🔙 Quay lại trang tải sách pdf ebook Giáo trình vật lý điện tử Ebooks Nhóm Zalo ■f PHUNG HO G I A O T R I N H D i E N T O n h A x u a t ban k h o a h o c v A k y t h u a t TRUING DAI HOC BACH KHOA HA N0I GS PHUNG HO g iAo t r i n h V A T LY D l l N TUf * * (In Ian thic2) NHA XUAT b An KHOA HOC VA KY THUAT HA NOI 2007 3 Muc luc Chiromg 1 Chuyen dong cua hat tich dien trong chan khong. 9 1.1 Phuang trinh chuyen dong trong dien trucmg va tCr trucmg .... 9 1.2 Chuyen dong cua hat tich dien trong dien trucmg d e u .................. 11 1.3 Chuyen dong cua hat tfch dien trong tijr trucmg d e u ...................... 14 1.3.1 Trucmg hop van toe ban d^u vuong goc vdri tOr trucmg. . . 15 1.3.2 Trucmg hop tong quat................................................................. 17 1.4 Chuyen dong duoi tac dung dong thcri cua dien va t€r trucmg . . 19 1.4.1 Dien truong va tvr truomg song song....................................... 19 1.4.2 Dien truomg va rtr trucmg vuong goc.......................................20 1.5 Sir tuong tu quang co ......................................................................... 23 1.6 Chuyen dong trong trucmg tinh dien khong deu, thSu kfnh tlnh d i e n ........................................................................................................... 28 1.7 Chuyen dong trong tvr trucmg tlnh khong deu, th£u kfnh tvr . . . 34 1.8 Chuyen dong vod van toe gdn bang v^n toe anh s a n g ...................36 Chiromg 2 Mot so van de vat ly lirqmg tir. 43 2.1 Luang tmh song hat cua cac he vi m o ...........................................43 2.1.1 Luang tinh song-hat cua hire xa dien rtr................................ 44 2.1.2 Luang tinh song-hat cua vat chat............................................ 48 2.2 Cac tfnh chat cua song De B ro g lie .................................................... 51 2.2.1 Ham song phang De Broglie.................................................... 51 2.2.2 Y nghia xac suat cua song De B roglie..................................53 2.2.3 Nhom song. Su lan truyen cua song De Broglie..................54 2.3 Nguyen ly bat dinh H eisenberg.......................................................... 58 2.3.1 He thtic bat dinh v6 toa do va xung luang cua hat..............58 2.3.2 He thurc bat dinh ve nang lugfng va thcri gian........................59 2.3.3 Nguyen ly bo sung. ................................................. 60 2.4 Phuang trinh ca ban cua ca hoc lirang tu .....................................60 4 Muc luc 2.4.1 Hat tu do.........................................................................................01 2.4.2 Hat trong mot trucmg lire............................................................^2 2.5 Toan tvr trong ca hoc lucmg t u .............................................................. ^3 2.5.1 Khai niem toan tu.........................................................................63 2.5.2 Toan tu va cac dai lucmg vat ly.................................................^6 2.6 Mot so bai toan don gian cua co hoc lucfng t u ................................70 2.6.1 Chuyen dong cua hat tu do........................................................ 70 2.6.2 Hat trong ho the nang..................................................................71 2.6.3 Hieu ting ducmg ngam (tunnel)................................................. 76 2.6.4 Dao dong tu dieu hoa (oscillator).............................................78 2.6.5 Quay tu ran.....................................................................................80 Chuomg 3 Pho nang luong cua cac he hat luomg tu. 85 3.1 Bai toan ve nguyen tu hydro va cac ion dong d a n g ...................... 86 3.2 He nhieu dien tu, nguyen ly loai trir Pauli ......................................95 3.2.1 He nhieu dien tu............................................................................95 3.2.2 Spin cua dien tu.............................................................................97 3.2.3 Nguyen ly khong the phan biet cac hat cung loai. Nguyen ly loai trU Pauli.............................................................................. 100 3.2.4 Cau hinh dien tu cua nguyen tvr................................................ 102 3.3 Trang thai dien tu cua nguyen t u .........................................................105 3.3.1 Trang thai dien tu cua nguyen tu hydro..................................108 3.3.2 Pho nang luong cua nguyen tu He, para-He va orto-He. .1 0 8 3.3.3 Pho nang luong cua cac nguyen to nhom II..........................I ll 3.3.4 Trang thai dien tu cua nguyen tu carbon................................114 3.3.5 Mot so quy luat trong pho nang luong cac nguyen tu . . . 114 3.3.6 Quy tac Hund.................................................................................115 3.3.7 Chuyen muc nang luong, quy tac chon loc............................116 3.4 Pho nang luong cua phan t u .................................................................. U 7 3.4.1 Trang thai nang luong dao dong............................................... 120 3.4.2 Trang thai nang luong quay........................................................12] 3.4.3 Quang pho cua phan tu............................................................. 12] 3.5 Pho nang luong cua dien tu trong tinh the chat r a n ................... j 23 3.5.1 Ham song dien tu trong trucmg tuan hoan..........................• 125 3.5.2 Phucmg trinh Schrodinger cua tinh th e ................................ ]26 3.5.3 Phuang phap g in dung liftn ket yeu.................................... 129 Muc luc 5 3.5.4 Phuang phap gin dung lien ket manh...................................130 3.5.5 Dieu kien bien ftiin hoan cua ham song............................... 132 3.5.6 Pho nang luong dien tu khi co tvr trucmg ngoai tac dung. Cac hien tuong cong hucmg trCr.................................................134 3.6 Nguyen ly hoat dong cua la s e r..........................................................140 3.6.1 Mo hinh hai mufc nang luong...................................................140 3.6.2 Nguyen ly hoat dong cua l a s e r ............................................146 C hoong 4 Mot so van de vat ly ban dan. 157 4.1 Dien tu trong tinh the ban d a n .......................................................... 157 4.1.1 Su thay doi muc nang luong trong dien trucmg ngoai. . . 157 4.1.2 Chuyen dong cua dien tu trong tinh the ban d in ................ 160 4.2 C iu true vung nang lucmg cua ch it ban d i n ..................................163 4.3 Nong do dien tvr va 16 trong trong dieu kien can b a n g ............... 165 4.3.1 Ban din rieng (ban din tinh khiet)......................................... 167 4.3.2 Ban din chua mot loai tap chit dono..................................... 168 4.3.3 Ban d in chua mot loai tap ch it axepto.................................. 171 4.3.4 Ban d in bu trvr, chuta hai loai tap chat....................................172 4.3.5 Ban d in suy bien.........................................................................173 4.4 Nong do hat din khong can bang .................................................... 173 4.5 Do d in dien cua chit ban d i n ............................................................. 176 4.6 Cac hien tuong phat xa dien tv r .......................................................... 179 4.7 Cac hieu ring tiep xuc ..........................................................................183 4.7.1 Hieu the tiep xuc ngoai..............................................................183 4.7.2 Hieu the tiep xuc trong giua hai kim loai..............................184 4.7.3 Hieu ung trucmg. Su uon cong vung nang luong cua ban d in trong dien truong................................................................. 184 4.7.4 Tiep xuc kim loai-ban din loai n ........................................... 188 4.7.5 Tiep xuc kim loai-ban din loai p ........................................... 190 4.7.6 Dac tuyen V-A cua cac tiep xuc dong kim loai-ban din. . 193 4.8 Ban d in khong dong nhit, chuyen tiep p - n ..................................... 194 4.8.1 Lop chuyen tiep p - n .................................................................195 4.8.2 Su phan bo dien tich trong chuyen tiep p-jz dot bien . . . 195 4.8.3 Dien trucmg trong ldp chuyen tiep p-n ...............................196 4.8.4 So do vung nang luong trong chuyen tiep p - n ...................198 4.8.5 Phan bo nong do hat din trong lop chuyen tiep p-n . . . . 199 6 Muc luc 4.8.6 Tinh chinh liru cua chuyen tiep p - n ......................... ' . 4.8.7 Be day 16p chuyen tiep p - n ........................................... • ' 4.8.8 Chuyen tiep di c h it...................................................... • • ' Chirong 5 X u ly tin hieu quang. ^ 5.1 Song dien tv r................................................................ ............................^ 5.1.1 He phuang trinh M axwell..........................................................^ 5.1.2 Phuang trinh song doi vdd trucmg dien tvr. ......................... 5.1.3 Song dien tvr phang...................................................................... ^ 5.2 Su lan truy£n cua anh sang trong linh kien d in song quang . • 218 5.2.1 He sai quang - mach quang tfch hop va nhung dac diem. 218 5.2.2 Kieu truyen song (mode) trong linh kien d in song quang. 220 5.2.3 Phuang thric truyen song trong linh kien d in song quang theo mo hinh quang hoc tia (ray-optics)................................ 224 5.2.4 C iu true linh kien d in song quang trong m ach quang tich hap .................................................................................................... 229 5.3 Sai q u a n g .................................................................................................... 232 5.3.1 Tinh chit,, phan loai va cong nghe che tao............................ 232 5.3.2 Cac mode d in truyen trong sod quang giat c ip ....................236 5.3.3 Svr han ch£' do rong dai thong do tan sac giua cac mode trong sod quang giat cap da m ode............................................ 240 5.3.4 Sai quang lien tuc da m ode........................................................242 5.3.5 Sai quang giat cap dan m ode.....................................................243 5.3.6 Ton hao trong sod quang............................................................. 