno

apotlesma.

Oi msec

timc thc ormc kai

thc jsewc tou swmatidou.

1

Ta JmataPanepistmio Patrn Tmma Majhmatik Antnhc Strklac Eisagwg sthn Kbantomhqanik Ptra 5/2/2008

J E M A T A

Jma 1.

(mon.1) Na gryete kai na analsete ta aximata thc Kban-

tomhqanikc. Na apodexete ti to

2

enai anexrthto ap ton qrno

(i h t

2

= 0).Ti onomzoume Na Na

Jma 2.

(mon.2) Na orsete ton qro tou Qlmpert.

unitary). gryete nan ermhtian kai nan monadiao pnaka diastsewc 2 2. exetsete an o diaforikc telestc d/dx enai ermhtianc.ermhtian telest kai poioc enai o monadiaoc telestc (

Jma 3. (mon.3) An megjh

T1

kai

T2

enai oi telestc pou antistoiqon se do

m1

kai

m2

tte na apodexete ti isqei h akloujh genikeumnh

sqsh abebaithtac

(m1 )(m2 )

1 [T1 , T2 ] 2i1 2

Pte isqei h isthta. Na efarmsete thn sqsh gia touc telestc thc jshc

x

kai thc kinhtikc enrgeiac

p2 . xkai

Jma 4. (mon.2) 'Ena swmatdio enai dsmio se na phgdi dunamiko me peira toiqmata

V (x) = 0

gia

a < x < a

V (a) = V (a) = .

Na brejon oi idiosunartseic kai oi idiotimc thc energeac tou.

Jma 5.

(mon.2) 'Ena swmatdio kinetai se mia distash sto disthma An h kumatosunrths tou enai

a/2 x a/2.

(x, t) = N eit cos (3kx) sin3 (kx)na brejon

) ) )

Oi dunatc timc tou

k.

Ta endeqmena apotelsmata thc ormc, thc energeac tou kai

oi pijanthtec gia kje endeqmeno apotlesma. Oi msec timc thc ormc kai thc jsewc tou swmatidou.

1

2

Oi ApantseicJma 1

Prto erthma. Ta aximata thc Kbantomhqanikc brskontai sthn pargrafo (4.2) tou biblou.

Axwma 1. To axwma anafretai sthn perigraf twn fusikn katastsewn. Sthn klassik fusik h katstash enc sustmatoc enai safc orismnh an xroume thn jsh kai thn orm tou. Sthn Kbantomhqanik

mia ttoia perigraf enai adnath lgw thc sqsewc abebaithtac tou Qizenmpergk. Ed h katstash tou sustmatoc perigrfetai ap mia sunrthsh

(r, t)

ap thn opoa mporome na proume lec tic plhroEnai na kma, onomzetai kumatosunrthsh

forec gia to ssthma.

kai ankei se nan katllhlo qro Qlmpert.

Axwma 2. jn.

To axwma anafretai sthn perigraf twn fusikn mege-

Sthn klassik fusik me ton ro fusik mgejoc ennoome mia

opoiadpote sunrthsh

F (r, p, t)

twn suntetagmnwn kai twn ormn.

Sthn Kbantomhqanik na mgejoc paristnetai ap nan grammik kai ermhtian telest. Ta apotelsmata twn metrsewn twn megejn enai oi idiotimc tou antstoiqou telest. Oi idiotimc autc enai pragmatikc gi aut kai o telestc prpei na enai ermhtianc.

Axwma 3. To axwma anafretai ston dunamik nmo thc jewrac. Epeid upologzoume msec timc gia thn exswsh kinsewc qoume do epilogc. Enai dunatn na jewrsoume ti h katstash

(r, t) exelssetai me ton

qrno en to mgejoc paramnei stajer. H exswsh kinsewc enai

i h

(r, t) t H = H(i , r, t) h

kai onomzetai exswsh tou Srntigker. O telestc enai o telestc tou Qmilton tou sustmatoc.

Enai dunatn na jewrsoume ti h katstash paramnei stajer en to mgejoc exelssetai me ton qrno. H exswsh kinsewc enai

i h

d T (r, t) = i T (r, t) + [T (r, t), H] h dt t

kai onomzetai exswsh tou Qizenmpergk.

