Chapter 1 The Dawn of Quantum Theory

Transcription

1 Chapt 1 Th Dawn of Quantum Thoy * By th Lat 18 s - Chmists had -- gnatd a mthod fo dtmining atomic masss -- gnatd th piodic tabl basd on mpiical obsvations -- solvd th stuctu of bnzn -- lucidatd th fundamntals of chmical actions - Physicists had -- gnatd th lationship btwn hat and wok -- dvlopd th fist two laws of thmodynamics -- dmonstatd th wavlik natu of light -- applid statistical mchanics to chmical systms * Sounds gat so what s th poblm? - Th gnal scintific community blivd: -- atoms a th basic constitunts of matt -- Nwton s Laws w univsal -- all th phnomnon in th wold is dtministic - Th w sval xpimnts which could not b xplaind basd on this dogma and h a a fw of thm: -- black body adiation -- th photolctic ffct -- disct atomic spcta - What conclusions do ths xpimnts lad to? -- atoms a not th smallst/most micoscopic objct -- w nd somthing bsid Nwtonian physics to xplain ths xpimnts * And thn cam quantum mchanics - xplains ths unsolvd issus - xplains bonding, stuctu and activity - uss pobability instad of dtminism - gnats uls fo lctons in atoms and molculs * Lt s talk about ths psnickty xpimnts - Black Body Radiation What is it? -- Objcts whn hatd will tun fom d to whit to blu which is an incas in ngy/fquncy -- th xact fquncy mittd is dpndnt upon th composition of th body -- an idal body absobs/mits all fquncis and hnc is also calld a blackbody and th adiation that is mittd blackbody adiation Classical Physics Bakdown -- classical physics assumd this mission of light was a sult of oscillating - s which act as antnna and can oscillat qually wll at any fquncy, -- Rayligh-Jans Law: usd classical physics to gnat th lationship btwn

2 spctal dnsity, (, T), and 8 kt B d (, T) ( T) d d ( ) T c wh ( T) d is th adiant ngy dnsity btwn and d J R 8.14 mol K J kb paticls 1.81 Boltzmann constant N K A 6.1 mol 8 m T absolut tmpatu (K) c.9981 spd of light s --- Expimntally w should s th gaphs blow: Simila to Figu 1.1 fom th txt As T is incasd you can s th colo chang --- Unfotunatly, w actually s a bak down with Rayligh-Jans Law calld th UV-Vis catastoph wh as ngy incasd/wavlngth ducd th RJL gos to infinity ath than back to zo th dashd lin is and is consistnt with th Rayligh-Jans Law at low T --- this lationship dos not wok at high tmpatus calld th UV catastoph --- classical physics failu! So, how do w fix this? Planck to sav th day --- Planck poposd th ngy of ths oscillating lctons fquncy o

3 E nh wh n1,, and h is popotionality constant ---- PLOT TWIST: Planck was on of th fist to cogniz vaiabls may not hav a continuum of valus but instad b quantizd ---- Blackbody adiation accoding to Planck 8h d(, T) ( T) d d ( T) h kbt c This xpssion can poduc th RJLaw at low fquncis o fo h << kbt x x x Rcall th Taylo Sis fo 1 fo - x 1!! h k 1 BT h h 11 1 kt B kt B! h k 1 BT h h h as h kt B kt B! kt B 8h 8 h kt B 8 kt B ( Td ) d d d RJLaw h kbt c 1 c h c - Photolctic Effct -- dfinition: lctons a mittd fom a mtallic sufac whn xposd to UV adiation -- Classical physics stats that light is an lctic fild, E, oscillating to its diction of popagation and th intnsity of th adiation E --- th - s should oscillat along with th fild and as th intnsity

4 incass so should ths oscillations which will vntually lad to th jction of an - fom th sufac of th mtal WRONG! --- th photolctic should occu fo any fquncy as long as th intnsity of th incidnt adiation is sufficintly high WRONG! -- Expimntally: --- th kintic ngy of th jctd - is indpndnt of th intnsity of th incidnt adiation --- th is a thshold fquncy,, which is dpndnt upon th mtal ---- blow this thshold no - s will b jctd fom th sufac ---- abov this thshold th K.E. of th - s vais linaly with -- Einstin to th scu, h poposd: --- light is mad up of ngy packts aka photons aka quanta --- th ngy of a photon is popotional to th light fquncy, E = h E = h K.E. = ½ m = h is calld th wok function and is analogous to th ionization ngy of an isolatd mtallic atom (mmb w a taking away an -) --- sinc ½ m must b, thn h o h = hnc K.E. = h - h --- th constant h that Einstin pdictd matchd that of Planck s SUCCESS! - Hydogn Atom Spctum -- In th 19 th cntuy scintists knw that ach atom possssd a chaactistic mission spctum Figu 1.6 fom txt --- ths a calld lin spcta sinc thy mit ngy at a slct numb of fquncis onc again th spctum is not continuous but disct -- quantizd -- Balm was th fist on to show that ths lin spcta followd a paticula pattn, n - wh n =, 4, 5, Figu 1.7 fom th txt dmonstating this lationship --- Fom this pattn h divd th lationship:

