Giáo trình Cơ học lượng tử

7/31/2019 Gio trnh C hc lng t

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TRNG AI HOC A LAT

GIAO TRNH

C HOC LNG T

ThS. NGUYEN DUY HNG

1998


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C hoc lng t 1 -

MUC LUCMUC LUC ............................................................................................................- 1 -LI NOI AU ......................................................................................................- 4 -BAI M AU ......................................................................................................- 5 -

1 LC O TONG QUAT CUA VAT LY HOC...........................................- 5 -2 BC TRANH THE GII CUA VAT LY HOC CO IEN.........................- 5 -I/ Hai y tng c ban cua vat ly hoc co ien :...............................................- 5 -II/ Hai bo phan chu yeu cua vat ly hoc co ien : ..........................................- 6 -III/ Hai dang vat chat c ban cua vat ly hoc co ien :...................................- 7 -IV/ Nhng quan niem c s cua vat ly hoc co ien : ....................................- 7 -

3 NHNG BE TAC CUA VAT LY HOC CO IEN VA NHNG Y TNGNA LNG T.............................................................................................- 8 -

CHNG I. C S CUA C HOC LNG T .............................................- 10 -1 C S VAT LY CUA C HOC LNG T ...........................................- 10 -

I/ The gii vi mo: ........................................................................................- 10 -2 HAI Y TNG C BAN CUA C HOC LNG T............................- 12 -

1/ Y tng lng t hoa. .............................................................................- 12 -2/ Y tng lng song hat. ..........................................................................- 13 -

3 C S TOAN HOC CUA C HOC LNG T....................................- 16 -I/.Toan t tuyen tnh : .................................................................................- 16 -II/ Giao hoan t va phan giao hoan t : ......................................................- 17 -III/. Bai toan tr rieng cua toan t tuyen tnh : ............................................- 18 -IV/. Mot so toan t at biet : .......................................................................- 19 -

V/.Toan t lien hp va toan t t lien hp (hecmitic) :...............................- 24 -4 TH NGHIEM QUAN TRONG TNH THONG KE CUA C HOC LNGT...................................................................................................................- 28 -

I/ Th nghiem hai lo : ..................................................................................- 28 -II/ Tnh thong ke cua CHLT: ......................................................................- 29 -

5 CAC BIEN ONG LC TRONG C HOC LNG T .......................- 30 -1/.Cac toan t toa o : .................................................................................- 30 -2/.Cac toan t xung lng : .........................................................................- 30 -3/. Cac toan t Moment xung lng L va toan t moment xung lng bnhphng L2 . .................................................................................................- 31 -4/.Toan t Hamilton H: ...............................................................................- 32 -

6 CAC HE THC BAT NH.....................................................................- 34 -I/ Y Tng Lng Song Hat Va cac He Thc Bat nh: ...........................- 34 -II/ Y ngha cua cac he thc bat nh: ..........................................................- 35 -III/ Mot so ket qua thu c t he thc bat nh : .......................................- 37 -IV . Xay dng he thc bat nh Heisenberg: ..............................................- 39 -

7 HAM SONG . NGUYEN LY CHONG CHAP TRANG THAI. ................- 44 -I/. Ham Song:..............................................................................................- 44 -II/. Nguyen ly chong chap trang thai: .........................................................- 45 -

8 PHNG TRNH SCHRODINGER...................................................- 46 -I/. Cach Thiet lap phng trnh: ..............................................................- 46 -

ThS. Nguyen Duy Hng Khoa Vat ly


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C hoc lng t 2 -

9 CAC PHNG TRNH CHUYEN ONG LNG T..........................- 49 -I/ Nhan xet chung : .....................................................................................- 49 -II/ Cac moc Poisson lng t : ....................................................................- 50 -III/ ao ham theo thi gian cua cac toan t :..............................................- 52 -

IV/ Cac phng trnh chuyen ong lng t . nh ly Ehrenfest . ..............- 54 -10 S LIEN HE GIA C HOC LNG T VI C HOC CO IEN VAQUANG HOC.................................................................................................- 56 -

I/ S chuyen t phng trnh Schrodinger hien ai ve phng trnh Hamilton Jacobi co ien : ...........................................................................................- 56 -II/ C hoc lng t va quang hoc : ..............................................................- 60 -

11 CAC CACH PHAT BIEU C HOC LNG T...................................- 61 -I/ C hoc lng t cua Schrodinger: ...........................................................- 61 -II/ C hoc ma tran Heisenberg: ..................................................................- 62 -III/ C hoc lng t cua P.Dirac : ...............................................................- 62 -

IV/ C hoc lng t cua R. Feynman :........................................................- 64 -12 CAC CACH MO TA S PHU THUOC THI GIAN CUA HE VI MO.- 65 -

1/ Bc tranh Schrodinger : ..........................................................................- 66 -2/ Bc tranh Heisenberg:............................................................................- 67 -3/ Bc tranh tng tac : ...............................................................................- 67 -4/ So sanh hai bc tranh c ban: (bc tranh Schrodinger va bc tranhHeisenberg ) . .............................................................................................- 67 -

13 CAC BIEU DIEN TRONG C HOC LNG T .............................- 68 -

I/ C hoc lng t trong F bieu dien : .......................................................- 68 -

II/ Vai bieu dien cu the:..............................................................................- 71 -CHNG II. MOT SO BAI TOAN C BAN CUA PHNG TRNHSCHRODINGER................................................................................................- 73 -

1 CHUYEN ONG T DO.........................................................................- 73 -2 BAI TOAN MOT CHIEU..........................................................................- 75 -

I/ Hat trong ho the ( gieng the ) sau vo han : ..............................................- 75 -II/ The bac thang: .......................................................................................- 76 -III/ S truyen qua hang rao the co be rong hu han: .................................. - 78 -IV/ Ho the nang co cac thanh cao hu han .................................................- 79 -

3 DAO ONG T IEU HOA ....................................................................- 82 -4 CHUYEN ONG TRONG TRNG XUYEN TAM. NGUYEN THYDRO..........................................................................................................- 84 -

I/ Momen goc : (Momen ong lng )........................................................- 84 -II/ Hat trong trng oi xng cau (chuyen ong xuyen tam)...................... - 87 -III/ Nguyen t Hydrogen : ..........................................................................- 89 -

CHNG III. LY THUYET SPIN CUA PAULI ...............................................- 92 -1 SPIN CUA ELECTRON............................................................................- 93 -

1/ Cac toan t spin : ....................................................................................- 93 -2. Cac tnh chat cua cac toan t Pauli .........................................................- 96 -

3. Vecto Spin ..............................................................................................- 96 -2 PHNG TRNH PAULI........................................................................- 100 -



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C hoc lng t 3 -

CHNG IV. PHNG PHAP NHIEU LOAN..............................................- 102 -1 LY THUYET NHIEU LOAN DNG KHONG SUY BIEN . .................- 102 -2 HIEU NG ZEEMANN ..........................................................................- 105 -3 LY THUYET NHIEU LOAN DNG CO SUY BIEN HIEU NG STARK ... -

108 -I/ Ly thuyet nhieu loan dng co suy bien : ...............................................- 108 -II/ Hieu ng Stark trong nguyen t Hydrogen : ........................................- 109 -

CHNG V. HE CAC HAT ONG NHAT .................................................... - 115 -1 TOAN T HOAN V. NGUYEN LY PAULI.......................................... - 115 -

1/ Toan t hoan v .....................................................................................- 115 -2/ Cac hat Bose va cac hat Fermi (cac boson va cac Fermion) ................- 116 -3/ Ham song cua he ong nhat khong tng tac. Nguyen ly Pauli. ..........- 116 -

2 NANG LUNG TRAO OI VA NGUYEN T HELI............................- 118 -I/. nh ngha nang lng trao oi : ..........................................................- 118 -

II/.Nguyen t Heli :...................................................................................- 118 -BAI KET ..........................................................................................................- 122 -

I/. Cac nguyen ly va bai toan c ban cua c hoc lng t : ..........................- 122 -1/. Ha m Song Va Nguyen Ly Chong Chat Trang Thai. ......................- 122 -2/ Gia Tr Trung Bnh: ..............................................................................- 123 -3/ Bai toan tr rieng va cac gia tr o c cua cac ai lng vat ly trong thcnghiem......................................................................................................- 123 -4/ Phng trnh c ban: .............................................................................- 123 -5/ Nguyen ly khong phan biet cac hat ong nhat:.....................................- 124 -

II/ Nhng chan tri mi hay la s phat trien tiep tuc cua C hoc lng t:- 125 -Tai lieu tham khao: ..........................................................................................- 127 -



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C hoc lng t 4 -

LI NOI AU

Cac ban sinh vien Vat Ly than men !

Cuon giao trnh c hoc lng t nay c soan danh cho cac ban. ay la schat loc t hau het cac giao trnh c hoc lng t a c soan bi nhng nha vat lynoi tieng tren the gii cung nh Viet Nam. Bi le c hoc lng t la thanh tu vai cua tr tue nhan loai the ky th 20,a va van ang la c s cua cac mui nhoncua vat ly hoc tham ch ngay ca trong the ky 21 sap ti.

Tuy nhien do: C hoc lng t la mon rat kho, rat phc tap tham ch kyquac, nen viec soan mot giao trnh cho de hieu sang sua trong khuon kho 100trang giay la ieu vo cung kho khan tham ch la ieu khong the thc hien t nhatla oi vi toi. Khong nhng the khuon kho cua giao trnh nay la tng ng vi 60

tiet hoc trong chng trnh hoc cua khoa vat ly hien nay. V vay mac du toi a hetsc co gang va bo nhieu cong sc cuon sach nay khong the c xem la mot giaotrnh hoan chnh va no cang khong the thay the cho viec nghe giang cua cac ban.

Toi xin lu y cac ban oi ieu ve ac iem cua giao trnh nay .Pham vi ngdung cua C hoc lng t la vo cung rong rai , v du nh : Hoa hoc lng t , Lythuyet trng Lng t , Ly thuyet hat nhan va Cau truc hat nhan nguyen t, Lythuyet chat ran , ien t hoc lng t va con nhieu nganh khac na .Chnh v thee hieu that s ay u ve C hoc lng t phai trnh bay trong nhng tai lieu khao so . Tuy nhien do nhieu ly do xuat phat t thc tien hoc tap va giang day cuachung ta hien nay , giao trnh nay ch chu yeu tra li cho cau hoi :C hoc lng t

ra i t au ?hay C s cua C hoc lng t la g ?(chng I) . Cong cu toanhoc cua C hoc lng t ch c gii thieu nhng net c s (toan t tuyen tnh tlien hp )va nhng tnh toan cho nhng van e cu the c xem la th yeu va oikhi ch trnh bay di dang gii thieu noi dung phng phap ch khong ap dung vaonhng bai toan cu the (Phng phap nhieu loan )

Toi that xuc ong khi nhng dong cuoi cung cua giao trnh nay c viet vaongay 20-11-1998- ngay ma t nhieu nam nay cac ban thng danh cho toi nhngli chuc mng tot ep, V the toi muon cac ban xem giao trnh nay nh la li camn cua toi en vi cac ban .Hn the na toi cung hy vong rang ay cung la bieu

hien cua long biet n chan thanh va sau sac cua toi oi vi biet bao thay co giao ,nhng ngi a thap len ngon la khat vong tm hieu the gii t nhien trong tamhon toi ma trong so o co khong t ngi ma vnh vien khong bao gi toi co the gaplai c na .

Chuc cac ban gat hai nhieu thanh cong trong hoc tap.

Ngay 20-11-1998.

Nguyen Duy Hng



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C hoc lng t 5 -

BAI M AU

1 LC O TONG QUAT CUA VAT LY HOC

Vat ly hoc cho ti ngay nay da tren hai hoc thuyet vat ly ln o la :Thuyettng oi cua Einstein va ly thuyet lng t.

1/ Ly thuyet tng oi cua Einstein c ac trng bi hang so c la van tocanh sang trong chan khong . Ve mat o ln c = 300000 km/s. Trong ly thuyet tngoi c la van toc truyen song ien t va cho ti nay no c xem la van toc gii hancua moi chuyen ong .