2

Axwma 4.

To axwma aut enai h statistik ermhnea thc parapnw

tupopohshc thc kbantomhqanikc. Upojtoume ti gnwrzoume plrwc thn katstash enc sustmatoc dhlad to na mgejoc

kai jloume na metrsoume

T.

Sumbolzoume me

tstoiqec idiotimc tou megjouc

n kai n tic idiosunartseic kai tic anT . Grfoume thn katstash tou sust-

matoc san na grammik sunduasm thn idiosunartsewn autn stw ti

=

n

an n .

H anptuxh enai dunat diti o telestc tou meg-

jouc enai ermhtianc. Upojtoume ti h kumatosunrthsh enai kanonikopoihmnh sthn monda, dhlad

|an |2 = 1. Tte kata thn 2 mtrhsh tou megjouc ja emfaniste h idiotim k me pijanthta |ak | . H msh tim twn metrsewn dnetai ap ton tpo < |T >= T dV pou me to smbolo < | > sumbolzoume to eswterik ginmeno tou =nqrou tou Qlmpert. Axwma 5. To axwma aut anafretai sthn kbantomhqanik mtrhsh.

2

Met ap mia mtrhsh h katstash tou sustmatoc enai h idiosunrthsh thc idiotimc pou brjhke kata thn mtrhsh. 'Etsi an metrsoume pli to dio mgejoc amswc met, brskoume pli thn dia idiotim. Dhlad ap to jroisma twn

k

idiosunarsewn katalgoume met thn mtrhsh

se mia mno idiosunrthsh kai gi aut to axwma onomzetai arq tou filtrarsmatoc. Detero erthma. H apnthsh brsketai sto tloc thc paragrfou (4.3) tou biblou.

i h

t

2

= i h

< | >=< i | > + < |i >= h h t t t

< H| > + < |H >= < H| > + < |H >= 0H teleutaa isthta isqei diti o telestc thc Qamiltonianc enai ermhtianc.

Jma 2Prto erthma. O orismc tou qrou Qlmpert tou ermhtiano telest kai tou monadiaou telest brskontai stic paragrfouc (2.4),(3.5),(3.6) antistoqwc. Paradegmata. 1. Ermhtianc pnakac. An

Aij

enai ta stoiqea enc pnaka tte gia na enai ermhtianc prpei

na isqei h sqsh

Aij = A . ji

' ra kpoia pardeigmata enai A

3

A=pou

0 0 , , kai

A=

0 i i 0

A=

i + i

enai pragmatiko arijmo.

2. Monadiaoc pnakac. An

Uij

enai ta stoiqea enc pnaka tte gia na enai monadiaoc prpei

na isqei h genik sqsh

2 k=1

Uki Ukj =

2 k=1

Uik Ujk = ij .

Oi sqseic

autc enai oi akloujec tsereic exisseic:

i = 1, i = 2, i = 1, i = 2,

j=1 j=2 j=2 j=1

U11 U11 + U21 U21 = 1 U12 U12 + U22 U22 = 1 U11 U12 + U21 U22 = 0 U12 U11 + U22 U21 = 0

Epilgoume gia thn eukola mac kpoia stoiqea na enai sa me to mhdn.

U11 = 0 h teleutaa dnei U22 U21 = 0 ap thn opoa epilgoume U22 = U22 = 0. 'Etsi h trth ikanopoietai kai oi do prtec dnoun U21 U21 = 1 kai U12 U12 = 1. Dhlad ta stoiqea U12 kai U21 enai iAn gia pardeigma migadiko arijmo me mtro thn monda thc morfc paradegmata enai

e

.

' ra kpoia A

A=

0 ei ei 0

A=

0 ei ei 0

A=

0 1 1 0 = 0,

Gia ton teleutao pnaka, pou bganei ap ton prohgomeno gia

fanetai amswc ti o suzugoanstrofc tou all kai o antstrofc tou enai kai oi do soi me ton pnaka

A=

0 1 1 0

.

' lloc aplosteroc pnakac enai o pnakac monda akma kai o pnakac A

A=

1 0 0 1

.

Mia aplosterh skyh gia na brome ttoia paradegmata enai o exc.

