1.1 CLASSICAL PHYSICS

Transcription

1 UIT 1 odern Pysis 1.1 CASSICA PHYSICS ewton s laws of motion are te basis of te most elementary priniples of lassial pysis. Equations based on tese laws are te simplest and tey are suitable for solution of simple dynamial problems, su as te motion of marosopi bodies, agrange s equations, Hamilton s equations and Hamilton s priniple are also fundamental priniples of lassial meanis, beause tey are onsistent wit ea oter and wit ewton s laws of motion. agrange s and Hamilton s equations are useful for soling many ompliated dynamial problems. In priniple, te properties of bulk matter must be deduible from te properties of eletrons and atomi nulei of wi it is omposed. Howeer, it is found tat many te obsered properties of matter annot be explained on te assumption tat te partiles obey te laws of lassial meanis. At te end of 19t entury and in te beginning of t entury, many new penomena su as potoeletri effet, x-rays, line spetra, nulear radiation were disoered wi wanted explanation on te basis of lassial pysis. aws of lassial meanis failed to explain te aboe said newly obsered properties of matter. Terefore te need of new onepts was felt in many areas of pysial sienes. Te onepts deeloped led to a new meanis alled quantum meanis. Anoter form of quantum meanis is alled wae meanis. Te matematial teory of tis meanis was deeloped by Erwin Sroedinger in 196. umerous problems of atomi pysis ae been soled by te appliation of quantum meanis. To understand te deelopment of wae meanis, we begin wit brief aount of blak body radiation, wi ould not be explained by lassial meanis. Tis is followed by desription of some penomena like te potoeletri effet, te Compton effet, et. Explanations of tese penomena are based on Plank s quantum ypotesis Blak Body Radiation A body wi ompletely absorbs radiations of all waelengts inident on it is alled a blak body, and te radiation emitted by su a body is alled blak body radiation or full radiation. Te nearest approa to a blak body is sown in Fig 1.1(a). It onsists of a porelain spere, aing a small opening.

2 Engineering Pysis 6 K (a) (b) Intensity 45 K 3 K in 1 14 Hz Fig. 1.1 (a) A blak body, (b) Spetral energy distribution Te inner surfae is oated wit lamp blak. Any radiation wi enters te spere troug te opening suffers a few refletions. At ea refletion about 98% of te inident radiation is absorbed. Tus after a few refletions at te inner surfae, te radiation is ompletely absorbed. If te area of te opening is ery small, te radiation annot be refleted out of te spere again. Te spere also emits radiant energy troug te opening. To study te distribution of radiant energy oer different waelengts, te blak body is maintained at a onstant temperature. By means of an infrared spetrometer and a bolometer te emissie powers of te blak body for different waelengts (or frequeny) are measured. Te results of te experiment onduted are illustrated in Fig. 1.1(b). Te inferene is tat at a gien temperature te radiation energy density initially inreases wit frequeny, ten peaks at around a partiular frequeny and after tat dereases finally to zero at ery ig frequenies. Blak body radiation is an important penomenon beause its properties ae a uniersal arater, being independent of te properties of any partiular material substane. Te oter onlusions are: (i) Te area under a ure wi measures te total energy of radiation at tat temperature, inreases aording to te fourt power of te absolute temperature. Tus, Stefan s law is erified. (ii) Te maximum energy peak sifts towards te sorter waelengt side wit te inrease in temperature of te body. Tis onfirms Wien s displaement law. (iii) Wien s energy distribution formula deeloped in 1983 agrees wit tese ures for sort waelengts only wile Rayleig-Jean s formula in 19 agrees for longer waelengts. 1. PHTEECTRIC EFFECT Potoeletri effet is probably one well-establised penomenon wit sound experimental erifiation of te existene of partile-like properties in radiations. In 1887 Hertz sowed tat a

3 odern Pysis 3 metalli surfae ould emit eletrons if illuminated wit ligt of ery sort waelengt. In 195 Einstein interpreted tis penomenon in terms of te energy relationsip, w = 1 m + ef = m + W (1.1) for wi e was awarded te obel prize. Te aboe equation an be written as p (pn) = n = 1 m + ef (1.) were W = e f is te work funtion of te metal orresponding to te eigt of a potential barrier at te surfae of te metal tat eletrons in te metal must oerome in order to esape into auum Experimental Arrangement for bsering Potoeletri Effet Here two eletrodes are losed in an eauated quartz bullbs (or glass). Te quartz ontainer does not absorb ultraiolet ligt. Te atode C is made of a potosensitie metal (zin, sodium, litium or esium). A potential diider is used to apply a p.d between anode A and atode C. Tis applied p.d say V olt is measured wit te elp of a oltmeter. Te urrent in te iruit is deteted wit te elp of a sensitie galanometer or miro ammeter. Tere are tree basi experimental ariables in te potoeletri experiment. Tey are (a) (b) (i) Intensity of ligt: Te grap between te potoeletri urrent and intensity is a straigt line. Tis sows tat te potoeletri urrent inreases linearly wit intensity. (ii) o detetable time lag as been measured between te switing of ligt and emission of potoeletrons. (i) Stopping potential: Tere exists a minimum potential V below wi tere is no potoeletri emission and ene no potoeletri urrent. Tis potential wi is just igt S C A V A ( ) Fig. 1. illikan s apparatus for studying potoeletri effet