245 5.4 Nguyen ly dieu bien q u a n g .................................................................. 246 5.4.1 Nguyen ly dieu bien dien quang...............................................248 5.4.2 Nguyen ly di6u bien ca-quang.................................................. 251 5.5 Nhung nguyen ly ca ban cua cac linh kien phat quang ban d in . 253 5.5.1 Dac diem cau true vung nang luong cua ch it ban din. . 253 5.5.2 Chuyen mtic nang luong, qua tnnh hap thu. 255 5.5.3 Tai hop bu:c xa tvr phat. Diot phat quang (LED). . 258 5.5.4 Tai hop burc xa kich thich. Laser ban din. . . 253 5.6 Nhung nguyen ly cO ban cua linh kien thu quang (photodetecta) 272 5.6.1 Linh kien quang d in hay quang tro......................... 5.6.2 Photodiot......................................................................... 275 7 L e r i n o i d a u Tvr lau vat ly da la ca so khoa hoc cua nhieu nganh ky thuat, dac biet la nganh ky thuat dien tu, viln thong. Cang ngay cang nhieu nhung thanh tuu khoa hoc vat ly duac nhanh chong ting dung vao ky thuat dien tvr, vi6n thong. Ky su dien tvr, han ai het, can co nhung hieu biet sau sac ve vat ly hien dai. Vi vay ngoai giao trinh "Vat ly dai cuang" chung cho tat ca cac nganh ky thuat, trong chuang trinh dao tao ky su dien tu duac bo sung them giao trinh "Vat ly dien tvr”. Giao trinh "Vat ly dien tu" nham muc dfch cung cap them cho sinh vien nhung hieu biet ve vat ly hien dai lien quan true tiep den nganh ky thuat dien tu, vifin thong. Nam 1981, Pho giao su Nguygn Minh Hien, vai muc dfch tren, da bien soan tap "Cac bai giang ve vat ly dien tu". Gan day do yeu cau hien dai hoa noi dung chuang trinh giang day, khoa Dien tvr - V iln thong trucmg Dai hoc Bach khoa Ha noi da xay dung mot d6 cuang mdfi cho mon hoc "Vat ly dien tu". "Giao trinh Vat ly dien tvr" nay duac bien soan dua theo de cuang do. TUy nhien, de cho giao trinh duac mach lac, su sap xep cac chuang muc trong giao trinh co it nhieu thay doi so vdi d l cuang mon hoc. Giao trinh duac chia lam nam chuang, gom ba dan vi hoc trinh vdi thcri lucmg 45 tie't. Chuang thur nhat trinh bay "Chuyen dong cua cac hat tich dien trong chan khong dudi tac dung cua dien trucmg va tir trucmg". Chuang thu: hai gidri thieu "Mot so v3n de vat ly luong tu", cung cap nhung hieu biet ca ban de tiep tuc nghien cuu nhung chuong sau. Chuang thur ba trinh bay "Pho nang luong cua cac he hat luong tu", tvr nguyen tu hydro don gian, nguyen tu co nhieu dien tvr, phan tvr va cuoi cung den tinh the chat ran. Chuang thur tu de cap "Mot so van de Vat ly ban dan", gicn thieu nhung khai niem can thiet nhat de nghien cuu vat ly linh kien dien tu, linh kien quang dien tu. Chuang thur nam gidi thieu "nhiing ca sa vat ly cua qua trinh truyen va xu ly tin hieu quang", de cap nhung co so vat ly cua cac linh kien quang dien tu nhu linh ., ‘ W g . ^ i Quang, dieu bien tin hieu quang, nguon phat quang, u in ieu quang, tach song quang. Cuoi m6i chuang co bai tap kem theo dap so. • Giao tnnh Vat ly dien tu nay da duac luu hanh noi bo tu nam 1976 va uyc tac gia cung mot so dong nghiep su dung de giang day cho sinh vien he c inh quy va hd tai chiic nganh dien tu-vien thong tnrcmg Dai hoc Bach khoa a n6i cung nhu cho sinh vien nganh dien tu-viin thong a m ot vai tnrcfng dai oc khac trong nude. Do yeu cau cap bach ve tai lieu giang day va hoc tap, nghien cuu chung toi xuat ban rong rai "Giao trinh Vat ly dien tu" nham phuc vu kip thcri dong dao ban doc. Chung toi rat mong duac su gop y cua cac ban doc ve nhung sai sot chac chan kho tranh khoi trong sach. Tac gia. Quran g 1 CHUYEN D 0 N G CUA CAC HAT TICH DIEN TRONG CHAN KHONG DUtil TAC DUNG CUA DI$N TRl/CfNG VA T l/T R U IN G Ngay nay rat nhieu nhung dung cu dien tu chan khong (thuong goi la den dien tu) da 16i thcri va khong duac su dung. TUy nhien vin con nhieu dung cu dien tu quan trong umg dung su chuyen dong cua cac hat tich dien dudi tac dung cua dien va tir trucmg nhu: ong tia dien tu cua may dao dong ky dien tir, ong hinh cua may thu hinh, kfnh hien vi dien tvr, may gia toe, khoi pho ke...Vi vay nhung kien thuc ve su chuyen dong cua dien tvr trong chan khong dudi tac dung cua dien va tir trucmg van rat can thiet cho nhung chuyen gia cua nganh dien tvr. 1.1 Phuang trinh chuyen dong cua hat tich dien trong dien trucmg va tir trucmg khong doi Gia sir trong khong gian ton tai mot dien trucmg tinh voi vec-to cuang do dien truofng £ va mot tir trucmg tinh voi vec-to cam ung tir B. Mot hat co khoi luong m, co dien tich q se chiu mot luc tac dung la F = q£ + q[v x B] ( 1. 1) Trong he SI, F do bang N, q do bang C, £ do bang V/m, v do bang m/s va B do bang T. Phuang trinh chuyen dong cua hat se la: d(mv) - dt = q£ + q[v x B] ( 1.2 ) 10 Chucmg 1. Chuyen dong cua hat rich dien trong chan khong Doi vdi cac hat co v « c ( c - van toe anh sang ) co the coi m la khong doi va phuang trinh ( 1.2) co dang: m ^ - = q£ + q[v x B] (1-3) at Nh&n vo hudmg hai ve cua (1.3) voi t?va bien doi ta co: = qv ■ £ (1 -4) m d(v2) 2 dt TCr day ta co nhan xet: a/ Su bien doi dong nang cua hat chi do tac dung cua dien truong, tu truong tlnh khong the lam thay doi dong nang cua hat, tu truong chi co the lam thay doi hudmg chuyen dong cua hat. b/ Gia su trong khoang thai gian tir tx den t2 hat chuyen dong tu diem A x den diem A2. Lay tich phan hai ve phuong trinh (1.4) doc theo quy dao A XA2 ta co: ^ ^ = q [ A* £ds = - q gradUds = q{Ux - U2) (1.5) L L J A , J A , v\,v2 - van toe cua hat tai A x va A2, tuong urng, UU U2 - dien the tai cac diem A x va A 2, tuong ting. Phuong tnnh (1.5) co the viet lai: mv\ TT mvo .. — qUi — —-----(- qU2 (1.6) 2 2 Vay nang luong toan phan cua hat tai A x va A2 bang nhau. c/ Neu cho van toe ban dau tai A x bang khong, tu (1.6) ta co 2 TTLV — =q(Ux - U 2) = qU (1.7) U = Ux - U2 la hieu dien the giua hai diem yliva A2. Ta di den ket luan khi dien tfch chuyen dong trong truong tinh dien su thay doi dong nang cua hat duoc xac dinh boi hieu dien the U. Dua vao day nguoi ta dua ra mot don vi nang luong goi la electron-volt (eV), la nang luong ma mot dien tich q = e = 1, G • l()-19C thu duoc khi di qua mot hieu dien the tang toe bang IV; leV = 1,6 • 10~19J; 1J = 6,24 ■ lO'^eV. Tu (1.7) khi biet dien tfch, kho'i luong cua hat ta co the tfnh duoc van toe v cua hat. n = A/ — doi vdi dien tu v{ — ) = G • 105\/U{V) (1 8 ) m s ' 1.2. Chuyen dong cua hat tich dien trong dien trucmg deu 11 ( H inh 1.1: Set dS mo ta chuyen dong cua dien tu trong ong tia dien tu dung tu dien lai tia. Q ta lim y rang dong nang cua dien tich thu duoc chi phu thuoc vao dien the diem ddu va diem cuoi cua chuyen dong, chinh dieu nay chung to trucmg tinh dien la mot trucmg the. Tir phuong trinh (1.3) neu biet su phu thuoc cua cac truong vao toa do, £(r),B(r), biet cac dieu kien ban ddu cua dien tich ta co the giai bai toan de tim duoc quy dao cung nhu cac thong so khac cua chuyen dong cua dien tich. Giai bai toan tong quat se phuc tap nen ta se xet mot so trucmg hop rieng sau day. 1.2 Chuyen dong cua hat tfch dien trong dien truomg deu Xet chuyen dong cua hat co dien tich q, co khoi lucmg m, trong tu dien phing voi dien truong dong deu E, ta co: dv =* ( l '9> Voi dieu kien bien la: tai thcri diem t = t0, hat co toa do la r0 va van toe la w0, lay tfch phan ( 1.9) ta co: * = * + j to ^ dt = m ^ { t ~ t o) +i J o = J t ( 1•1 ° ) 12 Chucmg 1. Chuyen dong cua hat tich dien trong chan khong Tfch phan (1.10) ta co f = fo + f t iJdt= ~ io)2 + M i ~ io) + ro r = ~^~£{t - t0)2 + v0{t - t0) + r 0 ( 1 - 1 1 ) Z m Chung ta xet trudng hop don gian va thong dung nhat, truong hop chuyen dong cua chum dien tu trong tu dien phang la hai ban lai tia cua ong tia dien tu trong may dao dong ky dien tu, nhu ve 6 hinh ( 1.1). C) day chung ta chon £ = (0,0, £z), chieu dai cua tu dien phang la U, khoang cach giua hai ban lai tia la d, van toe ban dau cua dien tvr song song voi true x, v0 = vXo, va duoc xac dinh bdi dien the cua cue gia toe dien tu U0, nghia la Uo = Vxo = ^J^Uo- Hieu dien the giua hai ban lai tia la UK, dien truong trong tu dien phang la £ = Chon goc toa do r0 = 0 tai diem dien tu bat dau di vao vung tac dong cua dien truong cua tu dien vao thcri diem t0 = 0. Tu nhung gia thiet tren va phuong trinh ( 1. 10) ta co: v = ——£t + uo va r — —^-£t2 + v0t (1-12) m 2m Viet (1.12) duoi dang cac thanh phan: \ v x = vZo = ^ U o ( x = vXot = Jm T o ■ t < vy = 0 y = 0 (1.13) vz = - ± £ zt = - ^ t { z = - £ - £ t 2 = - £ . U f t2 Chung ta thay qui dao cua dien tu nam trong m at phlng (xoz) va duoc bieu diSn boi cong thuc e UK x2 - U K 2 2 = -------- — T7-— = x 2m d u 0 TV 4d£/( o z = Tu day ta co the ket luan: - U K 2 4 d.Uo y = o ( i.i4 ) - Quy dao cua dien tu la mot duong parabol. - Voi UK va UQ cho trudc quy dao khong phu thuoc vao cac dac trim 2 Clj a hat (khoi luong, dien tich), di£u do co nghia la trucmg tinh dien khong co kha nang phan tach cac hat theo cac dac trung cua chung. 1.2. Chuyen dong cua hat tich dien trong dien trucmg deu 13 - Neil thay doi UK va U0 cung mot so ldn nhu nhau quy dao cua hat se khong doi. Khi ra khoi vung tac dung cua dien trucmg trong tu dien, dien tu se chuyen dong thing deu va dap len man huynh quang M. Chung ta co the tinh duoc do lech cua chum dien tu D so vdi tarn cua man, nhu ve o hinh (1.1) ta co: D = D , + D J > ,= » = h 11 do do D\ = —; e Uk - ~ Uk px Vxo y f W * D - - Uk l' U„ id TCr (1.15) ta th^y: !m J ® Uc id \[W « ( I J 5 ) - Do lech cua chum dien tu cang 16m khi dien ap tang toe cang nho, khoang each tu vung co dien trucmg den man hinh cang 1dm. Dua tren nguyen tac do nguoi ta thiet ke cac ong tia dien tu. - Do lech cua chum dien tir ty le thuan vdfi UK, hieu dien the giua hai ban lai tia. Trong ong tia dien tu, de dac trung cho su phu thuoc nay ngudi ta dua ra mot dai luong goi la do nhay S cua ong. Ve do 1dm, do nhay 5 bang do lech cua chum dien tu gay nen bdfi su thay doi dien ap lam lech (hieu the giua hai ban lai tia) co do 1dm IV. S = £ = -2w A +k) (,J6) Thong thucmg do nhay trung binh cua ong tia dien tir co S = 0 ,25mm/V. - Tren dudmg di cua chum tia dien tu chung ta co the dat hai cap ban lai tia (hai tu) vuong goc voi nhau. Tu co ban vuong goc vdi true 2 gay nen do lech theo true 2 (theo phuong thang dumg). Tu co ban vuong goc voi true y gay nen do lech theo true y (theo phuong nam ngang). - Neu dien ap tren hai ban lai tia la dien ap xoay chieu thi cac dien tir se dao dong lien tuc giua hai phien va do do diem sang tren man hinh cung chuyen dong lien tuc. Neu dien ap tren hai ban la "dien ap rang cua" thi diem 14 Chucmg 