1 1 enai o T = a a 0 pou a R. ' ra nac ermhtianc pnakac enai o A = A . 0 a 'Enac monadiaoc telestc me distash 1 1 enai o U = z pou z C kai |z| = 1. Gia pardeigma z = ei pou R. 'Ara nac monadiaoc pnakac ei 0 pwc kai prohgoumnwc enai o A = . 0 ei O diaforikc telestc T = d/dx den enai ermhtianc. Aut fanetai amswc ap ton telest thc ormc px = i(d/dx) pou enai ermhtianc. H Enai gnwst ti nac ermhtianc pnakac me distash 4

apdeixh petai

< 1 |T 2 >=< 1 |

d 2 >= dx

b a

1

d 2 dx = 1 2 |b a dx

b a

d 2 dx dx 1

pou gine mia kata pargontec oloklrwsh. faneiakc roc enai mhdn dhlad

An upojsoume ti o epi-

1 2 |b = 0 a

tte brskoume

< 1 |

d 2 >= 1 2 |b a dx

b a

d d 1 2 dx = < 1 |2 >=< T + 1 |2 > dx dxkai ra o telestc den enai

Epomnwc apodexame ti ermhtianc.

(d/dx)+ = d/dx

Jma 3To prto erthma enai to jerhma thc paragrfou (4.4). H isthta isqei tan h anisthta tou Sbartc isqei me to son, tan dhlad ta diansmata

kai h msh tim tou telest

|C1 > kai |C2 > enai sugrammik. Epiplon prpei A na enai sh me to mhdn dhlad < A >= 0.

Gia na apantsoume sto detero erthma brskoume prta ton metajth twn telestn

x

kai

p2 /2. x

1 1 1 1 h h h [x, p2 ] = [x, p2 ] = (px [x, px ] + [x, px ]px ) = (px (i ) + (i )px ) = i px x x 2 2 2 2Epomnwc brskoume

(m1 )(m2 )

1 (i px ) h 2i

=

h < px > 2

Jma 4To jma aut enai h skhsh me

5.1

tou biblou pou to

a/2

ed enai so

a.

Smfwna me thn skhsh aut (jtoume

a 2a)

h telik kfrash twn

idiosunartsewn kai twn idiotimn enai

n =

1 x cos ((2n + 1) ) a 2a n = 1 x sin ((2n) ) a 2a

En =

h2 2 (2n + 1)2 2 8ma h2 2 (2n)2 8ma2

n = 0, 1, 2,

En =

n = 1, 2,

5

Jma 5Apnthsh sto

)

erthma.

Epeid to ssthma den kintai xw ap to dedomno disthma h kumatosunrths tou enai sh me to mhdn. ' ra prpei kai sta shmea A

a/2kai

na

mhdenzetai tsi ste na enai suneqc. 'Ara

2 cos (3ka/2) = 0 3ka/2 = + k = 3a + 2 2 sin3 (ka/2) = 0 sin (ka/2) = 0 ka/2 = k = pou to = 1, 2, 3, enai nac fusikc arijmc.Apnthsh sto

2 a

)

erthma.

Gia na apantsoume sto erthma prpei na analsoume thn dedomnh kumatosunrthsh se na jroisma twn idiosunartsewn tou telest thc ormc. Enai gnwst ti oi idiosunartseic autc enai timc

k = eikx

me antstoiqec ido-

k = hk .

Ja qrhsimopoisoume touc akloujouc tpouc

cos (3kx) =

1 3ikx e + e3ikx 2

sin (kx) =

1 ikx e eikx 2i

Met ap aplc prxeic brskoume

=

N eit 3e2ikx + 3e2ikx + 3e4ikx 3e4ikx e6ikx + e6ikx 2(2i)3 M athematicame thn entol

Ed oi prxeic ginan me to prgramma

Expand[T rigT oExp[Cos[3kx]Sin[kx]3 ]]Epomnwc oi timc pou mpore na prei h orm enai oi akloujec

p1 = 2 k , p2 = 2 k , p3 = 4 k , p4 = 4 k , p5 = 6 k h h h h hOi stajerc anaptxewc enai

kai

p6 = 6 k . h

c1 = 3c, c2 = 3c, c3 = 3c, c4 = 3c, c5 = c kai c6 = c, it 1 . pou c = N e 2(2i)3 6 2 2 2 ' ra brskoume A k=1 |ck | = (9 + 9 + 9 + 9 + 1 + 1)|c| = 38|c|oi antstoiqec pijanthtec twn parapnw idiotimn enai

kai

P1 = P2 = P3 = P4 =so me thn monda.