2/ Ly thuyet lng t (M.plank,N.Bohr) c ac tng bihang so Plank . Hang so nay dung lam o o s phan lap cac gia tr kha d cac ailng vat ly va la cau noi hai mat song hat cua chuyen ong cua vat chat .

[h] = [nang lng] [thi gian] = [tac dung]Plank a goi h la lng t tac dung .

H = 6,62517.10-37 JsDa vao hai hang so nay ta co the thiet lap lc o tong quat cua vat ly hoc

nh sau .v


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C hoc lng t 6 -

- Y tng th nhat la y tng nguyen t .Theo y tng nay th vat chat ccau tao t nhng hat rat nho khong the phan chia c va c goi la nguyen t(Atomos).

- Y tng th hai la y tng ve s ton tai cua mot moi trng an hoi ac biet

cha ay khap ni trong khong gian va nh o ma cac nguyen t tng tac c vinhau .- Y tng nguyen t c e xuat t rat sm (the ky th IV trc cong nguyen

- Democrite)va cho ti the ky 20 a co nhng bang chng tuyet oi khong phunhan c ve s ton tai thc s cua nguyen t va phan t . Con ngi a co nhngdung cu dung e ghi lai c nhng ion rieng re - cac ong em - va nhng dung cuquan sat c quy ao cua cac hat (buong Wilson) Tham ch quan sat c bangmat va chup anh c nhng phan t ln rieng re cua mot vai hp chat hu c .

II/ Hai bo phan chu yeu cua vat ly hoc co ien :

1/ C hoc Newton :C hoc Newton - da tren ba nh luat cua Newton - la mot bo phan rat quantrong cua vat ly hoc co ien . Trong ba nh luat o nh luat th hai co vai tro acbiet . o la phng trnh :

amFrr

= hay la : Lc = khoi lng gia toc.Phng trnh nay co vai tro ac biet v ve trai la lc la nguyen nhan gay ra s

bien oi van ong , ve phai co khoi lng tc la thuoc tnh cua vat chat va gia toctc la he thc gia khong gian va thi gian .ieu nay co ngha la phng trnh nay

thiet lap moi quan he gia vat chat,van ong, Khong gian,thi gian va nguyen nhangay ra van ong.Cung vi nh luat hap dan, ba nh luat tren a cho phep con ngi giai thch

c mot cach thoa ang chuyen ong cua nhng he v mo trung hoa ien2/Ly thuyet ien t

Mot bo phan chu yeu khac cua vat ly hoc co ien nghien cu cac hien tng

ien va t . Cac hien tng nay c cho bang cac ai lng ien t.Cac ai lng nay c he thong nh he phng trnh Maxwell :

HBDErrrr

,,,

EJHBED

B

D

t

DJH

t

B

E

rrrrrr

r

r

rrr

rr

===

=

=

+=

=

,,

*

*

0

T he phng trnh nay co the giai quyet thoa ang tat ca cac bai toan ien t

a biet .



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C hoc lng t 8 -

3 NHNG BE TAC CUA VAT LY HOC CO IEN VA NHNGY TNG NA LNG T

Mat du giai quyet c mot so lng rat ln cac hien tng cua the gii vatchat , song cho ti cuoi the ky 19 vat ly hoc co ien a vap phai mot so hien tngma trong khuon kho cac nh luat a co vat ly hoc co ien khong the giai quyetc . o la cac hien tng sau :

1/ Bc xa cua vat en tuyet oi .2/ Hien tng quang ien .3/ Hieu ng Compton.

4/ Cau tao nguyen t va ly thuyet na lng t cua Bohr.e giai quyet nhng van e tren vat ly hoc phai a ra nhng quan niem mi

vt xa khuon kho cua nhng quan niem trc ay . Tng ng vi nhng hientng tren,o la nhng quan niem sau ay :

1/ Cac nguyen t cua vat chat khong hap thu va bc xa nang lng mot cachlien tuc ma ngc lai hap thu va bc xa mot cach gian oan cac lng t nanglng .

h== h h la hang so pho bien -lng t tac dung-hangso Plank2/ Anh sang la mot chum hat- lng t anh sang- hay photon. Cac photon co

nang lng xac nh va xung lng xac nh. h= , kP

rh

r=

He thc nay c goi la he thc Planck Einstein. Nh vay he thc nay a

lien he cac thong so cua hat vi cac thong so cua song ( va Pr

vi va kr

).3/ T ket qua thu c cua hieu ng Compton:

222 sinc=

Trong o: c la bc song Compton cua electron.Cong thc tren co ngha la: S thay oi cua bc song ch phu thuoc vao goc

tan xa ma khong phu thuoc vao tan so ban au (cua tia ti). Ket qua nay c giaithch de dang t s va cham cua mot photon vi mat electron ma khong the giaithch theo quan iem song.

4/ Cac ien t trong nguyen t khong chuyen ong tren nhng qu ao batkyma ch co the nhng qu ao xac nh goi la qu ao lng t. Cac qu ao nayc nhan sao cho moment xung lng M cua ien t thoa man he thc:

hnM = (n=1,2 . . .)Tren cac qu ao lng t cac electron co nang lng xac nh. Khi electron

chuyen t qu ao nay sang qu ao khac gan hat nhan hn nguyen t se phat ra



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C hoc lng t 9 -

mot photon va electron thc hien mot bc nhay lng t. Tan so cua photon phatra c tnh theo cong thc:

nmmn EE h= 5/ Nam 1927, C.Davison va L.Germer phat hien hien tng nhieu xa cua

electron. Hien tng nay a c DeBroglie tien oan t 1924. Hien tng nay chco the giai thch c bang nhng gia thiet hoan toan mi so vi nhng quan niemcu cua vat ly co ien o la viec tha nhan gia thiet cua De Broglie: hat electron vavi hat noi chung la co tnh chat lng nguyen:hay lng tnh song hat. Cu the la:

Moi hat t do co nang lng E va xung lng Pr

xac nh c bieu dien bi

mot song phang n sac co tan so vongva vector song kr

lien he vi E va Pr

bihe thc giong nh he thc Planck- Eistein oi vi photon :

h=E , kPr

hr

=

Song phang nay co dang :( )

[ ]

==

trkiEtrpi

AeeAtr

rrrr

hr

., va c goi la song De Broglie .



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C hoc lng t 10 -

CHNG I. C S CUA C HOC LNG T

1 C S VAT LY CUA C HOC LNG T

I/ The gii vi mo:1/ Hat vi mo:

Cac phan t, nguyen t, hat nhan nguyen t va cac hat c ban c goi la cachat vi mo. Danh sach cua cac hat c ban cho en nay a tr nen mot danh sach khao so nhng co the phan thanh ba loai la: cac lng t cua trng ien t - cacphoton, cac hardron va cac lepton. Cac hardron la cac hat tham gia tng tac manh(tng tac hat nhan) va ngc lai cac lepton bao gom: cac electron, cac muon vacac neutrion. Nhom cac hardron ong ao hn ca: chung bao gom cac nucleon

(proton va neutron), cac meson (cac hat nay nhe hn proton) va cac hyperon (cachat nang hn proton). Ngoai tr trng hp ac biet la photon va mot vai metontrung hoa cac hat con lai eu co phan hat.

Thuoc tnh rat quen thuoc cua hau het vi hat la co khoi lng ngh. V du khoi

lng ngh m cua electron la bang: g281019 ., , khoi lng cua photon la 1836m,

cua neutron la: 1839m, cua muon la 207m . Khoi lng ngh cua photon va cua tatca cac neutrino eu bang khong.

Khoi lng cua phan t, nguyen t va hat nhan nguyen t bang tong khoilng cua cac hat tao thanh tr i o hut khoi. o hut khoi bang nang lng dunge pha v vi hat thanh nhng hat cau thanh (thng c goi la nang lng lienket) chia cho bnh phng van toc anh sang. Cac nucleon trong hat nhan co nanglng lien ket ln nhat- o hut khoi cua moi nucleon ln hn 10m .

ien tch cua hat vi mo bang boi nguyen lan ien tch cua electron, tc la

bang boi lan .Ngoai nhng hat vi mo tch ien co nhieu hat vi motrung hoa ve ien. ien tch cua hat vi mo phc tap bang tong ai so cac ien tchcua cac hat thanh phan.

C191061 .,

2/ Spin cua vi hat.Spin la mot trong nhng thuoc tnh quan trong cua hat vi mo. Spin co the c

xem nh la moment c rieng cua hat. Bnh phng cua Spin bang: trongo s co the la so nguyen hay ban nguyen (s thng c goi la Spin), la hang sopho bien, ong vai tro ac biet quan trong trong c hoc lng t, gia tr cua

bang: . Spin cua photon bang 1, Spin cua electron bang 1 / 2 .

( 12 +ssh )h

h

sJ.., 3410051

Spin la mot thuoc tnh ac biet cua vi hat. Va do o no khong the co mo hnhtng t co ien. Viec giai thch Spin nh la moment c rieng cua vi hat tuy thuantien cho viec hnh dung nhng la khong ung vi thc te v khai niemvi hat quayquanh no la khong chap nhan c.



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C hoc lng t 11 -

Momemt goc cua vi hat co thuoc tnh khac thng. Cu the hnh chieu tren

phng bat ky cua no ch nhan cac gia tr gian oan: sh , 1sh ,, sh ,tc la(2s + 1) gia tr. ieu nay co the noi la vi hat co (2s + 1) trang thai Spin. Nh vay ston tai Spin oi vi vi hat a ti s xuat hien bac t do trong e mo ta no.

3/ Boson va Fermion:Tren c s Spin ngi ta chia vi hat thanh hai nhom. Nhom mot la cac vi hatco Spin nguyen hoac bang khong. Nhom hai la cac vi hat co Spin ban nguyen.

Nhom th nhat co ac iem la chung co the co vo so hat cung ton tai trongmot trang thai lng t, tuan theo thonh ke Bose - Einstein va do o nhom naythng c goi tat la cac Boson. Nh vay cac meson va photon la cac Boson.

Nhom th hai co ac iem la chung ch ton tai nhieu nhat mot hat trong moitrang thai lng t, tuan theo thong ke Fermi - Dirac va do o cac vi hat thuocnhom nay c goi tat la cac Fermion. Cac lepton - noi rieng la cac electron - cacnucleon va cac hyperon la cac Fermion.4/ Tnh khong ben cua vi hat:

Tat ca cac hat c ban - tc photon, electron, proton va neutrino eu khongben. ieu nay co ngha la cac hat o t phat phan ra thanh nhng hat vi mo khackhong can mot tac ong nao t ben ngoai. V du mot neutron t phan ra thanh motproton, mot electron va mot neutrino:

eepn ++

Chung ta khong the tien oan c vao thi iem nao s phan ra se xay ra

nhng t thc nghiem ngi ta a tm c quy luat phan ra.( ) ( )tNtN = exp

0 trong o la hang so ac trng cua neutron- c goi la thi gian song cua

neutron va bang s310 . Lng ( )texp xac nh xac xuat cua neutron khong

phan ra tai thi iem t .Cac hat vi mo khong ben c ac trng bang thi gian song cua no. Thi

gian song cua mot hat cang nho th xac xuat phan huy cua hat cang ln. V du: Thi

gian song cua muon la , thi gian song cuas61022 ., - meson dng la

, thi gian song cuas81062 ., - meson trung hoa la s1610 va cua hyperon la

s

10

10

. Mot nhom hat khac co thi gain song cc ky ngan vao khoang s22

10

en s

2310 . Cac hat nay c goi la cac hat cong hng.Mot ieu lu y la: cac meson va cac hyperon co the phan ra theo nhieu cach

khac nhau. V du cac - meson dng co the phan ra theo kieu nh sau:

e

e

e

e

++

+

+

++

++

++

0

Vi mot meson bat ky khong the tien oan c vao luc nao cung nh kieu

phan ra nao se xay ra. Tnh khong ben khong phai ch co oi vi cac hat c ban macon ung oi vi cac vi hat khac. Hien tng phong xa cho thay rang hat nhan



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C hoc lng t 12 -

nguyen t la khong ben vng. Cac nguyen tcung nh cac phan t khi trang thaikch thch cung khong he, chung se t phat tr ve trang thai ban au cua chung haychuyen sang trang thai kch thch thap hn.