1. Chuyen dong cua hat tich dien trong chan khong H inh 1.2: So do ong tia dien tu dung cuon day tu: trucmg lai tia. sang se chuyen dong d£u tren man hinh va no co the dong vai tro nhu mot "true thcri gian" tren man hinh. Bkig each dat tren hai tu lai tia nhung dien ap rang cua thich hop co the thuc hien viec quet toan bo man hinh theo timg dong mot. - Toe do cua tia dien tir r£t ldfn gSn nhu tire then vdi su thay doi dien ap, nen ong tia dien tu la m ot dung cu gdn nhu khong co quan tinh. Tuy nhien neu tin so cua dien ap qua cao ( / > 109Hz) thi nhan xet tren khong con dung nua. 1.3 Chuyen dong cua hat tich dien trong tir truomg deu Xet chuyen dong cua mot hat co dien tich q, khoi lucmg m trong khong gian co tir trucmg deu vcri vec-to cam ung tir B ta co m ^ = q [ v x B ] (1.17) Tir day chung ta co nhan xet: - Gia toe cua hat a = f - luon luon vuong goc vdi vec to van toe v. Vi vay nhu tren ta da nhan xet, tac dung cua tir trucmg khong lam thay doi dong nang hay do Ion cua van toe hat. 1.3. Chuyen dong cua hat tich dien trong tit trucmg deu 15 - Ta co the viet (1.17) dudi dang cac thanh phan (1.18) 1.3.1 Truomg hop van toe ban dau vuong goc voi tu truomg. Chung ta xet trucmg hop don gian va thucmg dung nhat, do la truong hop su dung cuon day tu truong nhu mot cuon lai tia trong ong tia dien tu, trong do vec-to cam ung tir vuong goc vdi vec-to van toe ban dau v0 = vXo = \ J ^ U 0 \B = (0, By, 0), chieu dai vung tac dung cua tir truomg theo true x\a lu khoang each giua vung tac dung cua tir truomg den man hinh la l2, nhu ve o hinh (1.2) Vi v0 = vXQ, nghia la vyo = 0, Bx = 0, B z = 0 cho nen ta co the ket luan vy = 0; y — 0 va quy dao cua dien tu la mot dudmg cong phang nam trong mat xOz vuong goc vdri dudng sire tir trucmg. Vi gia toe cua chuyen dong luon vuong goc vcri vec-to van toe va do 1dm cua van toe khong doi |i7| = |tT0| cho nen quy dao cua dien tu phai la mot dudmg tron, trong do lire tac dung cua tir truomg la luc hudmg tam, ta co W 2 - mvO r, /I im -— —evBy vi v = vo nen — ^-=evoB (1-19) R K Tir day ta rut ra duoc ban kfnh cua quy dao chuyen dong ( 1.20) Thcri gian di het mot vong cua quy dao, nghia la chu ky cua chuyen dong quay la (1.21) Tan so quay / va van toe goc cua chuyen dong la (1.22) Tan so quay cua dien tir trong tir trucmg vuong goc duoc goi la tan so cyclotron. Dien tir co the thuc hien duoc chuyen dong quay chi trong truomg hop vung tac dung cua tir truomg rong va tir truomg du 1cm. Trong trucmg hop chung ta xet trong ong tia dien tir vung tir tnrcmg /, hep va tir truomg nho, quy dao cua dien tir chi la mot cung tron, ra khoi vung tac dung cua tir trucmg dien tir se chuyen dong thang deu va dap len man hinh. 16 Chucfng I . Chuyen dong cua hat tich dien trong chan khong Hinh 1.3: a- cuon lai tia co loi sat, b- cuon lai tia khong co loi sdt. Neu do lech trong tu tritong khong lcm chung ta co the viet m ot cach gdn dung do lech cua dien tu trong vung tac dung cua tCt trucmg: ^ x _ x2eB _ eBx2 _ j e B x 2 2 R 2mv0 2m\J~^Uo • ‘i.mUo 2 Til (1.23) chung ta co nhung nhan xet sau: - W6\ x 0|| + Quy dao cua chuyen dong tong hop se la nhung dudmg Id xo co budrc thay doi ddn nhu bieu d iin o hinh (1.7). Quy dao cua chuyen dong tat nhien con phu thuoc vao chieu cua £ so vdri u0||. 1.4.2 Dien truomg va tur tririmg vuong goc. Xet chuyen dong cua mot hat co dien tfch q, khoi luong m, co van toe ban dau v0 - (vox,voy,0), dr thdri diem t - 0, f 0 - 0, trong dien trucmg £ = (<£TX, 0, 0) va tir truomg B = (0,0, B z), vuong goc vdri b i mat giay nhu dr hinh ve (1.8). Phuong trinh (1.3) co the viet lai duoi dang cac thanh phan 1.4. Chuyen ddng du&i tac dung dong th&i cua dien va tic truomg 21 Hinh 1.8 : Quy dao cua dien tich du&i tac dung cua dien va tic tru&ng vuong goc. Hay la — = 3-g+ dt2 m m dt “JL = (1.27) dt2 m dt d h , dt2 Neu gia thiet voz = 0; z0 = 0 chung ta co the thay quy dao la mot ducmg cong phang n k n trong mat phkig xOy (mat gi£y). Lay tfch phan phuong tnnh thu: 2 ta co: ^ ^ - - — X + V0y = -L O X + V 0y (1.28) voti u = Thay (1.28) vao phuong tnnh thu: nhat cua (1.27) ta co d2 x o a£ w + u x = - + wv„ (1 2 9 ) Nghiem tong quat cua (1.29) la x = Ci sin uit + c2 cos u t + -I- — , ( 1 30i mujz u> v > trong do cu c2 la cac hang so tich phan. Tir (1.30), ta co dx — — c\u cos u t — C2U)smut (1.31) 22 Chucmg 1. Chuyen dong cua hat tich dien trong chan khong Tvr cac d iiu kien bien: t — 0, X — 0, y — 0, Vx — Vox jV y — V0 ta co the tinh duoc ci,c2, ket qua tinh toan cho ta m qB , m m £ qB x w = — v „ s m — < + ( ^ » + 5 5 l ) ( 1 - <:° s V ‘ ) ( 1 3 2 ) S uiVnx qB , m m £ . . qB Neu cho v0 = 0, vox = voy = 0 ta co (1.33) voi 0.34) x(t) = ^C 1 - cos 0 = ^ ( 1 - cosw^ :i B B m . ^ m £ . qB £ y{t) = - —t + — pSin — t = - —« + B.0 sin ut B qB B m B Cac phuong trinh (1.33) la phuong trinh tham so bieu d iln m ot dudng cong goi la cycloit la quy dao cua dien tich. Ducmg cong cycloit chinh la quy dao cua m 6t diem tren ducmg tron co ban kfnh Rq n lm trong mat phang xay va lan tren true y vdfi van toe goc u, nhu bieu didn tren hinh ( 1.8). Phuong tnnh (1.33) cung cho ta thay van toe trung binh cua hat theo true y la u = va chuyen dong cua hat co the xem nhu gom hai dang: chuyen dong tinh tien theo true y vdri van toe trung binh u va chuyen dong quay vdd tan so goc u. Do cao cuc dai cua cycloit ( imax) dat duoc khi cosuit — — 1, u>t = (2k + l)n va bang Xmax = 2Ro Quy dao cua hat cat true y tai nhung diem cach deu nhau ung vdri x m i n = o khi cos tot — 1, uit = 2n. Khoang cach giua nhung diem cat nhau bang Ay = 2nE<). Dang chuyen dong cua dien tu tren day duoc sir dung nhu nguyen ly hoat dong cua mot loai dung cu dien tvr magnetron, mot loai den phat sieu cao tan. Tren hinh (1.9) ve so do mot loai magnetron: magnetron phang (nhu mot may phat sieu cao tan ung dung su tucmg tac giua chum dien tvr va song dien tir). M agnetron gom co catot phat dien tvr va anot co cac hoc cong hucmg. dat trong tir truong. Dien tu phat ra tir A' duoi tac dung cua dien va tir trudne 1.5. Su tucmg tu quang ca 23 Loi vao Loi ra A K Hinh 1.9: Nguyen ly hoat dong cua magnetron phdng. vuong goc vdi nhau se chuyen dong theo cac quy dao cycloit. Do cao cac cycloit phu thuoc,vao dien ap tren hai ban va tir trudng. Chinh quy dao dac biet nay cua dien tu da tao nen su tuong tac va trao doi nang luong giua chum dien tu va song dien tir trong magnetron va duoc su dung nhu nguyen ly hoat dong cua magnetron. 1.5 Sir tirang tu quang ca Chung ta biet rang trong quang hoc co mot nguyen ly quan trong do la nguyen ly Ferma hay Con goi la nguyen ly then gian cuc tieu. Theo nguyen ly nay, khi tia sang lan truyen tir diem A den diem B thi trong so' tat ca cac quy dao co the no se lan truyin theo quy dao ma thdi gian can thiet de di het quy dao do la cuc tieu. Nguyen ly do duoc diin ta bang toan hoc dudi dang mot bien phan nhu sau: r B r B I r B 6( dt)=5{ — ) = S( nds) = 0 (1.35) J A J A v J A trong do d la toe do lan truyin cua anh sang trong moi trudng co chiet sua't la n \ v = c la toe do lan truyin cua anh sang trong chan khong. Tir (1.35) nguyen ly Ferma cung co the phat bieu: quang lo cua tia sang theo quy dao that cua no bao gid cung la cuc tri. Trong co hoc ta cung co nguyen ly tuong tu nhu nguyen ly Ferma, do la nguyen ly tac dung cuc tieu, duoc bieu d iin bang toan hoc nhu sau: doi vdi mot hat chuyen dong trong trudng the ta luon co: r B r B mv2 J Tdt) = 6(J - j - d t ) = 0 (1.36) 24 Chucmg 1. Chuyen ddng cua hat tich dien trong chan khong trong do T la dong nang cua hat v la van toe cua hat. Vi dien trudng la mot trudng the nen chuyen dong cua hat tich dien trong trudng tTnh dien phai tuan theo nguyen ly tac dung cue tieu. Tur (1.36) ta co the viet lai: S(I " ' ^ f dt} = S(L \ m^ di ) = S(L hay la S ( J B vds) = 0; v = ^ U ; 6(J VUds) = 0 (1.37) Tir day chung ta thay rang co the xem quy dao cua hat tich dien trong trudng tinh dien nhu dudng di cua tia sang neu trong khong gian co the xem n tucmg tu nhu \ftJ, nghia la su thay doi gia tri can bac hai cua dien the trong khong gian giong nhu su thay ddi cua chiet suat trong moi trudng. Su tuong tu do goi la sir tuong tu quang co, no cho phep xay dung m ot thu quang hoc cac chum dien tu, chum ion giong nhu quang hoc cac tia sang. Nhu chung ta biet trong quang hoc co ba dinh luat co ban: 1. Trong moi trudng dong nhat (co chiet suat dong deu) anh sang bao gid cung truyen thing. 2. Khi tia sang di td moi trudng co chiet suat sang moi trudng co chiet suat n2, no bi khuc xa d mat gidi han giua hai moi trudng va ty so' giua goc tdi va goc khuc xa thoa man dieu kien sin a n2 sin/3 n i' 3. Goc phan xa cua tia sang tren mat phang gidi han giua hai moi trudng bang goc tdi a = a1 (xem hinh ( 1.10)). Doi vdi chuyen dong cua hat dien tfch trong trudng tlnh dien ta cung co :he phat bieu nhung dinh luat tuong tu: 1. Trong vung co dien the khong doi (U = const) hat dien tfch chuyen dong hang (chu y v ~ VU). 2. Neu chum hat tfch dien phan xa tren be mat ding the thi goc tdi bang loc phan xa. 7 , O day chung ta hay xet dieu kien phan xa mot chum dien tu chang han. ludng mot chum dien tu co van toe ban d£u bang {T0 vao mot be mat kim loai 1.5. Su tucmg tu quang ca 25 Hinh 1.10: Hien tuong phan xa va khuc xa anh sang. Hinh 1.11: Hien tuong phan xa tia dien tu. 'm ot colecto) co dien the Uc . De dien tir roi len colecto khi colecto tich dien 1m Uc < 0, thi van toe dien tu phai thoa man dieu kien m , J vox > eUc ; va\ true x vuong goc vdi be mat colecto nhu o hinh ve ( 1. 11). vox = v0 cos a va do do dieu kien tren co the viet lai: m 1 9 —v0 cos Q > eUc . V i ^ = eU0, f v% cos2 a = eU0 cos2 a = eU0{ l - sin2 a), nen dieu kien de dien u roi vao colecto la: |ef/0|(l - sin2 a) > \eUc \ 26 Chucmg 1. Chuyen ddng cua hat tich dien trong chan khong Hinh 1.12: Hien tuong khuc xa tia dien tu. hay sin a < yj 1 - \ j f t l Ngupc lai, diiu kien de chum tia phan xa trcr lai la sina > 1 - \^j-\ (1.38) Dieu kien (1.38) tuong tu nhu dieu kien phan xa toan phan trong quang hoc^ n2 sin a > — . ni 3. Khi hat dien tich chuyen dong tir vung co dien the U\ sang vung co dien the U2 thi hudng va dp Ion cua van toe cua hat se thay doi va dupe xac dinh bang dinh luat sina yfU~2 sin/3 y/Ul' Hien tupng khuc xa chum hat tich dien nay la do su tac dung cua luc dien trucmg ton tai a mot ldp mong phan cach giua hai vung co dien the khac nhau lam thay doi thanh phan v± (vuong goc vdi b i mat phan cach) cua van toe, xem hinh ( 1. 