4 4 Engineering Pysis () suffiient to stop te potoeletri urrent is alled stopping potential or te ut off potential. (ii) Te stopping potential V is independent of te intensity of te inident ligt. (i) Tresold frequeny: Tere exists a minimum frequeny n at wi te stopping potential is zero. Tis means tat no eletrons are emitted if te frequeny of te inident ligt is below n. Te minimum frequeny n below wi tere is no potoeletri urrent is alled tresold frequeny. (ii) Te stopping potential inreases linearly wit te frequeny. Sine V = 1 m max, tis means tat te maximum eloity of emitted eletrons inreases linearly wit frequeny. 1.. aws of Potoeletri Emission Te results of te experiment onduted by illikan are (i) Te potoeletri urrent inreases linearly wit intensity of ligt. (ii) Te maximum eloity of potoeletrons depends only on te frequeny of inident ligt and is independent of its intensity (iii) Te potoeletri effet does not our below a ertain frequeny alled te tresold frequeny. Tis frequeny depends on te metal used in poto atode. (i) Te emission of potoeletrons is an instantaneous proess. 1.3 EISTEI S PHTEECTRIC EUATI Wen ligt of frequeny n sines on a metal, an eletron instantly absorbs a poton of energy n. If n is greater tan te binding energy te absorbed poton will ejet te eletron from te metal surfae and will appear as a potoeletron. Te energy of a poton is used in two ways. (i) Te part of absorbed poton energy is used in releasing te eletron from te metal surfae (ii) Te balane of energy appears as te kineti energy of te eletron. Tus we an write Energy of inident poton = Binding energy of eletron + K.E of eletron i.e., n = 1 m + W Tus 1 m = n - W Wen W = W, te kineti energy of emitted eletrons will be maximum. Tus 1 m max = n - W...(1.3)

5 odern Pysis 5 Suppose a poton of frequeny n = n as just suffiient energy to remoe te least bound eletron from te metal. Ten te maximum kineti energy of emitted eletron is zero. Using tis fat we ae = n - W or, n = W...(1.4) Te frequeny n determines te tresold frequeny. It will depend on te work funtion of a partiular material. Substituting te alue of W from Eqn. (1.4) in Eqn. (1.3) we obtain 1 m max = n - n...(1.5) Tis equation is alled Einstein s potoeletri equation Conlusions 1. Wen we inrease te intensity of inident ligt more potons will strike potoatode. As a result of it te eletrons emitted will inrease. Tus an inrease in intensity auses an inrease in potoeletri urrent. Tis is te first law of potoeletri emission.. It is eident from Eqn. (1.5) tat te maximum eloity max of potoeletrons depends only on te frequeny of inident ligt and is independent of its intensity. Tis is preisely te seond law of potoeletri emission. 3. Equation (1.5) tell us tat if frequeny of ligt n is less tan n, te kineti energy beomes negatie. So no eletron emission an our. Tis is noting but te tird law of potoeletri emission. 4. Aording to Einstein s teory te ligt energy inident on te poto-atode is supplied in onentrated bundles of potons. Tese potons are immediately absorbed by some atoms. As a result te immediate emission of potoeletron starts. Tis is exatly te fourt law of potoeletri emission. 1.4 DUAIS Te optial penomena like refletion, refration, interferene, diffration and polarization of ligt ould be easily explained by wae teory of ligt. n te oter and, te wae teory of ligt failed ompletely to explain te potoeletri effet, Compton effet, absorption and emission of radiation by substanes. Tese penomena ould be easily explained by quantum teory of ligt. It appears as if ligt presents itself in su a form as to support wae teory at one and and quantum teory on te oter and. Tis omplex nature of ligt is said to be dual nature. In quantum teory of ligt, te energy is tougt to soot out from te soure in te form of energy paket, alled potons. Te energy arrying partile poton itself exibits diffration effets, tereby sowing tat poton is toug a partile, yet it possesses wae like araters. Te diffration effets due to single poton were experimentally obsered by a Cambridge student G.I. Taylor. Te details of te experiment ae been disussed in te next setion. But tis leads to an important onlusion tat