5 Hz and 1968 cm n, 4,5 n n --- This lationship will giv is to all of th visibl missions fo H, but what about th st? H coms Rydbg -- Rydbg dvlops a fomula which includs all of th possibl mission lins of hydogn RH cm whn n1 n1 n * Angula Momntum Spinning Right Round Figu 1.9 fom txt - If w otat a singl paticl aound a fixd point with adius w can wit th kintic ngy T as: T mv m I wh I is th momnt of intia p l lina : T angula : T m I - Whn w wit th K.E. in tms of momntum: * Boh, th hydogn atom and Rydbg - th hydogn atom -- consists of a massiv positiv nuclus and a small ngativ - which is in a fixd obit about th cntally locatd nuclus -- A tal of two focs --- Coulomb s law: foc of attaction btwn an - and a poton (th nuclus of hydogn) f 4 wh - is th chag of an -, th chag of a poton, is th adius and is th pmittivity C Jm mv --- Cntifugal foc: f wh m is th mass of an ths two must b qual in od to nsu th - dosn t spd towad nuclus ---- Solving fo w will poduc th Boh adius of hydogn:

6 4 mv Boh's obit - focing angula momntum to b quantizd: h n mv n n v m m n 4 m 4 4n n h m m mv fo n = 1, = 5.9 pm th Boh adius m m n 4 - Boh s Assumptions: -- th a stabl atomic stats in which atoms do not adiat --- ths stats a givn by En with n = 1,,, wh n = 1 is th lowst ngy stat o gound stat and is th most ngativ -- angula momntum is quantizd o ths stationay obits qui an intg numb of d Bogli wavlngths - Total E of ou -: P.E. fo an - and a poton spaatd by distanc is E mv 4 8 E K. E. PE.. T V mv mv call 4 4 nh m m 1 n m 8 hn 8 h n substituting yilds E wh n 1,,,... - Rlationship btwn En and Rydbg -- En = Ef - Ei = h = 4 1 h nf m m h 4 1 ni = h n i nf m 8 -- This xpssion looks suspiciously lik th Rydbg xpssion - Mt Rducd Mass 4 4 m 1 1 m h hc o R 8ch n i n f 8ch Figu 1.11 fom txt H

7 -- at th cnt of mass m 11 m 1 -- as w hav said pviously: T I -- wh ou momnt of intia can b wittn in tms of ducd mass: mm 1 ducd mass & I m m 1 -- looking back at ou H-atom th ducd mass tuns out to b m -- Ovall, th Boh modl woks gat fo any H-lik systm (H + o Li + ) - Limitations of this lovly dsciption -- dos not wok fo a systm containing mo than on - -- fails whn a magntic fild is applid to th systm * Wav-paticl duality h coms d Bogli - classical optics suppots th ida of light as a wav,.g. faction, tc. - th photolctic ffct suggsts that it can also b thought of as a paticl - nt d Bogli: h poposd that if light which is claly a wav can act as paticl than why can t a paticl act as a wav - Einstin povd that wavlngth,, and momntum, p, a invsly popotional: h p - d Bogli claimd matt would also follow this lationship -- fo matt p = mv wh m = mass and v = vlocity h -- thfo th d Bogli wavlngth is givn by mv -- but if matt acts lik a wav thn why an t w all oscillating? - Exampl: What is th d Bogli wavlngth of 75 kg boy and an lcton ach tavling at 1 mph? 4 4 kgm h Js s boy mils mils 1.69km hou 1m mv 75kg 1 75kg 1 hou hou mils 6s km Too small to b dtctabl 4 4 kgm h Js s 1 mils 1 mv mils 1.69km hou 1m kg kg 1 hou hou mils 6s km 1.61 On th od of UV * d Bogli Applid to Boh s H-atom Modl - Figu 1.1 txt: (a) psnts th Boh assumption and (b) 6 m 7 m

8 (d) show what happns if th intg assumption is not in plac th wav will vntually disappa h h nh n whn1,,,... sinc, thn n p p mv nh n v wh h m m * d Bogli In Ral Lif - X-ay Diffaction -- occus whn X-ays a fid at a cystallin substanc and is du to th intatomic spacing bing on th od of th X-ays -- this phnomnon is anoth xampl of wav-paticl duality - Elcton Micoscops -- Uss applid voltag though an lctomagntic fild and a abl to gnat a much shap imags than thi fofaths * Two-Slit Expimnts - a light wav is initially allowd to pass though on slit and ith hit o pass though two slits as shown in th figu to th lft - what sults in a pattn of bands in which th light spacs a wh th wav hav actd in a constuctiv way and th dak is wh thy hav actd dstuctivly as shown on th ight - Blow is a vido of what happns whn w allow only on paticl at tim to pass in ou xpimnt:

9 * Mo Unctainty - Hisnbg - Hisnbg unctainty pincipl: th xact momntum and th position of - cannot b know simultanously o xp h -- if w wish to know th location of an - within a ctain distanc x w nd a light souc whos solution is on th od of x o x -- unfotunatly as soon as w shin this light on ou - w chang its momntum, p -- using th d Bogli lationship w obtain th Hisnbg unctainty pincipl -- w will b visiting this lat - Consquncs of this unctainty -- w do not know what th vlocity is if w know th - is in th atom -- Boh assumd that th - was a paticl with known vlocity and position -- in od to complt th pictu w nd a tu wavlik dsciption of - s