12). Dien trucmg trong lop phan cach khong lam thay doi thanh phan v cua van toe: vi sina = v2sin/3; Dinh luat khuc xa tren day cho phep ta xac dinh dupe quy dao cua hat tich dien chuyin dong trong mot trucmg tlnh 1.5. Su tucmg tu quang ca 27 rfinh 1.13: Phuang phap xac dinh quy dao dien tich trong mot trudng the tinh dien idt ky. lien bat ky. Muon vay ta phai bieu dien trucmg tlnh dien bang tap hop rat nhieu cac mat dang the phan bo sat nhau de co the xem dien the trong khoang chong gian giua hai mat ding the la khong doi va bang U _ Uj + Uj+\ 2 Chung ta co the goi VU la chiet suat dien tu cua khong gian trong do co dien tich chuyen dong. Su dung dinh luat khuc xa qua mat thur i ta se nhan dupe sina; _ yjUj+\ + Uj sin/?.' ~ J U i-i + U i ^-39^ Quy dao that su cua hat dupe thay the mot each gan dung bang mot ducmg gay khuc gom nhieu doan thing noi vdi nhau. So mat ding nang cang ldn do chinh xac cua phep tinh cang cao (xem hinh (1.13)). Nhu da nhan xet trong muc §1-2, quy dao cua hat dien tich khong phu thuoc vao khoi lupng va dien tich cua no, quy luat nay dupe goi la quy luat dong dang cua cac hat chuyen dong trong trudng tTnh dien. 28 Chuang 1. Chuyen dong cua hat tich dien trong chan khong i i I I i ' l l i i i i \ \ 4 - + . . + . / / / i 1 !1 1 i 1 l l I I a) U, b) •'' 1 ' ' ' 7, i I \ ' » --•+ + V >- H -------- ' i i ' ' y\\1 1 ! ' ‘ i / !! 11 I i ' m ,' Tam khe U Hinh 1.14: a- thau kinh tinh dien mot mat phang, mot mat cong, b- thau kinh tlnh dien hai mat cong. 1.6 Chuyen dong cua hat tich dien trong trucmg tlnh dien khong deu, thau kinh tlnh dien Cung nhu trong quang hoc cua cac chum anh sang, trong quang hoc cua cac chum dien tu (ta goi la quang hoc dien tu), thau kinh la phdn tu quan trong nhat, no dung de hoi tu hay phan ky chum dien tu. Thau kinh dien tu co the tao duoc bang dien trucmg khong dong nhat co doi xung true, hoac tijf trucmg khong dong nhat co doi xung true. Trong muc nay ta se xet loai thau kinh thu nhat, dupe goi la thau kinh tinh dien. Tren hinh (1.14 a,b) bieu di6n hai loai thau kinh tinh dien: Thau kinh mot mat phang m ot m at cong (1.14 a) va thau kinh hai m at cong (1 .14b) trong do cac ducmg dut net bieu didn cac mat ding the. Cac thau kinh dupe cau tao tir hai dien cue co dien the khac nhau l \ nho han U2 va co hinh dang doi xung true. Trong hinh (1.14b), cac dien cue do la nhung hinh tru dong true, con a hinh (1.14a) cac dien cue co dang phing voi 16 tron a giua, goi la diapham (diaphragm). Dau tien ta khao sat mot cach dinh tinh quy dao cua dien tu qua thau kinh, di tir trai sang phai, tir mien U\ sang mien U2, U\ < U2. Cucmg do dien trucmg £ va do do luc tac dung len dien tu F co the’ phan tich thanh hai thanh phan Fz doc theo true va Fr vuong goc voi true. Tren hinh (1.14 b) chung ta nhan thay rin g cac mat ding the cong ve hai phia va co do cong khac nhau, co the tim dupe mot mat ding the gan giua khe co dang gan nhu phing, ta goi vi tri nay la tam khe giua hai cue hinh tru (ducmg 1.6. Chuyen ddng trong trucmg tinh dien khdng deu, thau kinh tlnh dien r Hinh 1.15: Gidi thich nguyen ly hoi tu chum dien tu. dtit net dam). D i dang thay rang khi dien tu chuyen dong tu trai sang phai va dang nam phia ben trai tarn khe, luc Fr co chiiu hudng vao true cua thau kinh nghia la co tac dung hoi tu, d phia ben phai tarn khe luc Fr co chieu ngupc lai huong tvr true ra ngoai, nghia la co tac dung phan ky. Thy nhien tac dung phan ky cua phan ben phai khong triet tieu hoan toan tac dung hoi tu cua phan ben trai, bdi vi d phan ben phai dien tu chuyen dong vdi van toe ldn hon nen thdi gian chiu tac dung nho hon. Ket qua la khi mot chum dien tu song song vdi true cua thau kinh di qua phan ben phai tarn khe, muc dp hoi tu cua chum dien tu do bi giam di chut it, nhung quy dao cua dien tir van cat true tai mot diem nao do, dupe xem la tieu diem phai cua thau kinh, khoang cach tvr tieu diem den tarn khe la tieu cu. Tieu cu phia trai va phia phai co the khac nhau. Chung ta tha'y rang de mot chum dien tu song song di qua thau kinh hoi tu tai tieu diem luc hudng true Fr tac dung vao dien tu phai ty le thuan vdi r, luc cang ldn khi hat cang d xa true, nhu minh hoa d hinh (1.15). Ro rang dien the' trong he thau kinh phai co doi xumg true: U{r) = U{-r). Ham phan bo dien the phai thoa man phuong tnnh Laplace ma trong toa do tru (va co doi xung true) co dang 30 Chucmg 1. Chuyen dong cua hat tich dien trong chan khong Phuong trinh (1.40) cung co the viet lai dudri dang -J L (r - — ) + = 0 (1-41) r dr dr dz2 Trong trucmg hop chum dien tu hep (r nho) ta co the bieu dien l T(r,z) ducri dang mot day cac ham mu cua r. n U(r, z) = y Un(z)rn = U0{z) + U\ (z)r + U2{z)r2 H---- o TUy nhien, do tinh doi xung true trong tong nay chi con lai cac so hang co s6 mu cua r la chan, nghia la U(r,z) = U0(z) + U2(z)r2 + U4(z)r4 + • • • (1.42) Neu ta dung cac ky hieu sau day cho dao ham r — — ■ r' = — (143) dt' dz' K 1 sti dung dieu kien phuefng trinh Laplace (1.41) va tinh doi xung true ta co the viet: U(r,z) = U0( z ) - j U Z + --- (1.44) trong do: U0{z) la dien the tren true th£u kinh (vdri r = 0); [/" la dao ham bac hai theo z cua C7(r, z) tren true thau kinh. Bieu thurc (1.44) nghiem dung (1.41) neu gia thiet U'0"(z) Uq (z). Chung ta hay tim cac luc cua dien trudng tac dung vao dien tu : Luc theo true z:dU Fz = m z = e — = eU0 (1.45) 6 day chung ta da bo qua la dai luong nho. Luc theo r: dU er, dr 2 Ft = mr = e— = ~ — U0 Ta thay rang luc Fr nay ty le thuan vdri r khoang cach den true. Co the viet mot each gdn dung rang theo true z van toe cua dien tu thay d6i theo cong thurc l^mz2 = eUo (1.46) 1.6. Chuyen ddng trong trucmg tlnh dien khong deu, thau kinh tinh dien 31 Tim dang cua quy dao nghia la tim r = r(z) (su phu thuoc cua r vao z) khi dien tu chuyen dong. Vi r = r(t);z = z(t)ta co the chung minh duoc cong thurc: toan hoc sau: ;2z‘r + zr (1.47) That vay ta co: dr dz ' . r — — — — r z dz dt f = ^ = i §t ^ + r' Wt ^ = kr"i + rli (1-48) Xac dinh cac gia tri rtr (1.45) va (1.46) r6i thay vao (1.48) ta co er TTII ^e/7 J' I e TJ't' 2m m m ~ U°r + ™ °r hay la " + w/' + w/=° ( l -4 9 ) Phuong trinh (1.49) duoc goi la phuong trinh co ban cua thau kinh tlnh dien. DSyila phuong trinh vi phan bac hai vdfi cac he so' bien doi, rat kho giai de tim nghiem tong quat. Tuy nhien tijr day ta cung co the rut ra mot so' nhan xetvi quy dao cua dien tvr hay hat tich dien noi chung: - Quy dao cua hat tich dien khong phu thuoc vao dien tich va khoi luong cua chung (trong cong thurc (1.49) khong co mat e/m). - Neu ta thay doi gia tri dien the cua cac diem tren true thau kinh (true z) mot so ldn nhu nhau thi quy dao cua hat tich dien khong doi, (do la vi cac ty so U'Q/Uo va UZ/Uo khong thay doi). - Neu biet cac gia tri cua dien the doc theo true thau kinh, nghia la neu biet cac ham U0, U^Uq thi ta co the tim dupe quy dao cua dien tu r = r(z). - Chung ta co the dung (1.49) de giai bai toan trong trucmg hop mot thau kinh tlnh dien mong va yeu. Ngucri ta goi thau kinh tlnh dien mong khi vung khong gian trong thau kinh co £ ± 0 la hep, trong vung do gia tri r cua dien tvr khong kip thay doi nhieu. De xac dinh ta xet mot chum dien tvr tvr diem s cach khe thau kinh mot khoang d (khoang cach vat) lam thanh voi true mot goc a, khi di qua thau kinh chum nay bi khuc xa va cat true tai diem su a khoang each anh d\, nhu 6 hinh (1.16). Chung ta co the viet lai phuong trinh (1.49): 32 Chucmg 1. Chuyen ddng cua hat tich dien trong chan khdng H inh 1.16: Gidi thich cach tinh tieu cU thau kinh tlnh dien. hay la d_ dz dr dz 1 Ud(z)r(z) (1.50) Tfch phan (1.50) theo z tvr S den Si, ta co dr S l TTII ugW - V w T j s, ~d~z - \ L V u & )r(z)dz (1.51) V> ham so' dudi d^u tfch phan chi khac khong trong m pt vung rat hep va trong vung do r thay doi rat ft r ~ r0. Khi do ta co the thay dr d z r0 dr = , g o = 7 ; &= tg/3 5! ro_ di Thay cac gia tri thu dupe vao (1.51) ta co roy/Uo(S\) roy/UoiS) _ r0 cx> UTT"q{z) (1.52) d\ - rn .: 0 day Uq(Si ) = Uo(z = Si); Uo{S) = Uo(z = S) ■ y /tW )1 Neu cho d = oo va d\ = f 2 va ta co the tfnh dupe tieu cu ben phai o o Tin 70(S,) L Uq(z) dz (1.53) h 4 x/C7o(SO J-oo y/U & j' Tuong tu nhu vay neu cho d\ = oo ta tfnh dupe tieu cu ben trai OO TTII UZ(z) f\ \ y / U Q{S) J-oo VUo(z)dz U0(S) L (1.54) 1.6. Chuyen ddng trong trudng tfnh dien khdng deu, thau kinh tinh dien 33 K - 11 1 1 U I c 1 1 Hinh 1.17: a- thau kinh tinh dien dcm gSm 1 diapham, b- he hai thau kinh dcfn. Ta nhan thay rang trong (1.53) va (1.54) / phu thuoc vao dau cua dao ham bac hai Uq { z ) . Neu U{f > 0 thi / se ldn hon khong ta duac thau kinh hoi tu, neu Uq < 0, / < 0 ta duac thau kinh phan ky. Ty le giua hai tieu cu h s/irx h y/UrfS) s t t(1.55) - Trong trudng hap dan gian nhat nhu ve d hinh (1.17 a), thau kinh tinh dien chi gom mpt tarn chan co cua so tron (diapham), giai gdn dung phuang trinh (1.54) cho ta cong thtic tinh tieu cu (vi du dien tvr di tur trai sang phai) / = 4 Ud Si-Sx trong do £2 = (UA - UD)/d2,£\ = (UD - UG)/di;UA- dien the anot, UG dien the ludi, t/D-dien the tai tam cua diapham (so vdi catot), d, khoang cach giua ludi va diaphan., d2- khoang cach giua diapham va anot, £u £2 dien trudng phia ben trai va ben phai cua diapham. - Cung co the’ co nhung he quang hoc dien tvr phtic tap han, vf du nhu he gom hai thau kinh d hinh (1.17b). Vi cvra so tren diapham rat nho so vdi khoang cach giua cac diapham nen tac dung cua tvmg diapham xem nhu doc lap vdi nhau. Dien trudng dupe tinh theo dien the d hai dien cue ben canh. 34 Chucmg 1. Chuyen ddng cua hat tich dien trong chan khong \ V N \ Q| ✓ z'' " T . " ’ s / / / . X _!