6 6 Engineering Pysis ligt possesses dual araters, i.e., at te same time it beaes wae and partile like Compton Effet Wen a monoromati beam of x-rays is sattered by a blok of paraffin or grapite te sattered x-rays onsist of two omponents: ne omponent aing te same waelengt as tat of te inident x-rays and te seond omponent aing a greater waelengt. Tis is known as Compton effet. Te former is alled te unmodified radiation and te latter te modified radiation. Te lassial eletromagneti teory explained te unmodified radiation but it totally failed to explain te presene of te modified radiation. Compton, oweer, gae a satisfatory explanation for te modified radiation on te basis of quantum teory General Teory Consider a poton of energy E = n olliding wit an eletron. It transfers some of its energy to te eletron. Tus te sattered poton will ae less energy tan te inident poton. Sine energy of te poton is diretly proportional to frequeny, te frequeny of sattered poton, will be smaller tan te frequeny of inident poton. Tis implies tat waelengt, l, of te sattered poton will be greater tan te waelengt l of te inident beam. Te energy assoiated wit te sattered poton E = n. et q be te angle of sattering of te poton after te ollision. As a result of ollision, te eletron aquires some eloity say r. et us say tat te eletron reoils at an angle f to te diretion of te inident beam. If m is te relatiisti mass of te eletron, ten momentum of te eletron after ollision is p e = m r...(1.6) Te total energy and te total momentum of te system will remain onsered as te ollision between poton and eletron is elasti. p Inident poton Sattered poton p Eletron at rest Reoil eletron p e Fig. 1.3 Sattering of poton by an eletron

7 odern Pysis aw of Conseration of Energy and aw of Conseration of omentum et E e and E e be te energies of te eletron before and after ollision. Te onseration of energy requires tat E + E e = E + E e i.e., n + E e = n + E e...(1.7) If m is te rest mass and m is te relatiisti mass, ten Eqn. (1.7) beomes n + m = n + m n - n + m = m...(1.8) Using n = l and m = m m were b = 1 - / 1 -b Eqn. (1.8) beomes Diiding ea term by e j = l l P + m = l l P + m = m 1 -b m 1 -b Squaring bot sides and simplifying, one gets l l l l 1 -b l P + m + m 1 1 P = m e m l l + - l l ll + - ll + l - ll + m - ll b g m = e1 -b j - m bl lg m m 1 = e j 1 - b - -b e j j...(1.9)...(1.1) l + l wit b = / - ll + m l -l ll b g = ( 1 -b ) m - m + mb

8 8 Engineering Pysis Tus l + l - ll + m l -l ll b g m = e 1 - / j...(1.11) Conseration of omentum Suppose p r r and p are te momenta of inident poton and sattered poton respetiely. Te onseration of momentum requires r r p = p + p r e r r p - p = p r e Take te dot produt of bot sides wit temseles. Tis gies r r r r r r p - p g p - p p p Tis yields b b g = b p + p - p r r p = p e Sine te angle between p r and p r is q, e e g Substituting for m, p + p - p p os q = m...(1.1) p + p - pp os q = m e 1 - / j...(1.13) omentum of Poton and Compton Waelengt Te relatiisti relation between energy and momentum aording to Einstein s teory is E =m 4 + p Sine a poton always traels wit te speed of ligt, its rest mass must be zero. Tat is m =., Hene E = p ; E = p and p = E Tis gies energy-momentum relation for a poton. Tis sows tat momentum of te poton is equal to its energy diided by speed of ligt. Tus momentum of te inident poton is p = n = l...(1.14)

9 odern Pysis 9 and momentum of sattered poton is P = n = l Substituting tese alues in Eqn. (1.13) one gets l + l - ll os q = m 1 - / e Subtrating Eqn. (1.15) from Eqn. (1.11) - + m bl - lg + osq = ll ll ll or - osq = m bl - lg ll ll ll (1 - os q) = m (l l) i.e., j...(1.14a)...(1.15) (l l) = ( 1 - os q) m l = l + ( 1 -os q )...(1.16) m l gies te waelengt of te modified line. We also infer te following from Eqn. (1.16). Te waelengt of te modified line: (i) depends on te waelengt of te inident radiation (ii) depends on te angle of sattering q (iii) is always greater ten te waelengt of inident radiation (i) is independent of te nature of te satterer. Te quantity as dimension of lengt and is known as Compton waelengt. Te m differene in waelengt between te sattered x-rays and te inident x-rays is alled Compton sift designated bydl. Tus Dl = l - l = m (1 - os q)...(1.17) 1.5 EXPERIETA VERIFICATI F CPT S THERY Te apparatus used by Compton to study is teory onsisted of an x-ray tube aing a molybdenum target (Fig. 1.4) giing a strong K a line. Te arateristi x-rays emerging from te tube were allowed to be inident on a small grapite piee C. Te x-ray tube and te grapite piee