Nhng can lu y rang tnh khong ben vng khong phai la ban chat chung cua

cac vi hat bi v ngoai cac hat khong ben co mot so hat vi mo khac la ben. V dunh: photon, electron, proton, neutrion, cac hat nhan nguyen t ben cung nh cacnguyen t va phan t trang thai c ban.

Ngoai tnh chat a neu tren hat c ban con co mot ac tnh ky quac the hientrong s bien oi qua lai cua chung. Tuy nhien ban oc co the tm hieu them van eo qua cac giao trnh khac chi tiet hn ve vat ly hat c ban.

2 HAI Y TNG C BAN CUA C HOC LNG T

1/ Y tng lng t hoa.Ban chat cua y tng lng t hoa la: mot so ai lng vat ly mo ta cac oi

tng vi mo trong nhng ieu kien thch hp ch co the nhan nhng gia tr ri racxac nh. Khi o ta noi rang cac ai lng ay b lng t hoa .

Y tng nay do M.Planck a ra nam 1900 khi nghien cu bc xa cua vat entuyet oi. Planck a gia thuyet rang nang lng cua bc xa ien t do vat phat rakhong phai la lien tuc ma la gian oan - theo tng lng t nang lng, nang lngcua moi lng t nay bang : h= . Gia thuyet nay a bao am cho s phu hp

cua ly thuyet va thc nghiem, ac biet la loai bo c kho khan khi chuyen lythuyet en vung tan so ln ma ngi ta goi la tai bien t ngoai .Tiep theo Planck, Bohr a ap dung y tng lng t hoa vao mau hanh tinh

nguyen t cua Rutherford e xay dng mau nguyen t mi cua ong, mau nguyen tBohr - hay mau nguyen t na lng t. Mac du ly thuyet Bohr ve nguyen t vanco nhng mau thuan noi tai song no c coi la bc au tien e xay dng ly thuyethoan chnh ve cau truc nguyen t.

e hieu noi dung cua y tng nay ta xet v du cac electron trong nguyen t. trong nguyen t nang lng cua electron nhan nhng gia tr ri rac nhng gia trnay tap hp thanh pho nang lng, nh hnh:

Gia s ta ch quan tam enhai mc 1E va 2E . Khi o electron E E1

ch co the co nang lng la 1E

hoac 2E ch khong the co nang E1

lng E thoa man: 1E


14/128

C hoc lng t 13 -

lng t sau nay. Electron co the chuyen t mc nang lng 1E sang mc

2E hoac ngc lai, trong hai qua trnh o electron nhan va phat ra mot photontng ng. Qua trnh o c goi la phep chuyen di lng t.

Nhng lu y rang: Nang lng electron khong phai khi nao cung b lng thoa. Ch cac electron trong trang thai lien ket (v du electron trong nguyen t) thnang lng cua no mi b lng t hoa. Cac electron trang thai t do th nanglng cua no khong b lng t hoa.

S lng t hoa khong phai ch xay ra vi nang lng ma la vi nhieu ailng khac. Cu the moment xung lng cua vi hat cung b lng t hoa. Khac vinang lng la ai lng ch b lng t hoa trong trang thai lien ket, moment xunglng luon b lng t hoa. Cac gia tr quan sat c cua bnh phng momentxung lngcua hat vi mo c cho bi cong thc:

( )122 += llhMtrong o l la so nguyen, ...2,1,0=l .Neu ta xet moment goc cua mot

electron trong nguyen t trang thai dng thn th sol nhan gia tr t 0 en (n-1).Ngi ta thng goi Moment xung lng (hay moment goc) mot cach n

gian la moment. Thc nghiem chng to rang: hnh chieu cua momentcua vi hat lenmot phng nhat nh nao o (ta thng goi o la hnh chieu tren truc oz cho tien)nhan cac gia tr:

mMz h= trong o llll ,1,...,1, +=m . Nh vay vi moi gia tr cua l , so m

nhan 12 +l gia tr gian oan phan tren ta a noi ve Spin cua vi hat va co the coi Spin nh la moment

noi tai cua vi hat. Ngi ta goi no la moment Spin e phan biet moment thng lamoment quy ao. Neu s la Spin cua vi hat th hnh chieu cua moment Spin se nhancac gia tr h trong o ssss ,,...,, 11 += .

Nh vay hnh chieu cua Spin cua mot electron se nhan hai gia tr: 2h

va

2h+

.

Cac so ,,, mn l xac nh cac gia tr gian oan khac nhau cua cac bienong lc lng t hoa va c goi la cac so lng t.

n la so lng t chnh.la so lng t quy ao.l

m la so lng t t. la so lng t Spin.

2/ Y tng lng song hat.Trong vat ly hoc co ien ngi ta coi cac khai niem song va hat la loai tr

nhau. Hat c ac trng bang khoi lng, co tnh nh x trong khong gian va co



15/128

C hoc lng t 14 -

the va cham vi nhau. Ngc lai song khong co tnh nh x nhng co kha nanggioa thoa, nhieu xa

Song nhng quan niem quen thuoc nay khong the chuyen qua c hoc lng tmot cach trc tiep c. mc o cac hien tng vi mo s phan nh ro rang hai

dang chuyen ong nay b phai m i mot cach can ban. Cu the la:Chuyen ong cua vi hat ong thi c ac trng bang ca tnh song lan tnhhat. Co the noi rang cac vi hat mc o nao o giong nh hat va mc o nao ogiong vi song. Cac mc o o phu thuoc vao ieu kien ma o ta xet hat vi mo.

Neu trong vat ly co ien hat va song la hai mat oi lap loai tr nhau- hoac lahat hoac la song th oi vi cac oi tng vi mo cac mat oi lap nay ket hp vinhau mot cach bien chng trong khuon kho mot oi tng vi mo thong nhat la tnhlng song hat.

Y tng lng song hat c Einstein ap dung cho bc xa ien t e giai thchhien tng quang ien. en 1924 De Broglie a m rong y tng lng song hat

khong ch cho bc xa ien t ma cho moi hat vi mo noi chung. Cu the moi hat vimo ong thi co nhng ac trng hat (nang lng va xung lng) va ca nhng actrng song(tan so va bc song). Moi lien he gia nhng ac trng o c chobang he thc Planck - Einstein:

h=E , kPr

hr

= (2.1)Nh vay la mot hat chuyen ong c lien he vi mot song ma bc song

c xac nh theo cong thc:

p

h=

Y tng nay a cho cac gia thiet cua Bohr mot c s vng chac. Chuyen ongcua electron trong nguyen t Hydro luc o se ben vng khi doc theo cac quy aoc quy nh la bang mot so nguyen lan bc song. Cac song doc theo quy aoc quy nh la cac song dng.

ieu nay tng ng vi quy tac lng t cua Bohr:

2h

nE=

trong ola tan so quay cua electron quanh hat nhan,tc la so vong quaycua electron quanh hat nhan. That vay, theo De Broglie o dai cua quy ao bang so

nguyen lan bc song:

rn 2= , mvh

p

h ==

T o suy ra :nhrmv =2

hay nhan ca hai ve vi va chia cho 2 ta co:

22

21

rmv

hn.=

Nhng e y rang :

vr =2



16/128

C hoc lng t 15 -

nen ta co :

hnEmv

21

22

==

ay chnh la quy tac lng t cua Bohr.

Van e at ra la vi cac hat vi mo nh hat bui chang han tai sao ta khongnhan thay tnh lng song hat cua no? Ly do n gian la t cong thc De Broglie tatnh c bc song cua no nho hn kch thc rat nhieu lan. Do o trong ieukien o ta khong the phat hien thay tnh song cua hat bui. Con khi hat co khoilng cang nho th bc song De Broglie cua hat cang ln khi o hat cang khac vihat co ien va cung v the hat vi mo cang giong vi mot song hn. Chu y la songDe Broglie khac vi nhng song co thc ma ta noi tren. Nh vay ta hnh dungquan he song hat nh the nao ? . Co hai cach hnh dung sai ve moi quan he nay.

Cach th nhat : la hat ta tren song. Cach nay la do chnh De Broglie e xuattrong thi ky au. Theo ong th hat ta nh ngoi tren song va song cuon no ti auth hat ti o- hay noi n gian la song chuyen ch hat. Nhng thc te la song cothe ton tai cung hat ngay ca trong chan khonghoan toan nh vay song co the la sanpham cua chnh hat. Nh vay hat tao ra song nh the nao? Song chia se so phancung vi hat nh the nao khi hat tng tac vi cac hat khac va vi moi trng? ola nhng cau hoi ma quan niem song chuyen ch hat khong tra li c.

Cach th hai : o la quan niem cho rang hat la san pham c tao nen t cacsong, la mot c cau ran ac do cac song ket lai hay noi cach khac hat nh la bosong la ni chong chat cac song co bc song rat ngan. Tuy nhien thc te cho thaycac song tham gia hien tng tan sac nh vay neu hat la bo song th hat phai b tan

ra khi nhng ieu kien cua s tan sac xuat hien. Nhng trong thc te hat khong heb tan trong nhng ieu kien o.Nh vay ca hai cach hnh dung tren la hai cach hnh dung may moc theo kieu

co ien eu khong chap nhan c.Can lu y rang tnh lng song hat khong phai ch co khi xet mot tap hp cac

hat ma ngay ca khi co mot hat duy nhat tnh lng song hat cua hat vi mo van bieulo ro ret.

Co nhieu cach giai thch y ngha cua song De Broglie nhng cha co cach naohoan chnh. Cach giai thch cua Born coi o la song xac xuat co ve nh cnhieu ngi chap nhan hn ca. Nhng song xac xuat van khong mang mot y nghavat ly cu the nh song c hoc (song am, song tren mat nc )Co cach giai thchrang hat vi mo luc th mang tnh song, luc th mang tnh hat. Cung co cach giai thchrang hat vi mo ong thi va la song va la hat.

ang chu y hn ca la quan niem rang : hat vi mo khong phai la song cungkhong phai la hat. Tnh lng song hat cua vi hat c hieu nh kha nang tiem tangcua the gii vi mo the hien nhng tnh chat khac biet cua no phu thuoc vao ieukien ben ngoai cu the la ieu kien quan sat no.



17/128

C hoc lng t 16 -

3 C S TOAN HOC CUA C HOC LNG T

Bat c ly thuyet vat ly nao cung la s ket hp gia cac y tng vat ly a ratren c s thc nghiem va mot cong cu toan hoc nhat nh .

Trong giai oan xay dng ly thuyet vat ly cong cu toan hoc co the la cha co -nh khi Newton xay dng ly thuyet cua ong , nhng khi xay dng c hoc lng tth cong cu toan hoc tng ng a c xay dng t trc . Cong cu o la ly thuyetcac toan t tuyen tnh .

I/.Toan t tuyen tnh :a/ Toan t va v du :Toan t la s tng ng gia cac phan t cua tap X vi cac phan t cua tap Y

nao .Cac thuat ng ong ngha vi toan t la :anh xa ,hay ham .Neu hai tap X va Yla hai tap hp so th ngi ta dung khai niem ham so . Vai trng hp at biet cua

toan t la :- Mot toan t t mot khong gian X vao chnh no goi la mot phep bien oi .- Mot toan t t mot khong gian ham so vo han chieu vao mot tap hp so c

goi la mot phiem ham .