_ \ \ '-V n s ' / \ X Hinh 1.18: Nguyen ly hoat dong cua thau kinh tit mong. 1.7 Chuyen dong cua hat tich dien trong tir trucmg tlnh khong deu, thau kinh tir Chung ta xet chuyen dong cua dien tvr co van toe ban ddu tT0 trong tvr trucmg cua mpt cuon day ngan, dupe xem la mot thau kinh tu mong. R6 rang tvr trucmg d day la khong deu nhung co doi xting true nhu dupe minh hoa tren hinh (1.18). Chung ta phan tich vec-to cam ting tvr B ra hai thanh phan: - Bz hudmg theo true cuon day, cung la true thau kinh - Br hudng theo phuong ban kinh Neu chi6u ducmg stic quy udc nhu tren hinh ve (1.18) thi thanh phan Br a ben trai co chieu hudmg vao true thau kinh, eon d ben phai co eh iiu hudng tvr true thau kinh ra ngoai. Gia svr van toe ban dau cua dien tu wo co hudmg tti trai sang phai va song song true z, v0 = \J^U q. Do tac dung cua thanh phan B r dien tu se chiu mot luc vuong goc vdi mat phang chtia B r va v. Fv = ev0B r Luc Fv lam cho dien tu co xu hudmg chuyen dong quanh true cua cuon day vdri thanh phan van toe tuong ting = j^B z, (d day vv = rip va co the chting minh rang < p = ^ B z). Den luot thanh phan van toe lai chiu tac dung cua thanh phan tu truong 1.7. Chuyen ddng trong tit trucmg tinh khong deu, thau kinh tic 35 Bz, lire tac dung vuong goc vdri vv va B z nen hucfng theo r ve phia true cua cuon day co gia tri Fr: Fr = evvB z = 6^ - B 2z (1.56) Chung ta nhan thay rang luc theo hudng ban kinh nay ty le vdi r, khoang cach tir vi tri cua dien tu den true cuon day giong nhu cong thiic (1.46) doi vdi thau kinh tTnh dien. Nhu vay tir trudng khong diu cua cuon day ngan da lam lech quy dao cua cac dien tu v6 phia true cuon day va do do sau khi ra khoi. cuon day cac dien tu nay dupe hoi tu tai mot diem tren true z, do chinh la tieu diem cua thau kinh tvr mong. Can luu y rang d nua ben phai nhu quy udc tren hinh (1.18) thanh phan B r cua tu trudng hudng rtf true ra ngoai, do do van toe vv cua dien tu se giam di nhung van con khac khong, ket qua la tinh hpi tu cua rtr trudng vln dupe dam bao. - Tren ca sd phuang trinh chuyen dong m ~ = - e l v x B ) (1.57) vdi gia thiet chum tia hep (r nho) tinh do'i xung true cua rtr trudng cuon day, rtr phuong trinh Maxwell divB — 0 hay viet trong he toa do doi xung true co the tinh gan dung phuong trinh quy dao cua dien tu d?r e Bzr)(z) -i~2 + — = o (1.59) azz m 8c/o Day la phuong trinh ca ban cua thau kinh rtr, trong do B z0{z) la thanh phan Bz (theo triic z) cua rtr trudng tai cac diem tren true thau kinh (r=0). Neu ta lay tich phan (1.59) va xu ly nhu doi vdi phuong trinh ca ban cua thau kinh tinh dien (1.49) ta co the di den cong thurc tieu cu cua thau kinh til 1 e f00 r w j _ j 2^ dz (1.60) 36 Chuang 1. Chuyen ddng cua hat tich dien trong chan khong Cong thtic (1.60) cho thay / luon duong va thau kfnh rtr luon la thau kfnh hoi tu. C in luu y rang do thanh phdn vv cua dien tu, nghia la do dien tu co xu hucmg quay quanh true thau kfnh nen anh cua vat qua thau kfnh se bi quay di mpt goc/OOB0(z)dz. 6='-2 V 2mUo -O O 1.8 Chuyen dong cua cac hat tich dien vdi van toe gan b&ng van toe anh sang Theo bieu thilc co dien cua dinh luat thu: hai cua Newton: Luc tong hop tac dung len mot vat bang dao ham dong lupng cua vat theo thai gian, nghia la: d-61) De bao gom ca cac hieu ling tuong doi tfnh chung ta can dua vao su kien la khoi lupng thay doi theo van toe cua vat. Til do suy ra rang bieu thurc cua dinh luat hai Newton m o rong cho ca thuyet tuong doi hep tro thanh: dt d_ m qu ^ l - i ;2/c2(1.62) dt Trong do m la kho'i lupng cua hat khi chuyen dong, m0 la kho'i lupng tlnh cua hat va m = 7 f ^ w ° ' 63) Trong co hoc co dien cung nhu co hoc tuong doi tfnh, dong nang T cua mot vat chuyen dong bang cong cua ngoai luc thuc hien de lam tang van toe cua vat rtr 0 den gia tri v cho truoc: Jvf=0FdS (1.64) Chung ta xet bai toan mot chieu F = Fx,dS = dx v a thay F rtr (1.62) vao (1.64) ta co: 1.8. Chuyen dong v&i van tdc gan hang van tdc anh sang 37 Tir (1.63) ta co m2 = l hay m2c2 — Tn~v~ = irinC~ Lay vi phan hai ve ( 1.66) ta co 2 me2 dm — m 22vdv — v22 mdm = 0 hay la ( 1.6 6 ) (mvdv + v2dm) = c2dm (1-67) Ve trai cua (1.67) chfnh la bieu thtic dudi d^u tfch phan cua (1.65) vi vay ta co:r V r m T = / (mvdv + v2dm) = / c2dm = c2 (m - mo) J v=0 J m=mo rr> 2 2 2 1 — me — moc = mocs j \ - v 2lc2- 1 ( 1.68) Khi v « c ta co 2 ? - mocmov T = moclv 2 Ta lai co bieu thurc dong nang trong co hoc co dien. Nhu vay khi hat chuyen dong vdi van tdc ldn khoi luong cua no thay doi, khoi lupng cua hat chuyen dong ldn hon khoi lupng nghi cua hat mot lupng la: Am = T-rc1 Dong nang cua hat bieu diln hieu so giua nang lupng toan phan cua hat E dang chuyen dong va nang lupng nghi E0 cua hat, do do E — Eq — me2 - moc2. Neu lay nang lupng nghi cua hat la E0 = m 0c2 chung ta nhan dupe he thuc Einstein E = me2 (1.69) He thtic nay neu len sir tuong duong giua nang lupng va khoi lupng . Bay gid chung ta ap dung nhung ket qua tren de xac dinh van tdc cua mot hat tfch dien dupe gia tdc bdi mot hieu dien the U, ta co’ E - E o ~m0c \/l - V2/ c2- m0c = qU (1.70) 38 Chucmg 1. Chuyen dong cua hat tich dien trong chan khong Tu day ta co: 1 --------- L ,— (1.71) Khi U —> oo ta thay v — > c, con khi U nho ta lai co cong thuc v = Doi voi dien tvr ta co the tfnh duoc: 771 6XJ LJ /* — "*-------2 = + mi Vn3 (1-72) mo 511.10J tuc la khi dien tu di qua hieu dien the U = 511 kV khoi lugfng cua no tang g£p doi. Ta co the tfnh duac nang luong tlnh cua dien tu : Eo = m0c2 = 9 ,1.10-31.(3.108)2 = 8 ,19.1(T14J = 0, 511MeV Vi dong luong duoc bao toan (chti khong phai van toe!), nen de tien loi ngucri ta thucmg bieu d iln nang lugfng cua vat dudi dang mpt ham cua dong lupng. De lam di6u do ta binh phuong bieu thuc cua m va nhan hai ve vdi c4 '1 - v2/c2] ta dupe2 4 2 2 2 2 4 m e — m v c = m%co1- Vi E = me2, E0 = moc2 va |P| = mv ta co E 2 = (P.c)2 + El hay E 2 = (me2)2 = (T + m0c2)2 = P 2c2 + (m0c2)2 (1-73) Tir p = mv ta cung tfnh dupe van toe cua hat Pc2 « = - £ - (1-74) Doi vdi hat chuyen dong vcri van tdc gan bang van toe anh sang su tuong tu quang-co van xay ra, tuy nhien bieu thuc cua he so khuc xa se thay ddi bcri vi trong truomg hop nay thay vi xet sir thay doi van tdc ta xet su thay dpi dong lupng, nhu d hinh (1.19). Khi hat tfch dien di qua giua hai mi6n co dien the khac nhau thanh phdn dong lupng song song vdi mat phan cach la khong ddi (chu khong phai thanh phan van tdc song song vdi mat phan cach khong doi). Dinh luat khuc xa se la: 1.8. Chuyen ddng v&i van toe gan bang van toe anh sang 39 Hinh 1.19: Khuc xa tia dien tu v&i van toe l&n. Ta lai co P 2 — }i{E2 - El). P 2 = ^ [(^o + qU)2 - E 2] = I {{qUf + 2qUE0\ . Thay vao (1.75) ta co: sum = (L?6) s f/> Nhu vay ta co trong truong hop nay chi so khuc xa trong quang hoc dien tu co dang n = u \l1+2~ i f o - 7 7 ) Chi so khuc xa khong nhung chi phu thuoc vao dien the U ma con phu thuoc vao ty so khoi luong tren dien tich ^ cua hat. Nhu vay la quy luat dong dang trong chuyen dong cua cac hat tfch dien khong con dung nua doi vdi cac hat tuong doi tfnh. Ngupc lai vi trong tu trucmg hat tfch dien khong thay doi dp ldn cua van toe ma chi thay doi hudng cua no, nen cac hat tuong ddi tfnh van chuyen dong binh thudng nhu hat cP dien va chi can thay m = ^ . Vf du ban 40 Chucmg 1. Chuyen ddng cua hat tich dien trong chan khong Hinh 1.20: So do nguyen ly may gia toe Cyclotron. De gia toe cac hat tich dien ta phai dung cac may gia toe. M pt trong nhung may gia toe dom gian la may cyclotron sur dung tit trucmg khong dpi va dien trudng thay doi vdri t£n so cao (ca 1.107 Hz). C£u tao cua cyclotron co dang nhu d hinh (1.20): Hai nua cua mot hinh tru det (hinh chu D) each dien vdi nhau bang mot khe ha noi vdi mot nguon dien cao tin. Hai dien cue nay dupe dat trong chan khong va tir trucmg deu vuong goc vdi mat day. Chung ta biet rang trong tu trucmg hat tfch dien co van toe v0 vuong goc voi ducmg sire B se chuyen dong theo quy dao tron ban kfnh R = vdi chu ky khong phu thuoc vao van tdc T = Do dien ap xoay chieu dat giua hai nua hinh tru, tren khe chan khong giua hai dien cue do dien trudng co chieu thay doi theo thdi gian vdi chu ky T0. Neu chu ky T0 bkig chu ky quay cua hat tfch dien T thi co the dat dupe dieu kien la khi hat di tu tren xuong dien trudng co dau thfch hop de gia tdc no lam cho no chuyen dong nhanh han vdi ban kfnh cong ldn han (nhung chu ky khong doi) den khi hat di qua khe hd tir dudi len dien trudng lai doi dau va gia tdc no. Cu nhu the hat se dupe tang tdc va m d rong dan ban kfnh quy dao. Can luu y rang trong buong gia tdc chan khong phai rat cao, ap suat khi phai rat thap (cd 10~9 - 10_lommHg) de tranh hat tfch dien bi va cham vao cac phan tu khf du. Bai tap chuang I (1-1) Tim bieu thuc va xac dinh do nhay cua mot ong tia dien tir t na may dao dong ky co cac thong so sau: Hieu dien the tang tdc chum dien tir la 900V', 1.8. Chuyen ddng v&i van tdc gan bang van tdc anh sang 41 Hinh 1.21: Cho bai tap (1-3). khoang cach tir diem giua ban lai tia den man huynh quang bang 20cm, chieu dai cac ban lai tia bang 2cm, khoang cach giua hai ban lai tia bang 0,5cm. (1-2) Tim bieu thurc va xac dinh do nhay cua mpt ong tia dien tu cua may dao dong ky co cac thong so sau: Hieu dien the tang toe chum dien tu la 900V. Khoang cach giua tam cua vung tvr trucmg lam lech tia dien tu den man huynh quang bang 35cm. He thong lai tia gom hai cuon day Helmholtz hinh tron co ban kfnh R. = 2 ,5cm dat each nhau mot khoang each 2z = 5cm. So vong day tren mPi cuon la 200 vong. Kich thudc vung tac dung cua tvr trucmg la 2R, trong do tir trucmg dupe xem la deu. Chu y: cudng do tir trudng trong cuon Helmholtz dupe tfnh bdi cong thtic H = 1 , w Rl Do tu tham cua chan khong = 1,26- 10_6Tm/A; i la dong * [n t Z j ' dien trong cuon day, W s6 vong day cua ca hai cuon. (1-3) Tren dudng di cua chum dien tir ngudi ta dat mot he gom hai ludi co dang mat cau dong tam ban kinh trung binh la R = 3cm. Dien the ludi ngoai la U\ - 300V, dien the ludi trong la U2 = 500V (so vdi catot phat xa dien tu). Chung minh rang he hai ludi nay co tinh hoi tu ddi vdi chum dien tu. Tinh tieu cu cua he. (Hinh (1.21)). (1-4) Mot he quang hoc dien tu gom mot catot phang K, mot ludi tang tdc G, co dien the UG bang 200V, mot tam ngan (diapham) co dien the UD va mot anot .4 dupe noi vdi diapham, hinh (1.22). Xac dinh dien the UD (so vdi catot) de chum dien tir song song hoi tu len anot. Cho biet khoang cach giua ludi tang tdc den diapham la 0,5cm, khoang cach giua diapham den anot la 0,8cm. (1-5). Xet mot he quang !.vc dien tu nhu d hinh (1.17-b) gom 1 catot K , 42 Chucmg 1. Chuyen ddng cua hat tich dien trong chan khdng K G D A Hinh 1.22: Cho bai tap (1 -4). hai diapham DUD2 va mpt anot A. Khoang cach giua cac dien cuc do bang nhau va bang 1cm. Dien the cua diapham thu 1 D x bang dien the anot A va bang U\ = 100V (so vdi dien the catot K). Xac dinh dien the cua diapham thu 2 D 2 de cho chum dien tu song song di qua he hoi tu d anot A. Dap so (1-1) S = 4,5 • 10-4 m/V; (1-2) 5 = 0,618m/A; (1-3) / = 13,25- lCT2m; (1-4) UD = 57V; (1-5) co hai nghiem U2 = 432V va U2 = 28V. 43 Chuang 2 M 0 T SO VAN DE VAT LY Ll/ONG TLf Ngay nay rat nhieu lTnh vuc khoa hoc, ky thuat, cong nghe can co nhung hieu biet ve cac he vi mo, trong do cac quy luat tuong tac, chuyen dong chi co the giai thfch dupe nho nhung kien thurc vat ly lupng tu. Vi vay, cac chuyen gia ve dien tir, nhung ngudi thiet ke, sir dung nhung linh kien ban dan, quang dien