9 kai 38

P5 = P6 =

1 . 38

Enai faner ti to jroisma lwn twn pijanottwn

9 1 (4 38 + 2 38 )

enai

H kinhtik enrgeia xroume ti qei tic diec idiosunartseic me thn orm

k = eikx diti oi do telestc enallssontai. Oi antstoiqec h 2 k2 idiotimc enai Ek = . To fsma twn idiotimn enai ekfulismno kai h 2m ikx idiotim aut qei do idiosunartseic, h deterh enai h k = e .dhlad tic Epomnwc oi idiotimc thc kinhtikc enrgeiac enai oi akloujec

E1 =

h 2 (2k)2 2m

=

4 2 k2 h , 2m

E2 =

h 2 (4k)2 2m

=6

16 2 k2 h kai 2m

E3 =

h 2 (6k)2 2m

=

36 2 k2 h 2m

Oi antstoiqec pijanthtec enai oi akloujec

R1 = R2 =

9 38

+

9 38

=

18 kai 38

R3 =

1 38

+

1 38

=

2 38

To jroisma twn pijanottwn enai so me thn monda

2

18 38

+

2 38

= 1.

To jma den dieukrinzei an zhtei thn kinhtik enrgeia thn sunolik enrgeia. H apnthsh gia thn sunolik enrgeia enai apl. Gnwrzoume ti h idiosunrthsh tou telest thc enrgeiac autc me antstoiqh idiotim

E = i t h

enai h

= eit

= h . h

Epeid h dedomnh kumatosunrthsh enai

idiosunrthsh tou telest thc enrgeiac, brskoume ti h monadik pijan tim thc enrgeiac enai sh me Apnthsh sto me pijanthta sh me thn monda profanc.

)

erthma.

Gia ton upologism thc mshc timc thc ormc qrhsimopoiome ton klassik orism thc statistikc. Brskoume

6

< p >=k=1

Pk pk = 2 k h

9 9 9 1 1 9 + 2 k 4 k + 4 k 6 k + 6 k = 0 h h h h h 38 38 38 38 38 38

Gia ton upologism thc mshc timc thc jshc qrhsimopoiome ton orism thc kbantomhqanikc. Brskoume

a/2

< x >=a/2

xdx =

a/2 a/2

x dx = |N |

a/2 a/2

x cos2 (3kx) sin6 (kx)dx

Parathrome ti h up oloklrwsh sunrthsh enai peritt kai epomnwc to oloklrwm thc, stw h sunrthsh

F (x),

enai rtia. Brskoume

< x >= F (x)|a/2 = F (a/2) F (a/2) = 0H teleutaa isthta enai o orismc thc rtiac sunrthshc. O upologismc me ton trpo aut den ja mporose na gnei an ta ria den tan summetrik wc proc to mhdn. O analutikc upologismc gnetai me thn bojeia tou upologist. Gia thn msh tim thc ormc h entol tou

a/2

M athematica enai Integrate[Cos[3kx]Sin[kx]3 (i )D[Cos[3kx]Sin[kx]3 , x], {x, a/2, a/2}] hkai gia thn msh tim thc jshc h entol enai

Integrate[xCos[3kx]2 Sin[kx]6 , {x, a/2, a/2}].Ta apotelsmata pou dnei o upologistc kai ta do enai sa me to mhdn.

7

Panepistmio Patrn Tmma Majhmatik Ptra 6/2/2012

Esagwg

sthn

Kbantomhqanik A. Strklac

J E

M A T

A

1.

(m. ko gia

2.5)

Pwc

perigrfoume Giat

thn h

katstash perigraf orzontai gryete

kai den h

ta

megjh na

enc

klassikai

sustmatoc. ta stoiqeidh

aut

mpore

epektaje kai ta

swmtia.