Vi mot toan t bat ky tac dung trong khong gian X va Y ta thng viet :(toan t la ch in co dau mu):

L

YxXxxxL = )(,)(),()( va noi rang : toan t tac dung len ham (hay vect ) (x).L

Cac v du ve toan t nh :Phep tch phan ,vi phan , phep lay can phep nanglen luy tha

b/ Toan t tuyen tnh :Trng hp at biet quan trong cua cac toan t o la cac toan t tuyen tnh .

o la cac toan t thoa man hai ieu kien :

+/ 2121 LLL)( +=+

+/ LL)( =

(oi hoi tnh tuyen tnh cua toan t co the xem nh bieu hien cua nguyen ly

chong chap trong c hoc lng t ).Co the de dang thay rang : Cac toan t ao ham , vi phan , tch phan la cactoan t tuyen tnh , con cac toan t nang len tha ,toan t khai can va toan tLogarit khong phai la toan t tuyen tnh .

Vi cac toan t tuyen tnh ta co mot so phep toan sau ay:

+ Toan t n v : c goi la toan t n v neu no thoa man ieu kien:LL = = 1..+ Phep cong cac toan t : Tong cua hai (hay nhieu) toan t va la mot toan t

ma ket qua tac dung cua no bang tong cac ket qua tac dung cua cac toan t thanh

phan , ngha la :



18/128

C hoc lng t 17 -

BAC += neu . BBC +=

+ Phep nhan cac toan t : c goi la tch cua hai toanC A va t va viet

la :

B

BAC . = neu co : .)( BAC =

lu y vi phep nhan cac toan t th t cua chung trong phep nhan la quantrong .

+ Toan t nghch ao : goi la nghch ao cua toan t1L L neu :

)()( xxL = th )()(1 xxL = .

+ Toan t lien hp phc : Toan t c goi la toan t lien hp phc cuaneu nh :

*L L

)()( xxL = th : )()( *** xxL =

+ Phep nang len luy tha :LLLLn ..... = (n: la tha so ).

II/ Giao hoan t va phan giao hoan t :

Giao hoan t cua hai toan t A va la : Vi cac toan t co hai bai toanquan trong can xet la giao hoan t va bai toan tr rieng

B

ABBABA =

.

Phan giao hoan t cua hai toan t A

va B

la :ABBABABA +==

+ .Lu y la giao hoan t cung nh phan giao hoan t cua hai toan t khong phai

luon bang khong .V du : cho

A =x, dxd

B =.

Khi o :

xdxd

dxdxABBABA == .

Hay :

[ ] )(, xdx

d

dx

dxx

dx

d

dx

dxBA ==

. ==

dx

dx

dx

dx

dx

dx

Vay: =

dxdx, hay1, =

dxdx .



19/128

C hoc lng t 18 -

III/. Bai toan tr rieng cua toan t tuyen tnh :

Gia s ta xet toan t , Neu co :L

)()( xxL = (=const, (x) 0).Th (x) c goi la ham rieng va c goi la cac tr rieng tng ng cua

toan t . Moi toan t co the co nhieu ham rieng va tr rieng tng ng ; khi o taviet :

L

)()( xxLnnn

=(n=0,1,2).

Tap hp cac gia tr { }n c goi la cac pho cac tr rieng cua toan t tuyen

tnh .L

Nh vay cac ham rieng cua toan t tuyen tnh la cac ham co tnh chat atbiet la no gi khong oi dang di tac dung cua toan t ngoai phep nhan vi mot trso (tr rieng ).

L

V du : gia s xiL

=

. ieu kien la ham (x) tuan hoan trong khoang[0,L]. Phng trnh tr rieng co dang :

)()(

xx

xi

=

.

Hay chnh xac hn :

)()(

xx

xi

nn

n =

.T phng trnh ta co ngay :

xin

nAex =)( T ieu kien tuan hoan ta co :

)()( LXx nn += .Mat khac lu y rang :

niLi ee n 21== Vay :

...)2,1,0(,2

== nL

nn

.Nh vay n lap thanh mot tap hp gian oan . khi L cac n cang sat lai

gan nhau hn hay chnh xac hn khoang cach gia cac n0. khi o cac ham riengtr thanh :

xiAex

=)( .Vi la mot bien lien tuc co the lay bat c gia tr nao . t ay ta cung nen lu

y mot ieu la :cac tr rieng phu thuoc vao ieu kien bien cua nghiem cua phngtrnh tr rieng , ngha la chung ch xac nh khi cho cac ieu kien bien .



20/128


21/128

C hoc lng t 20 -

dx

dx

dx

dLH 22

2

=

Th bang cach lay ao ham trc tiep ta co ket qua :

)(2)( xnHxHL nnH = (n=0,1,2)

2/Toan t LP:Ta a biet cac a thc Legendre.

21

23

1

11121

2210

2

===

=

xxPxxPxP

xxdx

dP

)(,)(,)(

),(l

l

ll l

11118

158

35863

83

415

835

23

25

355

244

33

==

+=

+==

)(,)()(

.)(

)(,)(

l

l

lPP

xxxxP

xxxPxxxP

Cac a thc Legendre co tnh chat :

( ) .0)()()12()(1/

.)(

21

1/

12

2)()(/

11

2

1

1

=+++

=+

+=

+

=

xPxxPxPc

rxP

rrx

b

dxxPxPa

o

mm

lll

l

l

l

ll

lll

l

Neu a vao toan t :

.2)1(2

22

dx

dx

dx

dxLP

Th ta thay cac a thc Legendre lien he vi Lp nh sau:

)()1()( xPxPLP ll ll += 3/ Toan t Laplace :Trong ly thuyet cac ham ac biet cac ham cau co vai tro ac biet quan trong .

o la cac ham :



22/128

C hoc lng t 21 -

)(cos)cos(

sin)!(2

)!)(12()1(

2

1),(

ll l

llPm

m

meY

m

mmimm

+

+=

Vi m 0.

Vi m 0, ham ( )*

)1(),( mmm YY = ll (dau * ch lien hp phc ).

Vi moi gia tr cua l=0,1,2,3 ta co 2l+1 gia tr cua m .m = -l,- l+1,- l+2,, l-2, l-1, l.

Mot so bieu thc cua ),( mYl vi l va m cu the :

i

eY

YY

=

==

sin

cos,

8

343

41

1

000

l

l

= 212

23

450

2 cosY

ieY = cossin8

1512

ieY = 222 2

1541

sin

Cac ham cau co tnh chat :

1

1

1

1

11

0''

2

0

'

'

sin/

/

sin),().(/

+

++

+

+=

+=

=

mmmi

mmm

mm

mm

dYCYYec

bYaYYCosb

ddYYa

lll

lll

llll

Trong o a , b , c, d la nhng hang so ch phu thuoc l , m. Neu a vao toan t

:

2

2

2sin

1)(sin

sin

1

+

=

Th cac ham cau lien he vi toan t nay nh sau :

),()1(),( mYY mm ll

ll +=

4/. Toan t LL:Cac a thc Lagendre co dang:



23/128

C hoc lng t 22 -

)(,)()( kjxLdx

dxL

kj

jjk

=

Vi :

( )xk

k

kx

k exdx

d

eL

= Ta co mot so a thc cu the :

....,9186)(

42)(

1)(,1)(

323

22

10

vvxxxxL

xxxL

xLL

+=

+=

==

Cac a thc Laguerre co tnh chat :

'

3

0' )!(

)!()()(

kk

xjj

k

jk jk

kdxexxLxL

=

Neu a vao toan t :

dx

dxj

dx

dxL

L)1(

2

2

++

Th ta co lien he cua LL vi cac a thc Laguerre nh sau

)(;)()()( kjxLjkxLL jk

jkL

=

5/.Toan t LF :Cac ham sieu boi suy bien la cac ham co dang :

...!3)2)(1(

)2)(1(

)1(

)1(

!1.1),,( 32 +

++

+++

+

+++= z

bbb

aaaz

bb

aaz

b

azbaF

Ta thay neu a la mot so nguyen am tc la a = -n (n la so nguyen dng )th

ham sieu boi F(-n,b,z)se la mot a thc cua z . Co the chng minh c rang :

zba eczzbaF ),,( khi Rez, c = const.azdzbaF )(),,( khi Rez, d = const.

Hn na da vao bieu thc nh ngha cua ham sieu boi , a thc Hermite vaa thc Legendre ta co cac he thc sau :



24/128

C hoc lng t 23 -

),1,()!1(

)!1()(

),23

,(!

)!12(2.)1()(

),21

,(!

)!2()1()(

212

22

xjkFj

kJxL

xnFn

nxH

xnFn

nxH

jk

nn

nn

++

++=

+=

=

+

(k nguyen ,khong am)

Ngoai ra neu cac ham Bessel viet dang :

++

=

=

+

1

2

2)!1(!

)1()(

k

kpk

P

x

pkkxJ

th ta co he thc :(n la so nguyen):

)2,21,2

1(

2)!1(

1)( ixnnFe

x

nxJ ix

n

n +++

=

Nh vay neu a vao toan t :

dz

dzb

dz

dzLF )(2

2

+

th ta thay cac ham sieu boi suy bien lien he vi toan t LF nh sau :

),,(),,( zbaaFzbaFLF = (5)

Neu b khong phai la mot so nguyen th nghiem th hai cua phng trnh (5)co dang:

),2,1(),,( 1 zbbaFzzbaG b =

6/.Toan t Laplace toan phan :Trong giao trnh phng trnh toan ly ta a gap nhieu lan toan t : (Trong he

toa oDescartes )

2

2

2

2

2

22

zyx +

+

(6)Trong cac he toa o suy rong q1, q2, q3:



25/128

C hoc lng t 24 -

( )( )3213

3212

3211

,,

,,

,,

qqqz

qqqy

qqq

===

Trong o 1, 2, 3 la nhng ham lien tuc va n tr . Neu at :

)3,2,1(;2

32

2

2

12 =

+

+

=

i

qqqH

iiii

va goi la cac he so Lame ,th ta co :

+

+

= 3321

322

13

211

32

1321

1

qH

HH

qqH

HH

qqH

HH

qHHH

Cu the vi he toa o cau q1 = r, q2 = , q3 = ta co: H1 = 1, H2 = r, H3 =

rsin. Do o :

2

2

2222

sin

1sin

sin

1

2

1

+

+

=

rrrr

rr

Vi he toa o truc : q1 = r ,q2 = ,q3 = z.Ta co :H1 = 1, H2 = r, H3 = 1.Nh vay trong he tru ta co :

zrrr

rr +

+

=

2

2

2

22

2

11

Nh lai bieu thc (3) ta co :

22

2 2

rrrr

++

=

V/.Toan t lien hp va toan t t lien hp (hecmitic) :1/ Tch vo hng cua hai ham :Gia s co hai ham (x) va (x) nao o ,khi o tch vo hng cua hai ham nay

c nh ngha bi:

=+

dxxx )()(),( *

trong o dau * ch lien hp phc .(Noi them : hai ham (x) va (x) phai bang

khong tai vo cung ).

Bieu thc :



26/128

C hoc lng t 25 -

( ) =+

dx

2,

c goi la chuan cua ham (x)Neu viet (x) dang cot :

=

)(

.....

)()(

)( 2

1

x

xx

x

n

va (x) dang ma tran hang :

)()....(),()( **2*1

* xxxx n = th :

( )

=+

nnn dxxx )()(,

*

.Neu 1(x), 2(x) n(x) la cac hang so C1, C2 Cn.con

)()....(),( **2*1 xxx n la cac hang so th tch vo hng (,) c

cho bang tch :

**2

*1 ..., nbbb

( ) =I

ii Cb*,

Trng hp : = (x,y,z); = (x,y,z) th :

( ) = dzdydxzyxzyx ),,(),,(, *

Co the chng minh tch vo hng nh ngha nh tren co cac tnh chat nh sau:

( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )

,,/

;,,/

,,,/

,,,/

,,/

*

2121

2121

*

=+

==+

+=+++=++

=+

const

(Cac chng minh nay khong kho khan lam ,ban oc co the xem nh mot bai

tap )

2/ Toan t tuyen tnh lien hp (Hecmitic)va toan t Unita :

Gia s A va la hai toan t c goi la lien hp cua nhau (hay lien hp vi

nhau ),va k hieu la:

B+= BA va += AB , dau + la dau lien hp .