tir... rat cdn thiet nhung hieu biet toi thieu ve co so cua thuyet lupng tir, v6 pho nang lupng cua cac he vi mo. 2.1 Luong tinh song hat cua cac he vi mo Tir lau chung ta da quen biet tinh chat song cua anh sang qua cac hien tupng giao thoa, nhigu xa va cung da quen biet tinh chat hat cua dien tir. Ddu the ky thti hai muoi, khoa hoc phat hien ra nhung hien tupng doi vdi anh sang (buc xa dien tir) khong the giai thfch dupe bang tfnh chat song cua no, ma chi co the giai thfch dupe neu gia thiet tia sang la mot chum hat. Ngupc lai ddi vdi chum dien tir ngudi ta cung phat hien ra nhung hien tupng chung to chung co tfnh cha't song. Theo quan niem co dien thi hat la mpt ddi tupng vat chat co vi tri, xunc lupng, dong nang, kho'i lupng, va dien tfch, con song lai co budc sons, tan so, van tdc, bien do, cudng do, nang lupng. Hat co the dinh xu: d mot ch6 , con song thi lai trai dai va lan rong do do choan mpt khoang khong gian rong ldn. Mot van de quan trong dupe dat ra vao dau the ky 20 la phai xay dung mot ly thuyet co the phan anh dupe hai ban chat trai ngupc nhau cua mot he vi mo: ban chat song va ban cha't hat, ma ta goi la luong tfnh song hat. 44 Chuang 2. Mot so van de vat ly lucmg tu Hinh 2.1: Sa do thi nghiem hieu ung quang dien. 2.1.1 Luang tinh song-hat cua bufc xa dien tir. Cuoi the ky 17 da xdy ra mot cuoc tranh luan soi noi va keo dai ve ban chat cua anh sang. Ddu tien Newton dua ra gia thuyet: anh sang la mpt dong hat chuyen dong r£t nhanh dupe phat ra tir mot nguon sang va lan truyen trong khong gian theo cac dinh luat quang hoc. Ngupc lai Huygens lai quan niem: anh sang hoan toan co tinh chat song. Su phat minh ra cac hien tupng giao thoa va nhilu xa anh sang da chung minh ro ret ban chat song cua anh sang va gia thuyet cua Newton bi lang quen ddn. Ddu the ky 20 sau khi phat minh ra cac hieu ung quang dien, hieu ung Compton ngudi ta mdi quay lai thuyet hat anh sang bdi vi cac hieu ung nay khong the giai thfch bang thuyet song anh sang. Chung ta hay xet hai hien tupng dien hinh phan anh tfnh chat hat cua anh sang, birc xa dien tir. 1. Hieu ting quang dien. Trong mot thf nghiem quang dien, ngudi ta cho anh sang chieu vao mot ban kim loai dat trong binh kfn da rut het khong khf, nhu d hinh (2.1). Tir be mat ban kim loai phat xa ra cac dien tir. Ngudi ta co the thay doi tan so v cua anh sang, hieu dien the U giua hai dien cuc emito va colecto (cuc phat va cuc gop) cung nhu vat lieu lam cuc phat. Neu cac dien tir co du nang lupng de thing dupe the ham thi chung tdi dupe colecto va tao thanh dong dien dupe ghi bang ampe-ke .4, muon the dien tir phai co dong nang ldn hon cong can do dien trudng gay nen. nghia la 1 - m v 2 > eU 2 Neu dong nang cua dien tir nho hon eU thi no khong tdi dupe cuc gop va khong tham gia vao dong dien. P = ii.A a/ 2.1. Luong tinh song hat cua cac he vi mo 45 ©'mo b/ Hinh 2.2: So do tucmg tdc photon-dien tu tu do: a- Trudc tucmg tdc, b- Sau tucmg tdc. Ngudi ta quan sat duac cac ket qua thuc nghiem sau: 1. Thuc te dong dien duac thiet lap g£n nhu ttrc thdi ngay ca khi cudng dp anh sang rSt yeu. Khoang thdi gian tir luc anh sang tdi den khi quan sat dupe cac dien tu vao cd l 0~9s va khong phu thuoc cudng do anh sang. 2. Khi giu tdn so va hieu dien the ham khong doi dong dien I ty le vdi cudng do birc xa tdi. 3. Khi giu tan so va cudng do sang khong doi, dong dien I se giam khi U tang va triet tieu khi U dat tdi mot gia tri U3 nao do khong phu thuoc vao ;udng do sang, goi la hieu dien the ham. 4. Ddi vdi mot vat lieu lam cuc phat cho trudc, hieu dien the ham U„ phu :huoc tuyen tfnh vao tan so theo bieu thirc: eUs = hu - W0 (2.1) Wo phu thuoc loai vat lieu, h la hang so Planck. 5. Ddi vdi mPi vat lieu cho trudc lai ton tai m ot tan so ngudng v3 sao cho leu anh sang tdi co tan so v < vs thi se khong burt dupe electron ra khoi vat ieu du cudng dp bu:c xa co gia tri nhu the nao. Mau sons anh sang chi co the giai thfch dupe mot ket qua, do la ket qua > ve tfnh ty le giua cudng do dong dien va cudng do burc xa tdi. Thuc vay, •udnc do burc xa cang ldn no cang mang nhieu nang lupng do do so dien tu >i but ra cang ldn. Trai lai, bon ket qua khac deu khong the giai thfch dupe heo mau song anh sang. De giai thfch ket qua thuc nghiem tren, Einstein dua ra mau lupng tir anh 46 Chuang 2. Mot so van de vat ly lucmg ti( sang. Theo mSu nay, buc xa dien tvr dupe tao thanh tir cac photon hay cac lupng tir. MPi photon co mot nang lupng E chi phu thuoc vao tin so u cua burc xa theo bieu thuc E = hu = /i y (2.2) Trong do h = 6,626 - 10-34Js la hang so' Planck. MPi photon se tuong tac hoan toan hoac khong tuong tac vdi vat chat nghia )& no hoac co the truyen toan bo nang lupng cua minh hoac khong truyen mot chut nang lupng nao cho vat chat. Theo quan diem do nang lupng do photon mang den bi hap thu bdi chi mot dien tir. Neu dien tir bi burt ra khoi kim loai thi no se co dong nang bang hieu so' giua nang lupng photon ma no hap thu va nang lupng lien ket dien tir vdi be mat kim loai. Nang lupng lien ket dien tir vdi be mat W0 co the khac nhau. Tuy nhien mPi vat lieu dupe dac trung bdi mot nang lupng lien ket cuc tieu. Nang lupng can thiet de giai phong cac dien tir lien ket yeu nhat goi la cong thoat = A(cos

Thay cac dai lupng d (2.16) vao (2.15) ta co: = A e -il* * -* ') (2.17) Nhu vay ham song (2.17) dupe dac trung bdi nang lupng hat E va vecto xung lupng P cua hat, vecto P co ba thanh phdn Px,Py,Pz va lien he vdi vecto song K biiig (2.16). Ham song (2.15) la mpt ham song phing vi mat song, noi ma song cd pha nhu nhau, la mpt mat phang vuong goc vdi phuong truyen song nhu bieu diin tren hinh (2.6). That v£y, tai mot thdi diem t cho trudc cac die’m co pha p bang nhau phai n im tren mat phing vuong goc vdi K, vi tren mat phing do t&t ca cac diem deu thoa man K f= const, nen = (u>t — Kr) = const. 2.2. Cac tinh chat cua song De Broglie 53 Hinh 2.6: Mat song cua ham song phang. Ham song (2.15) la mpt ham don sic bdi v! no dupe dac trung bdi mfit tin so v - £ = j- n h lt dinh. Song De Broglie dupe goi la song vat chit, no cung lan truyen giao thoa, nh iiu xa... nhu mot song phing. Bay gid chung ta se khao sat mot so nhung tinh ch it dac trung cua song De Broglie. 2.2.2 Y nghia xac suat cua song De Broglie. V ln de dat ra bay gid la tim hieu de biet dieu gi se xay ra khi mot dang vat ch it (nhu dien tir chang han) lai bieu hien cac tinh ch it song. Noi cach khac ta phai tim hieu y nghia cua ham song De Broglie. 1. Y nghia xac suat cua buc xa dien tic. Gia sir co anh giao thoa qua hai khe n im sat nhau cua anh sang (buc xa dien tir). Theo quan diem song, thi cudng do sang I la nang lupng tren mot don vi dien tfch, mpt don vi thdi gian tai mot diem tren man hinh co dang: I = e0c£2, (2.18) trong do £ la cudng do dien trudng tai diem do, e0 la hlng so dien, c van tdc anh sang. Mat khac theo quan diem hat thi cudng do blng: I = h v N (2.19) rong do hv la nang lupng mpt photon va N la thong lupng photon (so photon :ren mot don vi dien tfch, mot don vi thdi gian) tdi diem do. 54 Chucmg 2. Mot so van de vat ly luong tu Ta khong co cach nao de doan trudc noi ma mpt photon cho trudc se roi vao de sinh ra mot chdp sang. Nhung neu ta xet anh giao thoa gom cac van sang v^n toi xen ke nhau thi moi photon se co xac suSt ldn de rcri vao cho van sang va co xac suat bang khong de roi vao cho van toi. Nhu vay ta co the noi ring thong luong photon tdi moi diem tren man hinh cho ta so do xac suat de tim dupe mot photon d lan can diem do. Vi I = e0c£2 = hvN nen ta suy ra N ~ £2. Theo quan diem lupng tir thi dai lupng dao dong (d day la dien trudng) phai la mot ham ma binh phuong cua no cho ta xac suat tim thay photon tai mot diem cho trudc. 2. Y nghTa xac suat cua song vat chat. Anh giao thoa anh sang noi tren cung co the la anh giao thoa cua cac song vat chat. Khi do, y nghTa xac suat d.’/a tren luong tfnh song-hat cua anh sang dupe md rong cho ludng tfnh song hat cua vat chat. Thanh thir song ket hop vdi dien tir cho phep giai thfch dung anh nhidu xa dien tir, thi cung chfnh song do dupe bieu d iln bdi ham song ma gia tri tuyet ddi binh phuong |^ |2 cho ta xac suat tim th iy m ot dien tir tai mot diem cho trudc. Muon ket hop cac cach bieu diln song va hat, ta phai tir bo y nghi cho rang mPi hat cho trudc phai dupe dinh xir ro rang phai co quy dao xac dinh, ta chi dupe phep noi ve xac suat tim hat tai mot diem xac dinh d mot thdi diem da cho. Thong thudng ta ky hieu ham song De Broglie, ham song vat chat, bang . Song De Broglie $ cua mot photon la mot song dien tir, con ddi vdi mpt electron hay mot hat vat chat khac thi $ la mot song De Broglie phi dien tir. 2.2.3 Nhom song. Sir lan truyen cua song De Broglie. Ta xet mot song don sac lan truyen doc theo true x * { x , t ) = A e - « ut~kx) (2 .2 0 ) Dai lupng

d(huj) dE p ? 3 d K d(hK) dp dp m 2.3 Nguyen ly bat dinh Heisenberg 2.3.1 He thurc bat dinh ve toa do va xung lupng cua hat. Gia su ta muon xac dinh chfnh xac vi trf cua mpt doi tupng vat cha't nhu dien tu chang han, thi ta phai thuc hien mot thi nghiem nao do: vf du ta co the dat mot khe tren lo trinh cua hat chuyen dong theo phuong y, nghTa la vdi xung lupng P = Pv va vdi nang lupng cho trudc nhu d hinh (2.8). Neu hat de lai mot dau vet tren man chan dat sau khe thi ta biet chac hat da qua khe. Khi do ta xac dinh dupe vi trf cua hat tren true x vdi sai so bang do rong d cua khe. Noi each khac ta da xac dinh vi tri cua hat tai thdi diem (va ca trudc thdi diem) hat qua khe vdi do bat dinh Ax = d. Khe cang hep, do bat dinh Ax cang nho va vi tri cua hat theo true xdupc xac dinh cang chfnh xac. 2.3. Nguyen ly bat dinh Heisenberg 59 Do ban chat cua vat chat, ta biet ro hat se bi nhilu xa khi di qua khe. Ta khong the doan trudc hat se rcri vao chP nao tren man chin, nhung sau khi hat da roi vao mpt diem nao do tren man chin thi ta biet chdc hat da qua khe va co su thay doi cua xung lupng cua hat. Hien tupng nhilu xa da tac dong den xung lupng cua hat. Khi hat chua qua khe ta hoan toan khong biet vi tri cua no nhung lai biet xung lupng cua hat, ca ve dp ldn (vi da biet nang lupng hat) va phuong (vuong goc vdi khe). Khi hat qua khe thi ta co the xac dinh dupe vi tri cua hat nhung thanh phdn xung lupng Px theo phuong x cua hat trd nen khac khong vi hat chuyen dong lech vdi phuong ban d£u ve mot diem nao do tren anh nhiiu xa. VI ta khong biet hat roi vao d&u tren man hinh nen ta co mot do bat dinh tuong irng"APx ve thanh phan xung lupng theo phuong x. Theo ly thuyet nhilu xa thi