Pwc Na

katstasic exswsh

megjh tou

enc

kbantiko

susthmatoc.

thn

thc

knhshc

Srntigker kai na apodexete thn exswsh thc knhshc tou Qizenmpergk.

2.

(m.

2.5)

Na

orsete

ta

megjh

puknthta ti thc

pijanthtac

kai

puknthta su-

rematoc neqeac. stseic

pijanthtac. Poi enai h

Na

apodexete shmasa kai

ikanopoion exswshc

thn

exswsh Poic to

fusik

autc. an

kata-

onomzontai

stsimec

giat.

Na

exetsete

jroisma

do stsimwn

katastsewn enai stsimh.

3.

(m. st.

1.5)

Na

orsete

ton

ermhtian

kai

ton

monadiao

(

unitary

)

tele-

Na apodexete ti oi idiotimc tou prtou enai pragmatikc kai tou mkoc thn monda.

deterou migadiko arijmo me

4.

(m.

1.5)

Na

gryete thc

kai

na

lsete kai thc

tic

exisseic

thc

knhshc

twn

teletou .

stn

thc

jshc,

ormc

Qamiltonianc talantwt

sthn

parstash

Qizenmpergk, se

dunamik armoniko

pou

V (x) = kmx2

5.

(mon. An h

2) 'Ena swmtio kinetai se mia distash sto disthma kumatosunrths tou enai

a x a

.

(x, t) = N eit cos3 (kx) sin (3kx)na brejon

) )

Oi dunatc

timc tou

k

kai h

msh tim thc

jshc tou.

Ta endeqmena apotelsmata thc ormc kai thc kinhtikc enertou. Oi pijanthtec gia kje endeqmeno apotlesma kai oi

geac

msec timc

twn

megejn autn.

Panepistmio Patrn

Eisagwg sthn

Tmma Majhmatik

Kbantomhqanik

Antnhc Strklac

Ptra 13/9/2010

J E M A T A

1.

(m.3)

Pwc

perigrfoume

ta

fusik

sustmata

kai

pwc

ta

fusik

megjh

thc

kbantomhqanikc.

Poi

enai

h

antstoiqh

perigraf

thc

klassikc

fusikc

kai

giat den efarmzetai ston mikrkosmo.

Poic enai o kbantikc nmoc thc kinshc

twn

susthmtwn

kai

poic

twn

megejn.

Na

apodexete

ti

oi

do

auto

nmoi

enai isodnamoi.

2.

(m.1)

Na

gryete

touc

telestc

thc

jshc

kai

thc

ormc

sthn

p

parstash.

Na upologsete ton metajth

[q 2 , p3 ]

.

3.

(m.1) Na orsetai tic

katanomc

1 2a [(x

a) + (x + a)]sthn

kai

H(t |x|)morf

.

4.

(m.2)

Na

gryete

thn

kumatosunrthsh

polik

thc

dhlad

(r, t) = (r, t)ethn exswsh thc

i(r,t)

.

Na brete to rema puknthtac pijanthtac.

Na gryete

suneqeac

kai

na

ermhnesete

to

apotlesma

smfwna

me

thn

anlogh sqsh thc udrodunamikc.

5.

(m.2) 'Ena swmatdio kinetai ap aristra proc ta dexi se na phgdi dunamiko

V (x) = 0me

gia

enrgeia

a < x < a E >V. Na

kai

V (x) = V (x) = V > 0to prblhma smfwna

gia

x < a

kai

x>athn

lsete

me

thn

klassik

kai

kbantik mhqanik.

Poic enai oi diaforc twn sumperasmtwn twn do apyewn.

6.

(m.2) 'Ena swmatdio kinetai se mia distash sto disthma

a/2 x a/2

.

An h kumatosunrths tou enai

(x, t) = N eit cos (3kx) sin3 (kx)na brejon

) ) )

Oi dunatc timc tou

k

.

Ta

endeqmena

apotelsmata

thc

ormc,

thc

kinhtikc

energeac

tou,

oi

pijanthtec gia kje endeqmeno apotlesma kai oi msec timc touc.

H msh tim thc jsewc tou swmatidou.