10 1 Engineering Pysis are bot plaed inside a lead blok wit a slit S 1 in one wall. A furter system of slits S plaed beind S 1 allows te beam sattered at a definite angles q to pass troug tem. Te distribution of intensity wit waelengt in te x-rays sattered at arious angles q is measured by means of te Bragg s x-ray spetrometer. By tis metod, spetra of te molybdenum K a line after being sattered by te grapite C at different angles were obtained. If te sattered rays were of te same waelengt as te primary, te spetrum of te former sould be te same as tat of te latter. Compton, oweer, found in te spetrum obtained not only te K a line but anoter K a on te longer waelengt side as illustrated in Fig Furter wit q = 9 te waelengt of te unmodified line K a was found to be.78å, wile tat of te modified line K a is.73å, te Compton waelengt. Te ange in waelengt was also found to inrease rapidly as te angle of sattering was inreased. It was also establised tat it is independent of l. Compton and Rose, using potograpi metod found tat Dl for te K b line of molybdenum was te same as for K a line for a gien angle of sattering. oreoer anging te target in te x-ray tube sould not produe, aording to te teory, an alteration in te alue of Dl proided q is te same. In oter words different targets produe different primaries wit different alues of l, yet te ange in waelengt Dl was found to be te same for a gien angle of sattering. Te results for, 45, 9 and 135 as sattering angles are sown in Fig Te study of ompton effet leads to te onlusion tat in its interation wit matter, radiant energy beaes as a stream of disrete partiles (potons) ea of energy n and momentum n/. In oter words, radiant energy is quantised. Terefore te ompton effet is onsidered as a deisie penomenon in support of te quantisation of radiant energy. Crystal C S 1 S E - T Fig. 1.4 Experimental set up for te study of Compton effet

11 odern Pysis 11 U (unmodified line) Intensity = Waelengt U (modified line) Intensity = 45 Waelengt U = 9 Intensity Waelengt Fig. 1.5 Results of Compton effet 1.6 DUA ATURE AD DE BRGIE S HYPTHESIS ouis de Broglie in 194 enuniated a ypotesis on matter waes. Aording to tis onept eery moing partile as a wae-paket assoiated wit it. Te waelengt of su waes depends upon te momentum of te partile.

12 1 Engineering Pysis He gae an expression for te waelengt of su waes as l = m =...(1.18) p were m is te mass of te partile moing wit a eloity, te produt m = p is te momentum of te partile, is te Plank s radiation onstant. If we onsider Plank s teory of radiation, te energy of a quantum is gien by E = n =...(1.19) l were n is te frequeny, is te eloity of ligt in auum and l is te waelengt or l =...(1.) E If m is te mass of te partile onerted into energy, te equialent energy is gien by Einstein energy-mass relation as E = m Hene Eqn. (1.) an be written as l = m = =...(1.1) m p were m = p is te momentum assoiated wit quantum. Hene if a body of mass m moes wit a eloity, ten m and Eqn. (1.1) is written as l = = m p Te aboe relation may also be written in terms of kineti energy wi is gien as K= 1 m = 1 m m = p m p= mk Terefore l=...(1.) mk Wen a arged partile, arrying a arge e, is aelerated troug a potential differene of V olt, ten kineti energy K= ev l= =. 17 nm...(1.3) mev V Also, if te material partiles are in termal equilibrium at assoiated temperature T, ten K= 3 k BT and l = mev R S T 3kT B U V W 1

13 odern Pysis 13 or l = were k B = joule/kelin 3mk B T...(1.4) Experiments of Daisson and Germer Te experiments of Daisson and Germer were te first experimental eidene in support of matter waes. Tese two Amerian pysiists performed experiment on te diffration of eletron waes by a nikel target. Te eletron beam from an eletron gun is aelerated and ollimated to strike a nikel rystal. C is an ionisation amber for reeiing te eletron after tey ae been sattered by a nikel rystal. Te ionisation amber an be moed along a graduated irular sale so tat it is able to reeie te sattered eletrons at all angles between to 9 and teir intensity is measured by te galanometer urrent. Te wole assembly is plaed in a ery ig auum. Graps are drawn at arious oltages and te pronouned maximum obtained for 54 olt at f = 5. (Fig. 1.6b) f + q + q = 18 q = 18-5 = 13 q = 65 Te interplanar distane for nikel is.1 nm. Tus d sin q = l l =.1 sin 65 =.167 nm...(1.5) Eletron gun Ionisation amber G Primary eletron beam q ikel single rystal C Diffrated beam q (a) Bragg s plane 4 V 44V 48V 54 V 6 V 64 V (b) Fig. 1.6 Daisson-Germer experiment on diffration of eletron waes