vi tri cua van tdi thur nhat dupe xac dinh theo cong thurc sina = Mac d£u ta khong the biet chfnh xac diem roi tren man nhung vi tri co xac suat ldn nhat vln d lan can vung trung tam anh nhidu xa. VT vay ta co the coi Px nam trong khoang tvr 0 den P sin a , nghTa la APx = psina =p%- APx = = £. De giam dp bat dinh ve thanh phdn Px ta co the md rong khe d, nhung khi do lai tang do bat dinh Ax. Ta co he thiic AxA Px = h N hu vay cung mot thf nghiem ta khong the ve nguyen tac lam nho dong thdi do bat dinh vg vi tri Ax va vS thanh phdn xung lupng theo phuong x, APx. Vf du tren da minh hoa nguyen ly Heisenberg phat bieu nam 1927, goi la nguyen ly bat dinh Heisenberg. Co lupng tvr da chumg minh rang ddi vdi moi kieu thf nghiem cac do bat dinh Ax va APx lien he vdi nhau theo he thurc A xA Px > — 1 - 27r Can nhan manh rang he thuc nay co hieu luc ca trong ly thuyet, no co the dupe suy ra tvr he thuc AxAK s i da noi d tren, trong do Ax la kfch thuoc nhom song , AA' la gia so cua so song trong nhom song. 2.3.2 He thuc bat dinh ve nang lupng va thcri gian. Ta co the thiet lap he thtic bat dinh Heisenberg cho nhieu cap dai lupng khac nhau. Vf du muon do nang lupng E cua mot vat ta phai tien hanh trong mot thdi gian At. Nhu vay ta co the chung minh rang giura do bat dinh ve nang lupng va do bat dinh ve 60 Chucmg 2. Mot sd' van de vat ly lucmg tu thdi gian co he thurc A E A t > ~ ~ 2ir Ket qua la chi co the biet nang luong cua mot vat vdi do chfnh xac cao nhat (AE = 0) neu phep do dupe tien hanh trong khoang thdi gian vo han (At = oo). j Nguyen ly bat dinh nay keo theo mot he qua quan trong la cac he lupng tu nhu nguyen tu co the ton tai d mot trang thai trong mot thdi gian rat ngan, goi la thdi gian song r. Muon do nang lupng cua he thi phai do trudc khi trang thai do bi phan ra. Nhu vay do bat dinh cua he v6 nang lupng se la A E = va AE T > — 2?rr _ 2tr 2.3.3 Nguyen ly bo sung. Nguyen ly bat dinh da khang dinh la trong cung mot thf nghiem khong the dong thdi do dupe cac gia tri cua hai dai lupng goi la lien hop (nhu Px vdi x, E vdi t) vdi do chfnh xac tuy y. Tu do suy ra rang cac tfnh chat song va hat cua vat chat khong the xac dinh dong thdi trong cung mot thf nghiem. Vf du, neu ta lam thf nghiem de do cac dac trung hat cua mot ddi tupng thi d day nhat thiet phai co A x va At bang khong vi theo dinh nghTa mot hat co the dupe dinh xu vdi do chfnh xac cao vo han d bat ky thdi di£m nao. Con xung lupng va nang lupng la cac dac trung song (A = h/p, v = E/h) thi theo nguyen ly bat dinh, hoan toan khong biet. Nhu vay khi cac tfnh chat hat xuat hien thi cac tfnh chat song bi loai trir. Viec khong the quan sat dupe dong thdi cac tfnh chat hat va tfnh chat song cua vat chat da minh hoa mot nguyen ly do Niels Bohr dua ra vao nam 1928 ma ngudi ta goi la nguyen ly bo sung. Cac tfnh chat song va hat bo sung cho nhau theo y nghTa la ca hai mo hinh song, hat deu can thiet de ta hieu dupe d&y du cac tfnh chat cua vat chat my vln khong the quan sat dupe dong thdi cac dac trung song va hat. 2.4 Phuong trinh co ban cua co hoc luong tu. Phuong trinh Schrodinger De mo ta mot cach dinh lupng trang thai chuyen dong cua cac ddi tupng vi mo co luong tfnh song hat ta phai thiet lap mot phuong trinh the hien dupe tfnh chat do, giong nhu phuong trinh Newton ddi vdi cac vat vT m o hay nhu phuong trinh Maxwell ddi vdi song dien tir. 2.4. Phuang trinh ca ban cua ca hoc luang tu 61 Phuang trinh cdn tim phai dong then thoa man gia thuyet De Broglie va su )hu thuoc gitfa nang luong va tcin so thong qua hang so Planck. Ta xet mpt hat vi mo co khoi luong m chuyen dong vdi van toe ««c trong not trucmg luc U = U(x,y,z,t). Tinh chat hat co the dupe dac trung bang hai lai lupng E va P E — T + U vdi T = ^ - ( P 2 + p 2 + p 2 ) 2m 1 y z Tfnh chat song dupe dac trung bang budc song va tan sd lien he vdi cac dai upng tren bang cac cong thurc: ^ , h E = hu- p = - 1.4.1 Hat tir do. Day la trudng hop hat chuyen dong trong khong gian f / = 0. lam song De Broglie co dang: %(r,t) = A - e ~ ^ Et~pff) = Ae-%(Et- p*x- p'‘y- p' z'> (2.27) ^ y dao ham bieu thuc (2.27) theo thdi gian ta dupe: dip - - E * ur day ta nit ra dt h d% ih— = E * (2.28) ,ay dao ham bac hai (2.27) theo toa do x,y,z ta co the viet d2$> <92$ d2^ 1 r 2 T rp" 2777 + H----= 0 n2 h2 = T nay la vi hat tu do ta co E = T nen viet duac: 2m J f ^ E V = 0 h2 (2.29) 62 Chuang 2. Mot sd' van de vat ly lucmg tu 2.4.2 Hat trong mot truomg luc. Xu£t phat til phucmg trinh (2.29), Schrddinge da dua ra mpt gia dinh v i sau dupe thvra nhan nhu mot tien de cua co hoc lupng tu: chuyen dong cua mot vi hat bat ky trong mot truOng luc U ± 0 cung dupe mo ta bang phuong tnnh giong nhu (2.29) trong do T = E dupe thay th6 bdi T = E - U, nghTa la phuong trinh co dang: A# + ^ - ( E - U)^ = 0 (2.30) h Ket hop phuong trinh (2.30) vdi (2.28) ta dupe phuong trinh Schrodinger dang tong quat: ih ^- = - ^ A $ + £/* (2.31) ot 2m Neu the nang U la mot ham khong phu thuoc vao thdi gian thi nang lupng toan phan cua vi hat se bao toan, trang thai cua he trong do nang lupng co gia tri hoan toan xac dinh, khong doi, dupe goi la trang thai dimg. Trong trucmg hop nay ham song $ co the dupe bieu diln dudi dang tich cua hai thua so, trong do mot thvra sd chi phu thuoc vao toa do, thvra sd kia chi phu thuoc then gian, nghTa la ta co the viet $(x ,y,z,t) = $ (x , y, z)tp(t) (2.32) Thay (2.32) vao (2.31) ta dupe hai phuong trinh Q ih— [^(x, y, z)) Sau khi gian udc $ d phuong trinh dau,

(<)= (2.35) Phuong trinh (2.34) dupe goi la phuong trinh Schrodinger trang thai dung co nghiem la ^{x,y,z). Nghiem cua phuong trinh Schrodinger tong quat la: y> {x,y,z,t) = '& {x,y,z)e~, * t (2.36) 2.5. Toan tic trong co hoc luong tu 63 Gin luu y ring: ban than ham song ^ noi chung la ham phirc khong co y nghTa vat ly true tiep ma chi co binh phuong modun cua no mod co y nghTa vat ly. Dai lupng |$ |2 xac dinh khPng phai mat do mot dai lupng vat ly nao nhu trong ly thuyet co dien ma no chi xac dinh mat do xac suat dai lupng do, vf du mat do xac suat tim thay hat trong mot don vi the tfch tai mot thcri diem nhat dinh |\I>|2 dupe tfnh nhu sau |tf|2 = $ • >J* (2.37) Trong do ’£* la ham lien hop phirc cua '£. Vi xac suat tim thay hat trong toan bo khong gian bang 1 nen ta co dieu kien chu^n hoa cua ham song: J J J x ,y /// ,z\^\2dxdydz = 1 Mat khac tir (2.36) ta co : z) ■e~'^t = 'S>*(x,y,z)e1£ t. va ${x,y,z,t) ■ i*(x,y,z,t) = ^>{x,y,z) ■ ^*(x,y,z) nen ddi vdi trang thai dimg su phan bo xac mat tim thay hat khong phu thuoc vao thdi gian. Vdi nhung nhan xet tren day chung ta thay rang ve mat toan hoc, ham ;ong phai thoa man nhung dieu kien sau day: don tri, lien tuc, huu han. 1.5 Toan tu trong co hoc luong tu 5.5.1 K hai niem toan tur. Toan tir la su bieu di<§n tupng trung mot phep toan lao do (dai sd, vi phan, tfch phan...) dupe tien hanh ddi vdi mot ham sd de ihan dupe mot ham sd khac va dupe viet nhu sau: Z/’J' =

i va thi no cung co the d vao trang thai dupe xac dinh bang to hop tuyen tinh cua hai ham $ = c i $ ,+ c 2$ 2 (ci,c2 la cac hang sd). (2.40) Toan tu tuyen tfnh L dupe goi la tir lien hop neu thoa man dieu kien: f VLVdV = [ ^L*^*dV (2.41) Jv Jv Toan tu tu lien hop con goi la toan tir Hermit. Vf du toan tir ^ khong phai la toan tir tu lien hop, nhung toan tir lai la toan tir tu lien hop. That vay r+°° „ r+°° d<\> / VLVdx = i / V*^-dx J — 0 0 J — OO r+°° rfxb* = ^>~dx J-OO dX +°° _ , ’5—— dx —oo - - / : j — i + 00 dx L Chu y rang neu L = thi L* - - i — OO ^L*^*dx. Doi hoi cac toan tir trong co hoc lupng tir phai la cac toan tir tu lien hop xuat phat tir co sd vat ly la gia tri trung binh cua dai lupng vat ly L phai la mot gia tri thuc, turc la L = L*\ ma L = J ^*L^dV; L* = f^ I'^ 'd V . Do do j V ' L V d V = f V L * V mdV. Cac toan tir lien hop co hai tfnh chat co ban sau day: 2.5. Toan tu trong ccfhoc lucmg tu 65 1. Cac tri rieng cua toan tu tu lien hop bao gid cung la tri thuc Ln = L*n. 2. Cac ham rieng cua toan tir tu lien hop ung vdi cac tri rieng khac nhau, hop thanh mpt he ham true giao. Tinh chat nay dupe bieu di£n nhu sau: /= | ° n * m (2.42) I 1 neu n = m. Bay.gid ta xet hai toan tir L M. Neu L(M^) # M(L'J') thi ta noi L va M khong giao hoan vdi nhau. Neu L(M'Jt) = M(L^) thi ta noi L va M giao hoan vdi nhau. Vf du 1. Hai toan tu L = x- va M = ^ la khong giao hoan vdi nhau vi L(M A la Laplacian. hz2m O I O I u d x i d y 2 d z 2 4. Toan tunang luong toan phan, toan tu Hamiltonian. Toan tir nang lupng toan phdn dupe xac dinh dua tren cong thirc H - T + U — ► H = T + U H = - — A + U{x,y,z) 2m(2.47) Sir dung dinh nghTa toan tit Hamiltonian ta co the bieu d iln phuong trinh Schrodinger dudi dang toan tir H * = (2.48) Nhu vay nang lupng toan phan E cua he la tri rieng cua toan tir Hamiltonian. 5. Toan tu thanh phan moment dong luong. Toan tir moment dong lupng dupe xac dinh tir cong thtic co dien doi vdi moment dong lupng tuong ddi vdi mot diem cho trudc O. M = [rxP] TO day ta d i dang xac dinh toan tir hinh chieu cua moment dong lupng theo cac true toa do. Mr y P z z P y — i h ( y dz z Qy ) My = zPx - xPz = - i h { z £ - i h ( z £ - x £ ) - x-§-z ) } (2.49) m z = iPy - y h = -iHx-fc - y £ ) Ta cung co the dd dang xac dinh toan tir binh phuong moment dong krpng tir cong thirc: M 2 -- M l + M l + M? d \2 d \2 (2.50) 68 Chucmg 2. Mot sd van de vat ly lucmg tu Hinh 2.9: He toa do cdu. Ta co the chiing minh rin g cac toan tvr thanh phin moment dong luong la khong giao hoan vdi nhau, ta co [.Mx,M y\ = ihMz\ [My, Mz] = ihMx [Mz, Mx] = ihMy] [Mx ■ My - My ■ Mx}% = ihMz^>. Ttiy nhien m6i toan tu Mx,My,Mz lai giao hoan vdi toan tvr binh phuong moment dong luong. [.MX, M 2] = [Mv, M 2} = [MZ, M 2] = 0 6. Phuang trinh toan tu thanh phan moment dong luong: Mz'& = Mz'f> - i h ( x — - y — ) = M z* (2.51) De giai phuong tnnh nay ta sir dung he toa do c£u {r,6,) —> ekM,(v+2n) Dieu do co nghTa la e * M' 2* = 1 (2.54) va ^ phai la cac sd nguyen ^ = 0,±l,±2,--- = m (2.55) h Nhu vay ham rieng cua toan tu Mz la \& = c • eim*, con tri rieng cua no bang Mz = mh. Trong co hoc lupng tu hinh chieu cua moment dong lupng cua he vi mo nhan nhung gia tri gian doan bang sd nguyen ldn h. Trong trudng hop nay ngudi ta noi Mz da bi lupng tu hoa. Sd m xac dinh do ldn cua Mz dupe goi la sd lupng tu tvr. 7. Phuong trinh binh phuang moment dong luong: A,f2 = M 2'i! M 2 = M'2 + M 2 + M l, dupe xac dinh bang cong thuc (2.49), trong toa do cau toan tvr M 2 co dang M 2 = - h 2A e,v (2.56) Trong do la phan goc cua toan tvr Laplacian trong he toa do cic: 1 d . . Q d . 