V du 1 : Ta xet vi toan t C la mot so hay tm C+, theo nh ngha ta co :



27/128

C hoc lng t 26 -

( ) = ,)()(, * dxxCxC ( ) = ++ dxxxCC )()(, ** Hn the na :

=

+

dxxCxdxxxC )()()()(

***

Nh vay : C+*=C hay C+= C*.Ngha la toan t lien hp cua mot hang so la lien hp phc cua no .

V du 2 :Ta xet toan t nhan vi U(x) . Hay tm toan t U+(x).Lap luan tng t nh tren e thay rang :

U+(x)= U*(x)Hay tong quat hn : Neu L la mot toan t nhan th :L+= L*.

V du 3 : Hay tm

+

dxd .

Ta co :dxx

dx

dxdxxx

dx

d)()()()( *

*

=+

+

+

Tch phan tng phan ve phai ta c :

=+

+

+

dxx

dx

xdxxdxx

dx

dx )(

)()()()()(

***

V *(x) va(x) triet tieu tai nen ta co :

dx

d

dx

d =+

Nh vay ta co :

= ,,

dx

d

dx

d

(Luon phai lu y rang cac ham (x) va(x) triet tieu tai ).Hay tong quat

hn ta co :

= ,)1(,

n

nn

n

n

dx

d

dx

d

3/. Toan t t lien hp (Hecmitic)va toan t unita .

a/ Toan t t lien hp : Neu thoa man ieu kien :LL = L+.ngha la :

= dxLdxL ** )(

th L la toan t t lien hp (Hay Hecmitic )



28/128

C hoc lng t 27 -

Toan t Unita : Neu L thoa man :L L+ = L+L=1.Th L goi la toan t unita . Toan t unita thng c k hieu la U .

Co the chng minh rang : dx

diL =

la t lien hp .b/. Cac tnh chat cua toan t t lien hp (Hecmitic) :Tnh chat 1 : Mot toan t co cac tr rieng thc khi va ch khi no la toan t

Hecmitic .Chng minh :T bai toan tr rieng :L(x)=(x)Ta co :

=

=

dxxxdxxL

dxxxdxxL

)()()(

)()()(

****

**

Tr cho nhau :

= dxdxLdxL ***** )( do o :

= dxdxLL **** )()(

Nh vay ro rang =* khi va ch khi L la t lien hp L= L+.

Tnh chat 2 : Cac ham rieng cua mot toan t Hecmitic ng vi cac tr riengkhac nhau la trc giao .

Chng minh :Cung xuat phat t bai toan :L(x)=(x)ta coLn= nn

L*m*=m m*.

Hay viet cach khac : = dxdxL nmnnm ** Va :

= dxdxL mnmmn****

T o ta co :

=+ dxdxLL nmmnnm

*** )()(

Neu L la Hecmitic th L= L+.(va do o*mn = ) khi o :

0)(

**

= dxnmmn



29/128

C hoc lng t 28 -

V*mn (chung ta gia thiet cac tr rieng khac nhau )nen :

0* = dxnm .Ngha la cac n va m la trc giao vi nhau .

4 TH NGHIEM QUAN TRONG TNH THONG KE CUA CHOC LNG T

I/ Th nghiem hai lo :th nghiem ma ngi ta ngh ra e bao gom c moi b an cua CHLTva dan

dat ta en tat ca nhng nghch ly ,nhng b mat , nhng ieu ky la cua t nhien mot

cach ay u .Chnh trong th nghiem nay cha ng nhng ieu b an can ban.Nhan xet ve th nghiem nay Feynman a tng noi nh the .

1/ nguon la nhng vien an :

1 N1.

2 N2N12

S o th nghiem c bo th nh hnh ve .Nguon A la nguon phong an (vdusung tieu lien ) .Trong o N1 la so an ti bia khi bt kn lo 2 , N2 la so an tibia khi bt kn lo 1 va N12 la so an trung bnh cua ca 1 va 2 eu m (trong motn v thi gian).

Ket qua : N12= N1 + N22/ Nguon la hai song (v du song nc ):

1I1.

A

2 I2I12

ANguon

I12 Cng o song tren bia khi ca hai lo eu mI1 Cng o song tren bia khi lo 2 ong .

I2 Cng o song tren bia khi lo 1 ong.Ket qua : I12 I1 + I2 ( Co giao thoa song )



30/128

C hoc lng t 29 -

3/ Nguon la chum electron :

1 N1.

2 N1

N12Ket qua : N12 N1 + N2 (co giao thoa)Vi : I12 , I1 , I2 la cac xac xuat tm thay electron bia khi ong lo 2 ,

ong lo 1, m ca hai lo tng ng .Tom tat ba trng hp

an Song ElectronGian oan .

Xac xuat ti c o:N12 = N1 + N2Khong co giao thoa

Khong gian oanCng o song c

o: I12 I1 + I2Co giao thoa

Gian oanXac xuat ti c o :N12 N1 + N2Co giao thoa

Vi trng hp chum electron :cac electron i ti bia (hay may thu) theotng lng nho gian oan giong nh o la nhng hat nhng xac xuat e nhng hato ti bia lai c xac nh theo cung nhng quy luat nh quy luat xac nhcng o cua song nc .V vay co the noi rang cac electron va giong nh chum

hat va giong nh song . No ong thi la hai vat hoan toan khac nhau .II/ Tnh thong ke cua CHLT:Trong cac ieu kien ben ngoai cho trc ket qua cua s tng tac gia oi

tng vi mo va dung cu o noi chung khong the tien oan mot cach chnh xac , mach co the d kien vi mot xac xuat nao o .Tap hp cac ket qua nh vay a enthong ke tng ng vi phan bo nhat nh cua xac xuat. Nh vay ta phai a yeu toxac xuat vao cach mo ta oi tng vi mo va trang thai cua no .

Khai niem xac xuat cung c dung trong VLC nhng vi mot y ngha khac.Trong VLC xac xuat c a vao ch khi ieu kien cua bai toan khong c

biet mot cach ay u va phai lay trung bnh theo cac tham so cha biet . Song oa gia thiet rang ve nguyen tac th s trung bnh hoa la khong bat buoc va luonluon co the chnh xac hoa cac ieu kien e khang nh la mot trong so cac ket quakha d c xay ra hoan toan con cac ket qua khac se khong xay ra . Nh vay ,trong VLC xac xuat a phan anh s phat bieu khong ay u bai toan . S khongay u ay la khong the tranh c nhng ve nguyen tat la co the loai tr c .Trong vat ly co ien nguyen ly quyet nh luan Laplace a loai bo yeu to ngaunhien khi mo ta dang ieu cua tng oi tng rieng biet do o cac nh luat cua choc co ien la cac nh luat ong lc ch khong phai la cac nh luat thong ke .Tnh thong ke ch xuat hien khi nghien cu mot tap hp ln cac hat va la motphng phap co tnh thu thuat .



31/128

C hoc lng t 30 -

Ngc lai trong C hoc lng t yeu to ngau nhien co mat trong dang ieucua tng vi hat rieng biet . Yeu to ngau nhien la he qua cua s hu han cua hang soPlanck va nguyen ly bat nh Heisenberg . Do o c hoc lng t la mot ly thuyetthong ke ve mat nguyen tat . Va tnh xac xuat la mot trong nhng ac iem cua no .

5 CAC BIEN ONG LC TRONG C HOC LNG T

Trong vat ly hoc co ien trang thai va bien ong lc la hai khai niem c ban.Cac bien ong lc nh toa o, xung lng ,moment xung lng va nang lng trongvat ly hoc co ien la nhng ham so (thong thng ).

Nhng trong c hoc lng t van e khac mot cach c ban . Cu the la :- Trong c hoc lng t moi bien ong lc cua c hoc co ien c oi ng

vi mot toan t Hecmitic .- Co the chuyen cac he thc ong lc t c hoc co ien sang c hoc lng t

vi dang nh cu nhng trong o phai thay cac ai lng vat ly bang cac toan tHecmitic tng ng .

Noi cach khac : Cong cu cua c hoc lng t c xay dng hoan toan tngt nh cong cu c hoc co ien neu ta thay cac bien ong lc cua c hoc co ien (lacac ham ) bang cac toan t Hecmitic .

1/.Cac toan t toa o :),,(),,( 321 xxxzyxr =

r

o la cac toan t nhan ham f tren o cac toan t tac dung vi cac lngxi(i=1,2,3). Phep nhan hieu theo ngha thong thng , cac ham f=f(x,y,z).Ta co the de dang tm c cac giao hoan t cua cac toan t toa o la :[xi,xj]=0. (5.1)e y rang v cac lng xi la thc nen cac toan t nay eu la Hecmitic.

2/.Cac toan t xung lng :

),,(),,( 321 zyx PPPPPPP ==r

Cac toan t xung lng co dang :

)3,2,1(;, ===i

xiPiP

i

i hh

r

(5.2)Cac giao hoan t co dang :

)3,2,1,(;0, == jiPP ji (5.3)Ta cung co the tm c cac giao hoan t cua cac toan t nay vi cac toan t

toa o la :)3,2,1,(;, == jiiPx ijji h (5.4)

Cac toan t xung lng cung la cac toan t Hecmitic .



32/128

C hoc lng t 31 -

3/. Cac toan t Moment xung lng L va toan t moment xung lngbnh phng L2 .nh ngha :Toan t L c nh ngha nh sau :L=[r*p].

Nh vay ta co :

==

==

==

12

2112213

31

1331132

23

3223321

xx

xxipxpxL

xx

xxipxpxL

xx

xxipxpxL

h

h

h

Cac giao hoan t :Trc het ta co the tm giao hoan t cua L vi cac toan t x ; cu the ta co :

[ ] [ ] [ ][ ] [ ] [ ] [ ]

3

223223322232

2232322233221

xi

pxxxpxpxxxpx

pxxpxxpxpxxL

h=

+=

==

,,,,

,,,,

Hn na co the chng minh trng hp tong quat :

=k

kijkjiixL h

(5.5)Vi :

1,1 132213321312231123 ====== Cac thanh phan khac eu bang khong .

Giao hoan t cua Lr

vi bang cach tnh trc tiep ta co :pr

=k

kijkjipiL h,

(5.6)

Gia cac thanh phan cua L ta co :

[ ] [ ][ ] [ ] [ ] [ ]

32112

311311113131

131311

3113121

Lipxipxi

pLxpxLpLxpxL

pxLpxL

ppLLL

hhh =+

+=+=

=,,

,,

Vay: 321, iLLL = .Tong quat hn ta co :

= k kijkji LiL h, (5.7)



33/128

C hoc lng t 32 -

T cac toan t L1,L2,L3 ta a vao toan t :

23

22

21

2 LLLL ++= Goi la toan t moment bnh phng . Ta co the tm giao hoan t :

[ ] [ ] [ ] [ ] 0,,,,0,,,,,

313133212122

1231

221

211

21

22

211

2

=++++=++=++=

LLLLLLLLLLLL

LLLLLLLLLLLL

Tong quat ta co :

)3,2,1(;0,2 ==

iLL i

(5.8)Do cac toan t xi va pi la Hecmitic ta de dang thay rang cac toan t Li va L2

eu Hecmitic .

4/.Toan t Hamilton H:Tng ng vi bieu thc ham Hamilton trong c hoc co ien :

),,,(2

2

tzyxUM

pH +=

r

Ta a vao toan t Hamilton :

),,,(2

2

tzyxUm

H += h(5.9)

Trong he toa o cau (r,,) toan t hamilton co dang :

Urrrrm

H +

+

+

=

22

22 2

2 h

(5.10)Cac giao hoan t : Ta co the tnh trc tiep e tm cac giao hoan t cua H va xi

va pi nh sau :[ ][ ] 0,

0,

i

i

pH

xH

(i=1,2,3)

Ch khi U = 0 th ta mi co :[ ] 0, =ipH .Cac giao hoan t cua H vi cac toan t moment xung lng Va moment xung

lng bnh phng se c xet trong trng hp at biet khi U=U(r,t).Tc la tngng vi bai toan chuyen ong trong trng xuyen tam .