14 14 Engineering Pysis By de Broglie s ypotesis Eqn. (1.3) l = 17. nm =.167 nm...(1.6) 54 Tus we see 1% agreement in te alue. See Eqn. (1.5) and ompare wit Eqn. (1.6). 1.7 WAVE VECITY AD GRUP VECITY FR DE BRGIE WAVES Wae Veloity A wae is a disturbane from equilibrium ondition tat traels or propagates wit time from one region of spae to anoter Te original displaement gies rise to an elasti fore in te material adjaent to it, ten te next partile is displaed and ten te next and so on. Tus te motion is anded oer from one partile to te next. Terefore, eery partile begins its ibration a little later tan its predeessor. Tus tere is a progressie ange of pase from one partile to te next. Te pase relationsip of tese partiles is termed as wae and te eloity wit wi planes of onstant pase propagate troug te medium is known as wae eloity or pase eloity. Tus eloity of adanement of a monoromati wae (wae of single waelengt and frequeny) troug a medium is alled te wae eloity. For example, te equation of plane progressie wae is gien by y = a sin (wt k x) were w = p p = pn is te angular frequeny and k = is te propagation onstant. Te term T l (wt kx) represents te pase of te wae motion. Hene te planes of onstant pase are defined by (wt k x) = onstant Differentiating tis equation wit respet to time, w k dx dt = or dx dt = w k = u were u = dx is te pase eloity or wae eloity. Tus pase eloity (or wae eloity) is te dt ratio of angular frequeny w to te propagation onstant k and is te eloity wit wi a plane progressie wae front traels forward. If l is te waelengt and n te frequeny of te wae, ten pase eloity u = nl; n = u l...(1.7)

15 odern Pysis 15 If E is te energy of te wae, ten its frequeny n is gien by E = n; n = E Also from de-broglie teory, te waelengt of material partile l = m Terefore, te pase eloity of te assoiated de-broglie wae u = nl = E mp = E m But from Einstein s mass energy relation E = m, Terefore, u = m =...(1.8) m Sine >>, Eqn. (1.8) implies tat te pase eloity of te assoiated wae is greater tan, te eloity of ligt. It indiates tat te assoiated wae wit te partile traels faster tan te partile itself. Tus te partile will be left far beind. biously, a monoromati de Broglie wae an not transport a partile or arry energy. Te pase eloity is tus a purely matematial onept and represents merely te rate at wi a gien pase of a monoromati wae train adanes Group Veloity Te Conept of Wae Paket Te pase eloity of a wae assoiated wit a partile omes out to be greater tan te eloity of ligt. Tis diffiulty an be oerome by assuming ea moing partile of matter to onsist of a group of waes or a wae paket, rater tan a single wae train. A wae group orresponding to a ertain waelengt l onsists of a number of omponent waes of sligtly different waelengts in te neigbourood of l superimposed upon ea oter. Te mutual interferene between omponent waes results in te ariation of amplitude tat defines te sape of te wae paket. Te omponent waes interfere onstrutiely oer only a small region of spae, outside of wi tey interfere destrutiely and ene te amplitude redues to zero rapidly. Tus te resultant wae pattern onsists of points of maximum amplitude and points of minimum amplitude. Between any two onseutie minima, say A and C, tere is a position of maximum amplitude i.e., at B (mid way between A and C) as sown in Fig Te dotted loop represents a group of waes or a typial wae paket (Fig.1.7). Tis group of waes ( or te wae paket ) moes forward in te medium wit a eloity alled te group eloity. Tus group eloity is te eloity wit wi te slowly arying enelope of te modulated pattern due to a group of waes traels in a medium. Te importane of te group eloity lies in te fat tat it is eloity wit wi te energy in te wae group is transmitted.