1 d ^ = ^ r e M {sm er e ) + ^ r 6 a^ (2-57) 70 Chucmg 2. M6t so van de vat ly lucmg tu VT toan tir M 2 chi tac dung len phin goc 6 va - ^ t + k 2* = o (2.61) Nghiem rieng cua phuong trinh (2.61) co dang tf(x) = c - e ±iKx (2.62) VI dong nang cua mot hat tu do T = mg- luon luon ldn hon khong do do K luon luon la mot so thuc va nghiem (2.62) thoa man dieu kien lien tuc, don tri, huu han voi moi gia tri T > 0. Tham so K 6 day dong vai tro so song. Su phu thuoc cua nang lupng vao sd song K cd dang _ h2K 2 L = ------- 2m 2.6. M6t sS b a i toan dcm gian cua co hoc lucmg tu 71 U0 I JI m 0 CU X. H inh 2.10: Scr do ho the nang. Su phu thuoc giua E va K dupe goi la pho nang lupng cua hat. Trong trucmg hop nay pho nang lupng dupe xem la pho lien tuc va co dang ham parabol. Vi K = % nen ta co 'J'(x) = c • e± «Pl1 Ham song nay la ham rieng cua toan tu Px. Nhu vay hai toan tu H va Px co cung ham rieng nen chung giao hoan vdri nhau va do do nang lupng E va dong lupng P cua hat co the dupe do chfnh xac dong then. Con xac suSt tim hat tai mot vi tri nao do trong khong gian bang $*$ = \c\2e±iKx ■ eTiKx = |c|2 = const, khong phu thuoc vao toa do. Dieu nay chung to toa do cua hat la hoan toan bdt dinh. Ket qua nay phu hpp vdri nguyen ly bdt dinh Heisenberg. 2.6.2 Hat trong ho the nang. Ta xet chuyen dong cua hat trong mot vung the nang bien doi nhu sau: (Hinh (2.10)) U = U0 tai vung x < 0, U = 0 tai vung 0 < x < a, U = U0 tai vung x > a. Vung cd the nang bien doi nhu the nay dupe goi la mot ho' the nang, a dupe goi la be rong, U0 dupe goi la chilu cao ho the nang. Ta xet trucmg hpp nang lupng toan phdn E cua hat nho hon chieu cao ho the. Theo vat ly co dien trong truomg hpp nay hat chi cd the chuyen dong trong ho' the ma khong the nao vupt ra ngoai dupe. 1. Trucmg hap hang rao the vo han, U0 —> oo. Khi do ham song o vung ngoai ho phai bang khong, con ham song d trong ho the dupe xac dinh bdi phuong trinh song (2.61) d'2^ — - + K - * = 0, khi 0 < x < a (2.63) 72 Chucmg 2. Mot so van de vat ly lucmg tu Vdri dieu kien bien suy ra tut dieu kien lien tuc cua ham song ’f(O) = $(a) = 0 Nghiem tong quat cua phuong trinh (2.63) cd dang tf(z) = AeiKx + B e ~ iKx ung dung dieu kien bien ta cd x = 0 — * A + B = 0 A = —B x = a — > AeiKa + B e ~ iKa = 0 eiKa - e~iKa = 0 hay la sin Ka = 0 va K = ^ vdri n = l, 2,3, •• • Thay gia tri cua K vao bieu thi'rc nang luong ta cd E - h2Rn _ 71 2m 2ma2 (Chu y rin g d day khong ton tai trang thai ling vcri n = 0, K = 0 vi khi do ^(x) = 0.) Nhu v^y vi K nhan nhung gia tri gian doan nen nang luong E cung nhan nhung gia tri gian doan, noi cach khac pho nang luong cua hat khong lien tuc. Nang luong nho nhat ma hat trong ho the cd the cd la khong phai bang khong ma umg vdri n = l, K = J va h2n2 h2 E\ =2 m a 2 8 ma2 Khoang cach giua hai muc nang luong cho phep canh nhau tang theo so lupng tu n va ty le vdri hieu binh phuong giua chung A^ = ^ [ ( n + l)2 - n 2] - -n 2 m a 2 Be rong cua ho the nang cang lorn thi AEn cang nho. Doi vdri dien tu, khi a = 1cm E n = 3,37- 10_15n 2 ■ eV AEn = 0,7- 10_15n • eV. Khi a = 5A, E n = 0,68 n2 eV AEn = l,36neVr. 2.6. Mot so bai toan dcm gian cua ccf hoc lucmg tu 73 Hinh 2.11: a- Ham song cua vi hat trong ho the nang, Uo gi&i han, b- Xac suat tim hat Wn trong ho the nang, U0 gi&i han. BSy gid chung ta co the viet bieu thurc cua ham song ung vdi cac gia tri nang luong cho phep: TL7T $„(x) = A n \etKnX - e~'KnX1 = A n sin K nx = A n sin(— x) L J a Bien do An co the tim dupe tu dieu kien chudn hoa: [ |'J,n|2dx = 1 va bang A n = Jo Mat do xac suat tim thay hat Wn(x) = |# „(x )|2 = -s in 2( — x) (2.64) a a(2.65) Tren hinh (2.11) bieu d iin sir phu thuoc ham song va xac suat tim hat theo x. 2. Trifcmg hap hang rao theU0 la gi&i han, U - U0 < oo. Trong vung II hinh (2-10) nghiem tong quat phuong trinh van nhu cu co dang *(x) = AeiKx + Be -iKx (2.66) nhung se co cac dieu kien bien khac. Trong vung I va III phuong trinh Schrodinger co dang: h2 d2 0 ta phai cho C = 0, vay ta co trong vung I, x < 0, ’J'/ = CeK'x, trong vung III, x > a, ’J/h i = D ■ e~K'lX. Dieu kien bien doi hoi su lien tuc cua ham song va dao ham bac nhdt cua ham song, nghTa la ta co: 'r n t t d^!, d ^ , , .. , n Tai x = 0 = va —— = —-— (2.69) dx dx m • T T S dty I I d ^ I I I Tai x = a = va — — = — —!■- (2.70) dx dx Dieu kien (2.69) tai x — 0 cho ta 2 phuong trinh C ■ e° = Ae° + Be0 -> C = A + B K\ Ce° = KAe° - KBe° -> K XC = K A - K B Dieu kien (2.70) tai x = a cung cho ta hai phuong trinh tuong tu, cong them ta co phuong tnnh chudn hoa /OO\$\2dx = 1 -OO Tir 5 phuong trinh do chung ta co the tim duoc 5 dn so': A,B.C.D va nang luong E{K). Bo qua nhung tinh toan cong kenh chung ta co the dua ra mpt so' nhan xet v6 mot so trang thai thap nhat cung nhu ham song va xac sudt tim hat doi vdi cac trang thai do, tren hinh (2.12) bieu diin ham song va xac suat tim hat cua 3 trang thai thdp nhat. So sanh vdi cac ham song trong trudng hpp U0 = oo d hinh (2.11) ta thay d do trang thai thdp nhat ung vdi A = 2a, con trong trudng hpp U0 < oc ta thayA > 2a. Vi p = j ta thay trang thai thdp nhat khi U0 < oo ung vdi nang 2.6. M6t so bai toan dcm gian cua cd hoc luong tu 75 Hinh 2.12: a- Ham song cua vi hat trong ho the nang, v&i Uo - oo, b- Xac sudt tim hat Wn trong ho the nang, U0 = oo. y \ S ~j v v v V \ -/J Uo I I IE a Hinh 2.13: Ham song cua vi hat khi E > Uo. luong thdp hon so vdi khi U0 = oo. Phia ngoai ho the nang ta thdy ham song giam theo quy luat ham mu, nhung no khac khong trong mpt vung ngoai bd ho. Dieu do co nghTa la d ngoai bd ho noi ma E < U0 xac sudt tim thdy hat vdn khac khong. Do la bieu hien cua tfnh chdt song cua hat khong the giai thfch bdng vat ly co dien. Hien tupng tham nhap cua hat vao trong vung cam nhu the la trai vdi dinh luat bao toan nang lupng. Tuy nhien chung ta co the giai thfch hien tupng do tvr nguyen ly bat dinh Heisenberg: AE A t > h. That vay nang lupng la bat dinh, no co the khong tuan theo dinh luat bao toan trong thdi gian rdt ngdn r = At = h/AE va hat co the d ngoai bd ho trong thdi gian 3. Trucmg hap nang lucmg hat lan han U0, hat la tu do, khdp moi vung ham song co dang hinh sin, nhung sd song d vung II khac sd song d vung I 76 Chucmg 2. Mot so van de vat ly lucmg tu U0 E I n IE cu Hinh 2.14: Sex do hang rao the nang chu nhat. va vung III nhu trinh bay or hinh (2.13); Trong vung II: ^ _ h _ h p \j2mE Trong vung I va III \ — h _ s j2m (E - U0) Khi E > U0 pho nang lupng cua hat la lien tuc. 2.6.3 Hieu ung ducmg ngam (tunnel). Xet chuyen dong cua hat trong vung co the nang bien doi nhu 0 hinh (2.14), goi la hang rao the nang. khi x < 0 khi 0 < x < a khi x > a Theo vat ly co dien mot hat 6 trong vung I co nang lupng toan phan E nho hon chieu cao hang rao the (t/0) thi chi co the chuyen dong trong vung I, khong the vupt qua hang rao the vung II de sang vung III. Tuy nhien do tfnh chdt song cua hat vi mo, co hoc lupng tir tien doan ton tai mot xac sudt khac khong tim thdy hat trong vung III. Hien tupng do goi la hieu ung duong ngdm (tunnel). Ta xet bai toan cu the bang cach giai phuong tnnh Schrodinger doi vdi timg vung, trong truong hpp mpt chieu d- ^ + 2-^-E'Hx = 0 vcri x < 0 (2.71) dx1 h d^ L + ^ { E - U 0)$2 = 0 vdri 0 < £ < a (2.72) dx2 h 2.6. M6t so bai toan don gian cua cor hoc luong tu 77 d2 a (2.73) Chung ta dat K x = lV2mE- K 2 = \^/2m{U0 -E )- K 3 = \ ^2m E = K\ n n a Ta co the viet nghiem cua ba phuong tnnh (2.71), (2.72), (2.73), = A i e iK' x + B xe~iK' x = A 2e K' x + B 2e~K' x = A 3eiK' x + B 3e~iK' x (2.74) (2.75) (2.76) M6i nghiem dupe xem la gom hai song mot song lan truyin theo true x va mot song phan xa ngupc lai. Trong vung III khong the co song phan xa nen ta co the cho B3 = 0. NghTa la: $3 = A3ei K \ x(2.77) Tir dieu kien lien tuc cua ham song va dao ham bac nhdt cua ham song chung ta co : $ 1(0) = ^ 2(0); 2 i K xK 2e - iK' a .4i ( K 2 - K 2) sh(K2a) + 2i K \ K 2 ch(K2a) Ta co the tim he so truyin qua cua hang rao the D, nghTa la ty sd giua mat do hat di qua hang rao va mat do hat den hang rao. D i dang thay rang: D1-4.3 I 2 4 K 2K 2 -4, |2 (K2 + K 2)2 sh2(K 2a) + 4K \K \ (2.80) 78 Chuang 2. Mot sd'va'n de vat ly luong tu Hinh 2.15: So do hang rao the nang bai ky. Trong thuc ti6n chi quan tam den cac trudng hop ma trong do K 2a > 1 va ta co the viet shK 2a = | e K2“ va K\ - K 2, nen (2.80) co the viet gdn dung D = ------ A - ---- « e~2K2a 1 + ( \ e K*a)2 Thay K 2 = jry/2m(U0 - E) vao bieu thirc tren ta co D = e -TcV2rn(Uo-E) (2.81) Tir day chung ta co nhan xet: - Do truyen qua cua hang rao phu thuoc manh vao be rong hang rao, vdi (U0 - E) = 5eV ta thay D ddi vdi dien tu thay doi: D = e-2,3a(A)_ a(A) 1 1,5 2 2,5 10 D 0,100 0,032 0,010 0, -003 1,026 1 0 -10 - Hieu ung dudng ngam la hieu ung lucng tir, neu trong cong thuc ta cho h = 0 thi D = 0. - Ddi vdi loai hang rao the co dang bat ky ngudi ta dung cong thuc tfnh D nhu sau: D = C - e ~ 1^ ^ ' ^ ZS^ (2.82) Gia tri x\,x2 xac dinh nhu tren hinh ve (2.15). - Ung dung: Hien tucng tu phat xa, kfnh hien vi quet tunnel, diode tunnel 2.6.4 Dao dong tu dieu hoa (oscillator). Trong co hoc co dien, dao dong tu dieu hoa don gian nhat la mpt qua cdu co khoi lupng m treo tren m ot lo xo 2.6. Mdt so bdi toan dcm gian cua co hoc luong tu 79 khong trong lugfng co do cting K. Neu keo qua cdu lech khoi vi tri c&n bdng (dupe quy udc tai diem x = 0) doc theo true thing dumg x thi se sinh ra mot luc ty le vdri ly do x va ngupc chieu x, do la luc dan hoi: rr k d2x F = ~ K x = m dP Giai phuong trinh chuyen dong nay ta dupe x = A cos (ujt + a) A - bien do, u - tdn sd goc, a - pha ban ddu. Trong trudng hpp nay ta co the tfnh the nang cua qua cdu U(x) — - f Fdx = f Kxdx - J o io Nang lupng toan phdn bdng E = T + U = ^mv2 + K x 2 Theo quan diem co dien nang lupng cua dao dong tu dieu hoa co the nhan bdt ky gia tri nao, ke ca gia tri cue tieu E = 0 ung vdi trang thai nghi hoan toan. Neu xet dao dong tvr dieu hoa theo quan diem co lupng tu ta se thdy ket qua hoan toan khac. 6 day ta xet bai toan mot chieu ve chuyen dong cua mpt hat trong ho the nang parabol U{x) = Phuong trinh Schrodinger ddi vdi dao dong tu co dang £ + £ < * - ^ > * = ° (Z83) Giai phuong tnnh nay ta thdy: nghiem cua no thoa man cac dieu kien cua ham song khong phai vdi bdt ky gia tri nang lupng nao ma chi ddi vdi nhung gia tri nang lupng E xac dinh bdi bieu thurc sau: En =huj(n + 1/2) Vdi n = 0 ,1,2,3,... (2.84) N hu vay nang lupng cua dao dong tu dieu hoa lupng tu la mot dai lupng bi lupng tu hoa.