Trc het ta tm bieu thc cua Li va L2 trong he toa o cau .T moi lien hegia hai toa o :

cos

sinsin

cossin

rz

ry

r

=

==



34/128

C hoc lng t 33 -

Ta co : xy

yx

y

y

x

x

=

+

=

Ngha la ta co :

=h

iL3 (5.12)e tm cac bieu thc cua L1 va L2 ta e y rang :

x

ytg

rzyxr ==++= ;cos;222

.T o ta c :

sin

cos;0

sincos;

sin

sinsin,cos

ryz

ryrz

y

r

r

z

z

r

==

=

=

===

T bieu thc nh ngha :

( )

( ) ( )

+

+

+=

=

+

+

+

+

=

=

coscotsincossinsin

sinsincoscossinsin

cos

sinsin

g

rrri

zzrz

rr

zzrz

rri

yz

zyiL

22

1

h

h

h

Nh vay ta co :

= coscotsin1 giL h (5.13)Tng t ta tnh c :

=

coscotcos2 giL h

(5.14)T cac ket qua tren ta co :

+

==

2

2

2

222

sin

1sin

sin

1

hhL

(5.15)Ta xet trng hp ac biet khi U=U(r,t) .Khi o ta k hieu :



35/128

C hoc lng t 34 -

),(

2

2

0trU

mH += h

.T bieu thc nay va cac ket qua tren ta co :

Va [ ] 0,0,

20

30

=

=

LH

LH

(5.16)

Cac toan t ong lc co vai tro ac biet quan trong trong c hoc lng t bile no lien quan en cac gia tr o c trong thc nghiem . Phep chuyen t cac toant ti cac ai lng o c (hay quan sat c ) trong thc nghiem c thc hiennh bai toan tr rieng :

)()( xxL nnn = .Moi lien he o la :Neu he trang thai c mo ta bi he ham rieng n(x) cua

toan t oi xng vi bien ong lc nao o th khi o ai lng nay trang thain(x) ta se nhan c gia tr n nao o nam trong pho {n }.

6 CAC HE THC BAT NH

I/ Y Tng Lng Song Hat Va cac He Thc Bat nh:Ta xet tap hp cac song phang (co ien) lan truyen tren truc x. Ta gia s rangcac tan so song rai rac mot khoang nao o ,con cac gia tr cua vector

song trong khoang .xk

Neu chap tat ca cac songphang nay vi nhau th chung xta se nhan c bo song nh hnh ve. Vi bo song o ngi ta a tm c cac

he thc:

11

xk

t

x.

.

Nhng he thc nay rat quen thuoc trong vat ly co ien. Nhng ngi a biet

ve ky thuat radio eu biet rang: e tao ra tn hieu nh x hn th can phai laynhieu song phang vi cac tan so khac nhau hn. Hay noi cach khac: e giam x

va t th phai tang xk va .Mot cach hnh thc ta gia thiet rang cac he thc tren khong ch ung vi cac

song co ien ma con ung cho ca nhng ac trng song cua cac vi hat (Nhng lu y



36/128

C hoc lng t 35 -

rang ieu nay khong co ngha la chung ta mo hnh hoa vi hat di dang bo songnao o).

S dung cac he thc Planck Einstein:

h=E , kPr

hr

=

va chuyen sang bieu thc tng t cho cac ac tnh cua vi hat (cho nang lngva xung lng cua hat) ta co:

h tE. (6.2)h xP

x.

(6.3)Cac he thc nay c Heisenberg a ra lan au tien vao nam 1927 va c

ong goi la he thc bat nh (ngay nay ta thng goi chung la he thc bat nhHeisenberg).

Ngoai ra ta co the bo sung them he thc di ay:h

xx

M .

(6.4)trong o x la o bat nh cua toa o goc cua oi tng vi hat (xet phep

quay quanh truc x), con xM la o bat nh cua hnh chieu moment tren truc x.Cac he thc tren cung co the m rong cho cac trng hp tng t khac. Cu

the la ta co the viet:

h

h

z

y

Pz

Py

.

.

h

h

zz

yy

M

M

.

.

II/ Y ngha cua cac he thc bat nh:

Ta xet he thc:h xp

x.

trong o x la o bat nh cua toa o x, xp

la o bat nh cua hnh chieutren truc x cua xung lng cua vi hat. Nh vay neu vi hat c nh x tai mot iemxac nh x nao o th hnh chieu cua xung lng cua vi hat tren truc x co o bat

nh tuy y. Ngha la vi hat phai tan ra tren toan truc x.ieu nay co ngha la trong c hoc lng t vi hat khong the co ong thi toao xac nh va gia tr hnh chieu xung lng xac nh tren truc o. ieu nay chothay trong c hoc lng t hoan toan khong co khai niem quy ao. ay la mot hequa quan trong cua nguyen ly bat nh.

Bay gi ta xet he thc:h tE.

Vi he thc nay co nhieu cach giai thch khac nhau v t khong phai la obat nh theo thi gian (v thi gian la mot tham so ch khong phai mot bien onglc). Cu the la :



37/128

C hoc lng t 36 -

- Neu he trang thai kch thch trong khoang thi gian t th khi o hekhong the co nang lng xac nh va o bat nh cua nang lng la:

tE

h

Tuy nhien nang lng co o bat nh khong co ngha la nang lng khongc bao toan. Bi v neu he trang thai dng th nang lng cua he khong oitheo thi gian va khi o ta co the o nang lng trong khoang thi gian t lau tuyy. Ngha la neu =t th 0=E la khong co s sai lech nao ve tr so cua nanglng.

Nh he thch tE.

ta co the suy ra thi gian song cua hat vi mo, cothe tng tng ra nhng quatrnh ao e lam cong cu giai thch nhng hien tng ma khong vi pham nh luat

bao toan nang lng .Nhng ieu can lu y:- Th nhat: he thc bat nh nhan manh mot ieu la: s bat nh trong the gii

vi mo khong phai la nhng sai so ngau nhien cua phep o hay s khong hoan thiencua cac thiet b ky thuat ma xuat phat t ban chat lng song hat cua hat vi mo haynoi cach khac viec khong the xac nh ong thi toa o va xung lng la co tnhnguyen tac.

- Th hai : Cac ai lng ac trng cho song ( va ) khong phai la cac ailng nh x c. Do o xuat phat t he thc De Broglie ta co the rut ra he thcbat nh. Mat khac chuyen ong cua hat co khoi lng m va van toc v lien oi vimot song co bc song tnh theo cong thc:

mv

h=

trong o la bc song. ac trng cua song la s lan truyen vo han trongkhong gian cho nen bc song la mot ham so cua dang song ch khong phai la hamso cua mot iem cua khong gian. Cung do he thc De Broglie th van toc cua hatcung khong phai la ham so cua toa o cua hat. Noi tom lai van toc va toa o cuahat khong the ong thi co gia tr xac nh. He thc bat nh la mot he thc c bannhat cua c hoc lng t, mot trong nhng he qua quan trong nhat cua gia thiet De- Broglie ve tnh lng song hat cua vi hat.

- Th ba: Nhn cac he thc bat nh ta thay cau truc cua chung phai nh thenao e trong moi he thc gom hai ai lng lien hp chnh tac vi nhau theo yngha co ien. Do o ta co the suy ra rang oi vi cac cap ai lng chnh tac thcac he thc tng t van ung. Ve phng dien ly thuyet tng oi cac he thc batnh Heisenberg co tnh chat oi xng: cu the la ba thanh phan cua xung lng va

nang lngEzpypxp ,,, tao thanh mot vector 4 chieu, con ba toa o va thi

gian (x,y,z,t) tao thanh mot vector khac



38/128

C hoc lng t 37 -

III/ Mot so ket qua thu c t he thc bat nh :a/ c lng xac nh nang lng cua trang thai c ban cua nguyen t hiro.Trong vat ly hoc co ien bie thc nang lng cua hat tch ien trong mot

trng coulomb la :

re

mPE 222

=(6.5)

trong o m va e la khoi lng va ien tch cua electron .e s dung cong thctren trong c hoc lng t ta phai xem Pva r trong bieu thc o nh la x bat nhcua xung lng va toa o cua electron . Theo he thc bat nh gia xung lng vatoa o :

h xP * ta i ti he thc tng ng : hrp.

hay n gian hn la :

h=rp.(6.6)Thay r tnh c t (6.6) vao (6.5) ta co :

h

pe

m

ppE

22

2=)(

(6.7)T ket qua nay ta de dang thay rang E(p) co mot cc tieu tai p=p1.Ta ky

hieu nang lng cc tieu o la E1 .Lng E1chnh la nang lng cua trang thai c

ban cua nguyen t Hyro con 11 p

rh=

la kch thc c lng cua nguyen t

(trong ly thuyet cua Bohr kch thc c lng cua nguyen t la ban knh cua quy

ao th nhat trong nguyen t) . Co the de dang xac nh c h

2

1me

p =. Cuoi

cung ta co:

2

4

12

21

2hh me

Eme

r == ,

(6.8)Nhng ket qua nay hoan toan phu hp vi nhng ly thuyet chnh xac ,chac

che co ien .

b/ c lng nang lng cua dao ong tai iem khong cua mot dao ong t .Nang lng cua mot dao ong t ieu hoa mot chieu co ien c cho bangbieu thc :

22222 mx

m

PE x +=

(6.9)Xem Px va x nh la s bat nh cua xung lng va toa o cua vi hat dao ong

va s dung phng trnh :h=xPx .

nh la mot trng hp cua he thc bat nh , ta co :



39/128

C hoc lng t 38 -

2

222

22 xx

xP

m

m

PPE

h+=)(

Bang cach giai phng trnh :

0=)( xx PEdPd

Ta tm c : hmPo = la gia tr tai o E(Px)cc tieu .De dang t osuy ra :

h== )( oo PEE (6.9)ay la mot ket qua rat ang e y . Bi le no chng to rang : Trong c hoc

lng t nang lng cua mot dao ong t khong the triet tieu va co gia tr lah .Nang lng nay c goi la nang lng cua dao ong khong cua dao ong t .

Lu y rang c lng (6.9) khac vi bieu thc chnh xac cua nang lng cua dao

ong khong cua dao ong t mot tha so 21

(gia tr chnh xac la :h21 ).

S dung ket qua nay vao cac tnh toan nang lng cua chuyen ong ieu hoacua nguyen t trong mot tinh the ngi ta i en mot ket qua thu v bat ng la :Nang lng o khong triet tieu ngay ca o khong tuyet oi.

Dao ong khong cua mot dao ong co nang lng khac khong chng to mot skien c ban la : Khong the tm thay mot vi hat ay cua mot ho the hay noi cachkhac hat vi mo khong the ri xuong ay cua mot ho the, du cho ho the co dang nhthe nao. S d nh vay la do s ri xuong ay ho thecua vi hat la khong cho

phep do tnh triet tieu cua xung lng , cung nh tnh bat nh cua toa o cua vi hat.c/ Giai thch tai sao electron khong ri vao hat nhan .Khi a ra gia thuyet ve trang thai dng trong ly thuyet nguyen t , Bohr

khong giai thch v sao cac electron chuyen ong co gia toc xung quanh hat nhanlai khong bc xa va ri vao hat nhan . Nhng he thc (6.3) lai cho phep ta giai thchieu nay kha de dang . Electron ri vao hat nhan ong ngha vi viet thay oi angke toa o cua no . Gia s trc khi ri vao hat nhan electron cach hat nhan vao c

cm

me

r 82

2

1 10=h

.Khi ri vao nhan electron se nam trong vung co kchthc c 10-12cm . Theo he thc bat nh (6.3) tnh nh x cang cao th x nhoecua xung lng cang ln. Do o trc khi ri vao hat nhan xung lng cua electronphai tang len ieu nay co ngha la electron phai tieu ton mot nang lng .Thanh ths co gang gicho electron khong ri vao hat nhan la s nh x cho electron vung rat gan hat nhan.