16 16 Engineering Pysis B A C Fig. 1.7 Wae paket and pase eloity Expression for Group Veloity To derie an expression for group eloity, onsider a group of waes wi onsists of only two omponents of equal amplitude a but sligtly different angular frequeny w 1 and w and propagation onstants k 1 and k. Teir separate displaements may be represented by te equations y 1 = a sin (w 1 t - k 1 x) y = a sin (w t - k x) Te resultant amplitude due to superposition is gien by y = y 1 + y = a sin (w 1 t - k 1 x) + a sin (w t - k x) or y = a[sin (w 1 t - k 1 x) + sin (w t - k x)] i.e., y = a sin P ( w1 + w) ( k1 + k) t - x os or y = a os os ( w w ) ( ) 1 - t k1 + k - x sin Tis equation represents a wae system of amplitude A = a os ( w1 - w) t ( k1 - k) x - P P é( w1 - w) t ( k1 - k) ù ê - xú ë û ( w1 + w) t ( k1 + k) x - wi is modulated bot in spae and time. Eqn. (1.9) an be written in modified form as y = a sin (wt kx) os P...(1.9) Dw Dk t - x...(1.3) P

17 odern Pysis 17 were w = ( w w ) 1 +, k = k + k 1 and D w = w 1 - w, D k = k 1 - k Te resultant wae (Eqn.1.3) tus as two parts, (i) A wae of frequeny w, propagation onstant k and eloity wi is te pase eloity or wae eloity. u = w k = pn pl / = n l Dw (ii) Anoter wae frequeny, propagation onstant D k and eloity G = D w D k, superimposed upon te first wae. It represents a ery slowly moing enelope of frequeny Dw = ( w w ) 1 - and propagation onstant D k = ( k 1 - k ). Tis enelope is represented by te dotted ure in Fig.1.7 and moes wit a eloity G = w k - w 1 - k 1 = D w D k known as te group eloity. If a group ontains a number of frequeny omponents in a ery small frequeny interal, ten te aboe expression may be written as G = w k p n G = 1 p l Tis is te expression for group eloity. F H G I K J = - l n l...(1.3a) Relation Between Pase Veloity and Group Veloity If u is te pase (wae) eloity, ten, sine u = w k, te group eloity is gien by G = d w d = dk dk ( uk) = u + k du dk But k = p l, \ dk = - (p/l )d l

18 18 Engineering Pysis Hene Terefore, group eloity is gien by k dk = - l d l G = u + - G = u - P l d l du du P l...(1.31) d l Te relation sows tat group eloity G is less tan te pase eloity u in a dispersie medium i.e., wen u is funtion of l. In a non-dispersie medium, waes of all waelengt trael wit te same speed i.e., du = and ten G = u. Tis is true for eletromagneti waes in auum d l and elasti waes in omogeneous medium Relation Between Group Veloity and Partile Veloity Consider a material partile of rest mass m. et its mass be m wen moing wit a eloity. Ten its total energy E is gien by Its momentum is gien by E = m = m 1 - P p = m = 1 - m Te frequeny of te assoiated de Broglie wae is gien by 1 P /...(1.3) n = E = R S T m F 1 - HG IU KJ V W \ w = pn = R S T pm F HG 1 - IU KJ V W

19 odern Pysis 19 r d w = pm 1 - P / (d )...(1.33) Te waelengt of te assoiated de Broglie waelengt is gien by (See Eqn. 1.3) Hene, propagation onstant l = p = 1 k = p l = pm 1 - dk = pm P /...(1.34) m 1 P / P F + H I 1/ -1/ 1- d 1- K 1 - P F HG 1 dk = p m 1- d + 1- KJ dk = pm I P P F HG P -1/ -3/ - 3/ d -1 pm dk = P / T d R S F HG I KJ + F H G P d I KJ -3/ d IU KJ V W P d P d P

20 Engineering Pysis or dk = pm d 1-3 P /...(1.35) Sine group eloity, G = d w, terefore diiding Eqn. (1.33) by Eqn. (1.35), one gets group dk eloity, G = d w =, te partile eloity. Tus te wae group assoiated wit a moing material dk partile traels wit te same eloity as te partile. It proes tat a material partile in motion is equialent to a group of waes or a wae paket. Table 1.1 Potoeletri tresold frequeny and work funtion for some metals. Potoeletri work Tresold frequeny Tresold waelengt etal funtion, W n = W l = n (in ev) in 1 15 Hzin nm Calium Gold Iron ikel Platinum Rodium Siler Sodium Tantalum Tungsten Table 1. Waelengts of eletrons under seleted oltages Voltage applied in olt Waelengt, l in nm

21 odern Pysis 1 BJECTIVE UESTIS 1.1 Te potoeletri effet ours wit (a) only free eletrons (b) only bound eletrons () bot bound and free eletrons 1. Te maximum eloity of potoeletrons (a) depends on te frequeny of te inident radiation (b) depends on te intensity of te inident radiation () independent of te waelengt of te inident radiation (d) all are true 1.3 Te eloity of poton in te isible region is (a) (b) were is te eloity of ligt () 1 (d) 1.4 Te potoeletri effet is obsered only if te waelengt of ligt is (a) aboe tresold waelengt (b) below tresold waelengt () zero (d) equal to tresold waelengt 1.5 uartz bulb is used for irradiating te potosensitie atode beause quartz (a) absorbs ultraiolet rays easily (b) does not absorb ultraiolet rays () reflets ligt radiation easily (d) is a polarizer 1.6 At wat angle of sattering te waelengt of te sattered poton will be minimum? (a) (b) 3 () 6 (d) At wat angle of sattering te waelengt of te sattered poton will be maximum? (a) 9 (b) 4 () 6 (d) 1.8 Te de-broglie waelengt of a partile at rest is (a) zero (b) infinite () p (d) 1.9 Te penomenon wi points toward te partile nature of eletromagneti radiation is (a) diffration (b) interferene () Compton effet (d) none of tese 1.1 Can te waelengt of de Broglie wae of a partile of mass m and energy E be represented by l = me (a) yes (b) no

22 Engineering Pysis 1.11 In wae meanis te group eloity is equal to partile eloity (a) true (b) false 1.1 An eletron, neutron and a proton ae te same de Broglie waelengt. Wi partile as greater eloity? (a) proton (b) neutron () eletron 1.13 Te group eloity is less tan te pase eloity (a) true (b) false SHRT UESTIS 1.1 Explain blakbody radiation spetrum. 1. Wat are te laws of potoeletri effet? 1.3 Explain te meaning of te following terms: (i) Work funtion (ii) Stopping potential 1.4 Wat are matter waes? 1.5 Explain de Broglie s ypotesis. 1.6 Explain pase eloity and group eloity. 1.7 Wat is Compton effet? 1.8 Explain te results one infers from te study of Compton effet. 1.9 Gie te basi priniple of Daisson-Germer experiment. 1.1 State te most important experimental obserations about te potoeletri effet. REVIEW UESTIS 1.1 Wat is potoeletri effet? Gie an aount of te potoeletri emission of te eletrons. Gie Einstein s interpretation for te same. 1. ention te important experimental obserations obsered in te potoeletri emission of eletrons and explain tese on te basis of Einstein s teory. 1.3 Gie te laws of potoeletri emission. Derie Einstein s potoeletri equation and sow ow tis equation explains te laws. 1.4 Write down Einstein s potoeletri equation and use it to explain te following. In an experiment on potoeletri effet of measuring te energy and number of potoeletrons wat appens if: (i) Te frequeny of inident ligt is anged, target material and intensity of ligt being kept onstant; (ii) Te target material is anged, te frequeny and intensity of ligt being kept onstant;

23 odern Pysis 3 (iii) Te intensity of ligt is anged, te frequeny of ligt and target material being kept onstant. 1.5 Write a note on blak body radiation. Desribe an experiment to demonstrate potoeletri effet. 1.6 Wat is Compton sattering? How does it onfirm te orpusular nature of radiation? 1.7 Wat is Compton effet? Derie an expression for te waelengt of sattered poton. At wat angle of sattering te waelengt will be maximum? 1.8 Desribe an experiment to study Compton effet. btain an expression for Compton sift. 1.9 Disuss te dual nature of matter waes. Derie an expression for te de Broglie waelengt. 1.1 Explain de Broglie ypotesis. Desribe and explain Daisson and Germer experiment for te onfirmation of de Broglie ypotesis Explain learly te wae eloity and te group eloity. Sow tat te pase eloity of te wae is greater tan te speed of ligt wile te group eloity of te eletron is equal to eletron s eloity. 1.1 Disuss te onept of group eloity. btain te expression for group eloity. PRBES AD SUTIS 1.1 Calulate te number of potons emitted in 3 ours by a 6 watt sodium lamp. Gien l = 589.3nm Solution Energy of poton = n = l = = J umber of potons emitted by te sodium lamp in one seond = \ Te number of potons emitted by te sodium lamp in 3 ours is = Ans. 1. igt of wae lengt 447Å falls on a potoeletri ell wit a sodium atode. It is found tat te potoeletri urrent eases wen a retarding potential of 1. olt is applied. Calulate te work funtion of te sodium atode.