Trong khi xet dao ong khong tren ta thay rang vi hat trong mot ho the luonco mot nang lng cc tieu khac khong Eo . o ln cua Eo noi chung phu thuocvao o rong cua gieng .S dung cac ket qua o ta de dang co the suy ra la :

2

2

maEoh

= . Nh vay neu a giam th Eo se tang len .Nh vay vi a u nho , nang



40/128

C hoc lng t 39 -

lng Eo co the tr nen ln hn o sau cua ho the. Ro rang mot gieng nh the sekhong the gi noi mot hat nao trong no .

S ri vao hat nhan cua electron se tng ng vi s giam be rong cua ho

the t 10-8cm xuong 10-12cm .Tham ch thap hn . Theo cong thc : 2

2

ma

Eoh

,khi o cc tieu nang lng Eo se tang t 10 ev en109ev va tham ch hn na . ketqua la nang lng toi thieu cua electron vao c ln hn nang lng lien ket cuacac nucleon trong hat nhan nguyen t (nang lng lien ket nay vao c109 ev). Lapluan ay khong nhng tra li cho cau hoi v sao electron khong ri vao hat nhanma con tra li cho mot cau hoi rat quan trong la : V sao trong hat nhan nguyen tkhong co electron .

IV . Xay dng he thc bat nh Heisenberg:

1. Gia tr trung bnh:a/ Ky vong toan hoc trong ly thuyet xac xuat : Gia s ta co mot ai lng

ngau nhien l co nhan cac gia tr 1 , 2 , k cung vi xac xuat tng ng ; 1 ,2 , k va

1 + 2 + .. + .. = 1Ky vong toan hoc hay trung bnh thong ke cua ai lung ngau nhien l la

tong cac tch cua tng gia tr nay nhan v xac xuat xuat hien cua no :

= KKl

oi vi cac ai lng lien tuc ngha la lng l bien oi lien tuc mot khoangnao o , Xac xuat e cho l nam trong khoang va ( + d) se phu thuoc vao vaty le vi d . Khi o ky vong toan hoc cua ai lng l se co dang :

( )= dl Trong o : ) la mat o xac xuatb/ ky vong toan hoc trong C hoc lng t :

Trong C hoc lng t trang thai cua vi hat c mo ta bi ham song .Neu la ham rieng cua toan t L th khi o ai lng bat ky neu nhan c gia trrieng , ta co the bieu dien bang cong thc:

=

= LdV

dVL*

*

.

Ky hieuL

ch rang cua ai lng L trong trang thai .Nh vay vi bien ong lc toa o ta co :



41/128

C hoc lng t 40 -

=

dV

dVx*

* x

Neu ta gia s co he thc :

= 1* dV th khi o :

= dVxx * (6.12)

Tng t ta co :

dXdx

d

i

dXXPxP

x

Xx

=

=)(*

* )()(

h

(6.13)

Tong quat hoa cho toan t bat ky :F

= dVFF * (6.14)

Vi ieu kien ham song chuan hoa ve n v .Gia tr trung bnh co mot so tnh chat nh sau :

+/ gia tr trung bnh cua cac ai lung chnh xac bang chnh gia tr cua ailng chnh xac o .

Neu trong trang thai gia tr cua toan t L bang th ta co :

= L = dVL*l

+ Neu trong cung mot trang thai gia tr trung bnh cua laF F , cua G la

G th gia tr trung bnh cua tong thoa man :

+=+ dVGFGF )(*

= + dVGdVF

**

= GF+ 2/ ieu kien e hai toan t F va G giao hoan vi nhau :

mot van e quan trong cua C hoc lng t la kha nang o ong thi cac giatr cua cac ai lng vat ly gan lien vi c he lng t . vi van e nay ta co nh lysau:

nh ly : Muon cho hai ai lng F va G co cung gia tr xac nh trong mot

trang thai nao o mo ta bi ham song

)(Xn

th ham song nay phai ong thi laham rieng cua toan t F va G .



42/128

C hoc lng t 41 -

Chng minh :)()( XXF nn =

tac dung tiep len o toan t G th ta co :)()( XXGF nn = (a)

Vi toan t G :)()( XXG nn =

Tac dung tiep toan t F len o ta c :

)()( XXGF nn = (b)Lay (a) tr i (b) ta co :

[ ] 0)(,)()( == XFGXFGGF nn

ieu nay co nhga la : Neu hai ai lng co the ong thi co cac gia tr xacnh th hai toan t tng ng vi chung phai giao hoan vi nhau .

nh ly ao : Neu hai toan t F va G gia ohoan vi nhau th chung co chungmot he ham rieng . Ngha la neu :

[ F , G] = 0th he ham 1 (X) thoa man :F1 (X) =1 (X)Va he ham 2 (X) thoa man :

G2 (X) =2 (X)Se hoan toan trung nhau . Ngha la : 1 (X) 2 (X) (X).Chnh minh nh ly nay khong kho lam cac ban co the xem nh bai tap ;

hoac xem trong cac tai lieu tham khao.3/ khai niem tap hp u cac ai lung vat ly :

Ta can lam ro hn thuat ng cho trc trang thai cua he Trong C hoclung t .

Trang thai cua he a cho neu cho biet ham song mo ta he . Song ta khongthe o trc tiep uc ham song trong bat ky ieu kien nao, ch co bnh phngmodun ham song co y ngha vat ly nh la mat o xac xuat tm thay hat mot v trxac ng trong khong gian . V vay trong C hoc lng t ngi ta quan niem rang :

Khi ta noi rang cho trc trang thai cua he lng t th ieu o co ngha lacho trc gia tr cua tap hp xac nh cua cac ai lung C hoc lng t . Tap hccac ai lung nay c goi la he u cac ai lung vat ly .

V du:Tap hp u cac ai lung vat ly xac nh trang thai cua electron co Spin

chang han co the la mot trong ba tap hp sau :x ,y ,z ,

Px , Py , Pz ,E , , m , l



43/128

C hoc lng t 42 -

Trong o : , m , la so lng t qu ao , so lung t t va so lung t Spin. Cac toa o va cac thanh cua xung lung ri vao hai tap hp khac nhau v cac ailung nay khong the o c ong thi . Do o cac he thc trong vat ly co ien :

l

)(2

2

rum

PE

r+=

= PrL X

rrr

Khong co y ngha khi chuyen sang C hoc lung t v cha ca xung lung va

toa o .4/ Xay dng he thc bat nh :Trong phan tren ta a tm c ieu kien ma trong o co the o ong thi hai

ai lung vat ly . Bay gi gia thiet ta co hai ai lung vat ly G va F khong co ongthi cac gia tr xac nh khi o cac toan t nay khong gia hoan vi nhau .

Gia thiet chung thoa man he thc :FG GF = iBTrong o B la mot toan t Hecmitic nao o . Neu chon G = PX , F = x , th B

= v :h

x PX - PX x = i .hNh vay van e at ra la : Khi ong thi o hai ai lung nay th s khong

chnh xac ( hay sai so ) se la bao nhieu ?.e giai quyet van e nay ta phai xet them hai khai niem la thang gian trung

bnh va thang gian toan phng .Thang gian la mc o sai lech gia gia tr trung bnh va gia tr thc .

Mc sai lech gia gia tr chnh xac a va gia tr trung bnh a la :aa = a - a

trong o :=i ii

aa

Thang giang trung bnh la : a . Lu y rang thang giang trung bnh luonbang khong . That vay :

( ) ( ) = == daaaaaaa )(

0)()( === aadaaadaaa Mat du vay ca lu y la co the a khac rat xa vi gia tr chnh xac a .V le o phai a vao mot ai lng khac e anh gia mc o thang giang o

la khai niem thang giang toan phng .Thang giang toan phng c nh ngha nh sau :

( ) 222 )(aaaTGTPa ==

Bay gi ta tm cac suy ra he thc bat nh mot cach chat che .Xuat phat t bat ang thc hien nhien :



44/128

C hoc lng t 43 -

02

)( + dxdx

xdx

trong o la so thc . Bieu dien di dau tch phan co the bien oi nh sau :

=++=+

dx

dx

dx

dx

dx

dx

*2

=++=

dx

dx

dx

dx

**

dx

d

dx

d

dx

dxx

++= 2222

.

Neu at :

=

=

=

dxdx

d

dx

dC

dxdx

dxB

dxxA

.

,

*

2

22

th bieu thc tren co dang :

22

2

4040

BACacBCBA

= +

Mat khac e y rang :

=== dxxdxxxA 2*222

( )

( ) 1**

*2

=+=

==

dxx

dxdx

dxdx

dx

dxB

2

2

2**

*

2

1X

P

dxdx

d

dx

ddx

dx

d

dx

dC

h

=

==+

Vay ta co :

1.

4 222 XPxh



45/128

C hoc lng t 44 -

hay :

4.

222 hXPx

Neu ta chon goc toa o tai iem vi toa o x th 0=x . Va gia s rang he

chuyen ong vi van toc mPX th 0=XP . Nh vay :

2222

2222

)(

)(

XXPPPP

xxxx

==

==

do o ta co :

4

222 h

XPx

hay :

2

h XPx.

Thong thng he thc nay viet gon la :

2.

h XPx

ay la bieu thc chnh xac cua he thc bat nh ma y ngha cua no ta a thaoluan phan tren .

7 HAM SONG . NGUYEN LY CHONG CHAP TRANG THAI.

I/. Ham Song:Thc chat cua viec mo ta mot c he nao o gom hai noi dung sau:- Mot la: Mo ta trang thai cua he c mot thi iem co nh a cho- Hai la: Mo ta s bien oi cua trang thai theo thi gian, hay noi cach khac la

mo ta chuyen ong cua c he.Nh vay c hoc co ien thc chat chu yeu ch nghien cu mat th hai cua c

he, tc la nghien cu chuyen ong cua no, con kha canh th nhat c thc hienbang viec xac nh toa o (suy rong) va van toc (suy rong) cua no. Tuy bai toan cuthe ma van toc hay toa o c xem xet mot cach oc lap.

Khac biet vi c hoc co ien, c hoc lng t cho phep xem xet van e thnhat mot cach ky lng hn.(cung can lu y rang toa o va van toc trong c hoc coien la cac ham do o chung giao hoan vi nhau). Trong c hoc lng t xac nhong thi toa o va xung lng la khong the c do o e mo ta trang thai cua c

hoc lng t ngi ta dung mot khai niem mi o la ham song ( )tr,r

. Khi o ||2



46/128

C hoc lng t 45 -

ty le vi mat o xac xuat tm thay hat cho xac nh nao o trong khong gian. Xacxuat tm thay hat tai the tch dV nao o cha iem (x,y.z)

d (x,y,z)= (x,y,z) 2dxdydzHam song phai thoa man bon ieu kien sau:

- Lien tuc- Hu han- n tr

- chuan hoa bang n v; ngha la: =

+

0

21dV

ieu nay co ngha la viec tm thay hat trong toan khong gian la chac chan.

II/. Nguyen ly chong chap trang thai:Nguyen ly chong chap trang thai la mot trong so cac nguyen ly c s cua vat

ly hoc. Trong vat ly co ien- cu the trong ly thuyet song nhat la trong song ien t-

nguyen ly nay co vai tro ac biet quan trong.Trong c hoc lng t nguyen ly nay con chiem mot vai tro quan trong hn

lien quan en kha nang mo ta c he lng t, kha nang giai thch y ngha vat ly cuacong cu cua c hoc lng t ly thuyet cac toan t Hecmitic va tnh oc lap cua che lng t vi dung cu o. Noi dung cua nguyen ly chong chap la:

Neu ton tai cac trang thai mo ta bang cac ham song trn ,r

th cung tontai trang thai:

( ) ( )tratrn

n

n,,rr

=

=1

S la chon tap hp cac hamtr

n,

r

gan lien vi s la chon cac ai lng

ac trng xac nh cua he lng t. Cac he so xac nh phan bo xac xuat cuacac ai lng tng ng.

na

Ve phng dien toan hoc so vi vat ly co ien nguyen ly chong chap trangthai lng t hoan toan tng t, ngha la phng trnh chuyen ong xac nh cactrang thai lng t cung la phng trnh tuyen tnh.