24 4 Engineering Pysis et te stopping potential be V olt and te work funtion be f. It is gien tat te potoeletri urrent eases wen te retarding potential is 1. olt i.e., V s = 1. olt aximum kineti energy of potoeletrons = 1. ev We ae n = f + E max But n = l = n = J = = 3.1 ev f = n E max = f =.1 ev Ans. 1.3 A poton of energy n is sattered troug an angle q by a free eletron originally at rest. Sow tat te ratio of kineti energy of te reoil eletron to te energy of inident poton is a( 1 - os q) 1+ a( 1- os q) n were a = m m is te rest mass of te eletron and is te eloity of ligt. Solution Te kineti energy of te reoil eletron is te differene between te inident poton energy and sattered poton energy. We, terefore, ae K = n - n F HG R S T = 1-1 l l K = l -l ll Substituting (l' - l ) from Eqn. (1.17) and l' from Eqn. (1.16) in te aboe equation, I K J U V W 1 - os q m K = R lsl + b1 - os q m T b g gu V W

25 odern Pysis 5 or It is assumed F HG K /l K= R I KJ = S T R S T K n = l R P S T 1 - os q m 1+ b1-os q m l ( 1 - os q) ml 1+ m / l 1-osq b gb g b g g P U V W n ( 1 - os q) m 1+ ( 1- os q)( n/ m ) U V W l U V W were 1 l = n a = n m K n = a( 1 - os q) Ans. 1+ a( 1- os q) 1.4 X-rays of waelengt.1 nm from a arbon blok are sattered in a diretion making 6 wit te inident beam. How mu kineti energy is imparted to te reoiling eletron? Solution Te Compton ange in waelengt is gien by l' - l = m sin (q/) Dl = R S T =.114 nm l' = l + Dl =.1114 nm Energy of inident X-ray poton = n = l U V W (.5) Energy sattered X-ray poton = n = l

26 6 Engineering Pysis Hene energy imparted to te reoiling eletron = l l = ( l - l ) ll = ( Dl) ll = (. 1 1 ) ( ) = J = = 149 ev 149 ev 1.5 A beam of x-rays of waelengt I Å is inident on a arbon target. Te sattered x-rays are deteted at an angle of 6 in te diretion of te inident beam. Find te waelengt of te sattered x-rays. Solution l' = l + m (1 - os q) But = =.43 nm m Tus l' = (1.5) l' =.111 Ans. 1.6 Te most rapidly moing alene eletron in metalli sodium at absolute zero temperature, as a kineti energy 3 ev. Sow tat te de Broglie waelengt is 7Å. Solution For te eletron -34 K = 3 ev = J m = (3 1 8 ) = J = = ev Terefore, kineti energy of te eletron is small ompared wit m. Hene de-broglie waelengt is gien by

27 odern Pysis 7 But l = m 1 m = ev or = ev m Hene But K = J, l = m P 1 = = ev m ev m l = 7 Å Ans. 1.7 Calulate te momentum of an eletron possessing te de Broglie waelengt m. Solution l = p p = l = - 11 = 1-3 kg.m/s p = 1-3 kg.m/s Ans. 1.8 Find te pase and group eloities of an eletron wose de-broglie waelengt is.1 nm. Solution l = m Te pase eloity of te wae is gien by p = w k...(1) It an be written as p = w k Using te relations E = w and p = k Tus p = E p...() Also E = 1 m = m p

28 8 Engineering Pysis Substitute tis in Eqn. () p = p m...(3) Using de-broglie relation l = p in Eqn. (3) p = -34 ml = p = m/s Te group eloity is always equal to te eloity of te partile. Tus g = = p m Using Eqn. (3), it beomes g = p = m/s Ans. EXERCISE 1.1 Ea of a poton and an eletron ae an energy of 1 eletron olt. Calulate teir orresponding waelengt. (Ans: l p = 1.4 Å. l e ª.4 nm) 1. Eletrons are emitted wit zero eloity from a ertain metal surfae wen it is exposed to radiation of 68 nm. Calulate te tresold frequeny and te work funtion of te metal. (Ans: Hz, 1.83 ev) 1.3 Te waelengt of te inident poton in Compton sattering is.4 nm. If sin q is.5, alulate te waelengt of sattered radiation. Gien l of te inident poton is 4 nm. (Ans:.415 nm) 1.4 Calulate te waelengt of a 1 kg objet wose eloity is 1 m/s and ompare it wit te waelengt of an eletron aelerated by 1 olt (l = m, l e = m and l e = l ). ASWERS T BJECTIVE UESTIS 1.1 (b) 1. (a) 1.3 (d) 1.4 (b) 1.5 (b) 1.6 (a) 1.7 (a) 1.8 (b) 1.9 () 1.1 (b) 1.11 (a) 1.1 () 1.13 (a)