Mac du vay, ve noi dung nguyen ly chong chap lng t co nhng khac biet

quan trong,khac han vi nguyen ly chong chap trong vat ly co ien. e thay roieu nay ta xet mot v du. Gia s ham 1(x), 2(x) mo ta trang thai ma o chungta o ai lng A va nhan c gia tr xac nh A1 va A2 tng ng. Chong chaphai trang thai nay ta c trang thai mi:

( ) ( ) ( )xaxax 221112 += Trong vat ly co ien khi o ai lng A trang thai 12 (x) th ta nhan c

gia tr (A1+A2 )/2 nao o ngha la gia tr nhan c la trung bnh cong cua cactrang thai thanh phan.

Trong c hoc lng t tnh trang hoan toan khac: khi ta o ai lng A trangthai 12 (x) th ta nhan c gia tr hoac A1 hoac A2. Khong nhng the ta conkhong the tien oan chnh xac gia tr A1 hay A2 se xuat hien trong phep o cu the



47/128

C hoc lng t 46 -

ma ta ch co the nhan c xac xuat nhan c A1 hay A2 tng ng la2

1a va2

2a Nh vay co ngha la: S phan lap cac gia tr cua cac ai lng vat ly va s bat

nh trong ket qua o la ac trng c ban cua nguyen ly chong chap lng t.Giai thch ac iem trang thai cua vi hat Bohr a phat bieu nh sau: Trang

thai cua vi hat khong the xet mot cach tach biet vi phng tien quan sat, ma phaike en ieu ay, cac trang thai cua oi tng vi mo c nghien cu nhat thiet nhcac thiet b v mo ma chnh cac thiet b ay tuan theo cac nh luat cua vat ly coien.

Cung noi ve nguyen ly chong chap trang thai P. Dirac a viet: Khi t boquyet nh luan cua vat ly co ien chung ta a phc tap hoa s mo ta thien nhien,song s phc tap hoa nh vay c en bu mot cach xng ang bi s n gian hoa

do nguyen ly chong chap trang thai mang lai .

8 PHNG TRNH SCHRODINGER

Phng trnh Schrodinger la phng trnh c ban cua c hoc lng t vai trocua no trong c hoc lng giong nh vai tro cua phng trnh Newton trong c hocco ien. Phng trnh Schrodinger la phng trnh co dang:

Ht

i =

h

.trong o:H la toan t Hamilton hay Hamiltonien. Phng trnh Schrodinger

thiet lap moi quan he gia trang thai va bien ong lc-hai khai niem c ban cua choc lng t.

I/. Cach Thiet lap phng trnh:1/. Phng trnh Schrodinger cho hat t do:T dang song mo ta chuyen ong cua hat t do:

( )

[ ]Etrpi

Aetr

=

rr

hr

, . (1)

Ta co: tiEE

i

t ==

hh

( ) 22222

2

2

2

2

2 1zyx PPP

zyx++=

+

+

h

t he thc nang lng: mP

E2

2

=ta suy ra



48/128

C hoc lng t 47 -

=

mt

i2

2hh

(2)

Phng trnh Schrodinger co nhng tnh chat sau:

+/ Phng trnh ao ham rieng tuyen tnh (lien quan en nguyen ly chongchap trang thai).+/ Phng trnh ch cha ao ham bac nhat theo thi gian v no gan lien vi s

bieu hien cua nguyen ly nhan qua. (T 0,rr

suy ra ( )tr,r

mot cach n tr).Nhng lu y rang do y ngha cua ham song chung ta ch co the hieu nguyen ly nhanqua theo ngha thong ke.

Nghiem cua phng trnh (2) co dang:

( ) ( )

Eti

ertrh

r

= ., (3)thay (3) vao (2) ta co phng trnh cho ham ch phu thuoc vao toa o la:

( ) ( ) 02 =+ rmEr h (4)

Phng trnh (4) xac nh ham (r) cho hat t do.2/. Phng trnh Schrodinger cho hat chuyen ong trong trng lc:Thay E bang (E-U) trong o U la the nang cua hat khi o phng trnh (4) tr

thanh:

( ) [ ] ( ) 0)(2 =+ rrUEmr h (5)Phng trnh (5)la phng trnh Schrodinger cho hat chuyen ong trong

trng the tuy y khong phu thuoc thi gian.Tong quat hn ta thay:

tiE

= h

vao phng trnh (5) va nh vay ta co phng trnh:

( )trrUmt

i ,)(2

2 rhh

+=

(6)

Neu at :U

mH +=

2

2h

th phng trnh co dang gon hn la:

Ht

i =

h

(7)Phng trnh lien hp phc cua (7) la:

**

Ht

i =

h

3/ Gia thiet ve y ngha xac xuat cua ham song cua M.Born:

e n gian ta xet bai toan mot chieu:



49/128

C hoc lng t 48 -

( )( ) ( )

( ) ( ) ( )txxUxmt

txi

txxUxmt

txi

,2

,

,2

,

*2

22*

2

22

+=

+

=

hh

hh

Nhan phng trnh au vi *(x,t) va phng trnh sau vi (x,t) roi tr i

nhau ta co:

( )

=

=

+

ttxmti

txxxmtt

i

**

22

2

*2

2

2*

2**

2

,2

hh

hh

hay:

02

**2 =

xxxmit

h

Tong quat hoa, trong trng hp ba chieu:

( ) 02

**2 =+

mit

h

(8)Nh vay neu at:2= *= la mat o xac xuat

Va:( )**

2 =

mij

hr

la mat o dong xac xuat th (8) codang het nh phng trnh lien tuc:

0=+

jt

r

trong ien ong lc hoc.(gia thiet Born).Neu ta chuan hoa: 2dr=1 th xac xuat cua hat nam trong dv la:

d = *(x,y,z,t) (x,y,z.t)dvMot ieu can lu y la trong trang thai dng mat o xac xuat khong phu thuoc

vao thi gian. Cung t cac ket qua nay ta suy ra rang: C hoc lng t ch nghiencu cac qua trnh trong o so hat la bao toan.

4/ Phng trnh Schrodinger dng:Ta xet trng hp the nang U khong phu thuoc thi gian khi o phng trnh:

+=

)(2

2

rUmt

ih

h

(9)

co the c giai bang phng phap phan ly bien so. That vay at:)()(, rtAtr =r



50/128

C hoc lng t 49 -

roi thay vao phng trnh ta co the tach (9) thanh hai phng trnh:

( ) ( )

)()(

tEAdt

tdAi

rErH

=

=

h

trong o E la hang so tach (-tr rieng). Phng trnh theo thi gian co nghiem

la:Et

i

CetA h

=)( Phng trnh th nhat cho cac nghiem (ham rieng) la:1,2 ,3 n va

cac tr rieng tng ng: E1,E2 .. En Nh vay nghiem cua (9) la:

( )tE

i

nn

n

ertr hrr

= )(, .hay tong quat hn la ham:

( ) =

nn

tEi

nnzyxeCtr

n

),,(, hr

Xac xuat tm thay hat tai iem nao o cua khong gian se khong phu thuoc thi

gian v ta co:

( ) ( ) ( )0,,,2

rtrtr nnn ==r

v

1= tEi n

e h

9 CAC PHNG TRNH CHUYEN ONG LNG T

I/ Nhan xet chung :Ta biet rang vi C hoc co hai khai niem c ban o la trang thai va cac bien

ong lc . Hai khai nien nay gan lien vi nhau mot cach chat che cu the la neu tabiet c cac ai lng ong lc th co the biet c trang thai cua he va ngc lai

. Khi mo ta s tien trien cua he theo thi gian ta co the dung cac phng trnhchuyen ong v vay cac phng trnh chuyen ong co vai tro quan trong .Trong C hoc co ien cac phng trnh chuyen ong c viet qua cac bien

ong la cac phng trnh Newton , phng trnh Hamilton Jacobi , phng trnhchnh tac Hamilton va qua ham trang thai la phng trnh Lagrange. Giai cacphng trnh nay chung ta se tm c cac ai lng ong lc la nghiem cua cacphng trnh o la cac ham so cua thi gian :

)(trrrr

= , )(tpprr

= , E = E(t) , )(tMMrr

= .



51/128

C hoc lng t 50 -

Tuy nhien e y rang trong cac bien ong lc o th rr

va pr

co vai tro c ban

hn so vi E va Mr

v hai ai lng sau co the bieu dien qua hai ai lng rr

va pr

,

v vay tap hp prrr

, c goi la trang thai cua c he :

Trong hnh thc luan Hamilton , ma c s cua no la phng trnh chnh tatHamilton :

i

i

i

i

q

H

dt

dp

p

H

dt

dq

=

= ;

(9.1)(Neu a vao moc Poisson th ta co the viet cac phng trnh nay di dang

oi xng cho ca cac toa o va cac xung lng lien hp )Bay gi ta xet en C hoc lng t .

Trong C hoc lng t trang thai c mo ta bang ham song tr,r

. Bietc ham song chung ta se thu c gia tr trung bnh cua bat c ai lng nao tcla biet c ay u ve he . Phng trnh chuyen ong c viet qua ham songchnh la phng trnh Schrodinger . Nh ta a biet :

Ht

i =

h

Nh vay xuat hien mot van e la : trong C hoc lng t co the viet c cac

phng trnh hoan toan tng t nh cac phng trnh chuyen ong cua c hoc coien cho cac bien ong lc hay khong ? Cau tra li la co the c . Tuy nhien elam c nh vay trc het ta phai xay dng vai nh ngha mi o la cac mocPoisson lng t va khai niem ao ham theo thi gian cua cac toan t .

II/ Cac moc Poisson lng t :1/ Cac moc Poisson trong c hoc co ien :ao ham toan phan theo thi gian cua mot ham F(q,p,t) nao o (phu

thuoc toa o q , xung lng p va thi gian t ) trong vat ly hoc co ien co dang :

+

+

=

K

K

K

K

Kt

p

p

F

t

q

q

F

t

F

dt

dF..

(9.2)Neu dung cac phng trnh chnh tat Hamilton trong o H = H(q.p) la ham

Hamilton nang lng toan phan trong vat ly hoc co ien bieu dien qua xunglng va toa o , th khi o ta co the viet :

=

+

=

KKKKK

p

F

q

H

q

H

p

H

t

F

dt

dF..

hay),( FH

dt

dF

dt

dF +=.

Trong o (H,F) la ky hieu cua tong ben ve phai c goi la moc Poisson coien .



52/128

C hoc lng t 51 -

Tong quat hn moc Poisson khong phai ch co the nh ngha cho H va F maco the nh ngha cho hai ham bat ky F(p , q) va G(p . q) tc la cho cac ham phuthuoc vao xung lng va toa o nh sau :

+

=

K KKKK p

F

q

G

p

G

q

FGF ),(

(9.3)Neu bien ong lc F khong phu thuoc tng minh vao thi gian th :

)0:(),( ==dt

dFFH

dt

dFdo

(9.4)

Bang ky hieu moc Poisson ta co the viet lai phng trnh Hamilton co dang :

),(),( KK pHdtK

dpqH

dtK

dq

== (9.5)

e y rang so vi (7.1) cac phng trnh (6.5) hoan toan oi xng oi vi toao va xung lng .

Ta con co vai moc Poisson ac biet khac na la :(qi , qj) = 0 ; (pi , qj) = ij ; (pi , pj) = 0 (9.6)T nh ngha moc Poisson ta suy ra cac he qua sau ay :a/ Tnh phai xng :

(F,G) = -(F,G)T o suy ra : (F,F) = 0b/ Neu G = const th :

(F,G) = (F,C) = 0c/ Tuyen tnh theo cac phan t :

(F1+F2 , G) = (F1,G)F2 + F1(F2,G)d/ Tnh phan bo oi vi phep nhan :(tng t quy tac lay vi phan mot tch ) .

(F1F2 , G) = (F1,G)F2 + F1(F2,G)e/ Thoa man ong nhat thc Jacobi :

(F,(G,H)) + (G,(H,F)) +(H,(F,G)) = 0

2/ Moc Poisson lng t :Ta vhuyen sang bai toan C hoc lng t

Giáo trình Cơ học lượng tử

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