Chapter 3. Problem Solutions

Transcription

1 Capter. Proble Solutions. A poton and a partile ave te sae wavelengt. Can anyting be said about ow teir linear oenta opare? About ow te poton's energy opares wit te partile's total energy? About ow te poton s energy opares wit te partile's kineti energy? Fro Equation (., any partile s wavelengt is deterined by its oentu, and ene partiles wit te sae wavelengt ave te sae oenta. Wit a oon oentu p, te poton s energy is p, and te partile s energy is ( p (, wi is neessarily greater tan p for a assive partile. Te partile s kineti energy is K E ( p ( For low values of p (p<< for a nonrelativisti assive partile, te kineti energy is K p /, wi is neessarily less tan p. For a relativisti assive partile, K p, and K is less tan te poton energy. Te kineti energy of a assive partile will always be less tan p, as an be seen by using E (p ( to obtain ( p K K. Ina University Departent of Pysis

2 Capter. Proble Solutions. Find te de Broglie wavelengt of a.0-g grain of sand blown by te wind at a speed of 0 /s. For tis nonrelativisti ase, J s 9 λ. 0 ; v 6 (. 0 0 kg(0 /s quantu effets ertainly would not be notied for su an objet. 5. By wat perentage will a nonrelativistle alulation of te de Broglie wavelengt of a 00-keV eletron be in error? Beause te de Broglie wavelengt depends only on te eletron's oentu, te perentage error in te wavelengt will be te sae as te perentage error in te reiproal of te oentu, wit te nonrelativisti alulation giving te iger wavelengt due to a lower alulated oentu. Te nonrelativisti oentu is p nr K. 7 0 and te relativisti oentu is ( 9. 0 kg / s, kg(00 0 ev( J/eV p r ( K ( ( ( 0. 5 MeV / kg /s, Ina University Departent of Pysis

3 Capter. Proble Solutions keeping extra figures in te interediate alulations. Te perentage error in te oputed de Broglie wavelengt is ten ( / pnr ( / pr p r pnr 4. 8 %. / p p Te atoi spaing in rok salt, NaCl, is 0.8 n. Find te kineti energy (in ev of a neutron wit a de Broglie wavelengt of 0.8 n. Is a relativisti alulation needed? Su neutrons an be used to study rystal struture. A nonrelativisti alulation gives r ( / λ nr p ( (. 4 0 ev K. 0 0 ev 6-9 λ ( ev(0.8 0 (Note tat in te above alulation, ultipliation of nuerator and denoinator by and use of te produt in ters of eletronvolts avoided furter unit onversion. Tis energy is u less tan te neutron's rest energy, and so te nonrelativisti alulation is opletely valid. 6 Ina University Departent of Pysis

4 Capter. Proble Solutions 9. Green ligt as a wavelengt of about 550 n. Troug wat potential differene ust an eletron be aelerated to ave tis wavelengt? A nonrelativisti alulation gives 6 p ( / λ ( (. 4 0 ev 6 K ev, -9 ( λ ( 5 0 ev(550 0 so te eletron would ave to be aelerated troug a potential differene of 5.0 x 0-6 V 5.0 µv. Note tat te kineti energy is very sall opared to te eletron rest energy, so te nonrelativisti alulation is valid. (In te above alulation, ultipliation of nuerator and denoinator by and use of te produt e in ters of eletronvolts avoided furter unit onversion.. Sow tat if te total energy of a oving partile greatly exeeds its rest energy, its de Broglie wavelengt is nearly te sae as te wavelengt of a poton wit te sae total energy. If E (p ( >> (, ten p >> and E p. For a poton wit te sae energy, E p, so te oentu of su a partile would be nearly te sae as a poton wit te sae energy, and so te de Broglie wavelengts would be te sae. Ina University Departent of Pysis

5 Capter. Proble Solutions. An eletron and a proton ave te sae veloity Copare te wavelengts and te pase and group veloities of teir de Broglie waves. For assive partiles of te sae speed, relativisti or nonrelativisti, te oentu will be proportional to te ass, and so te de Broglie wavelengt will be inversely proportional to te ass; te eletron will ave te longer wavelengt by a fator of ( p / e 88. Fro Equation (. te partiles ave te sae pase veloity and fro Equation (.6 tey ave te sae group veloity. 5. Verify te stateent in te text tat, if te pase veloity is te sae for all wavelengts of a ertain wave penoenon (tat is, tere is no dispersion, te group and pase veloities are te sae. Suppose tat te pase veloity is independent of wavelengt, and ene independent of te wave nuber k; ten, fro Equation (., te pase veloity v p (ω/k u, a onstant. It follows tat beause ω uk, d v g ω u v p. dk Ina University Departent of Pysis

6 Capter. Proble Solutions 7. Te pase veloity of oean waves is gλ/ π, were g is te aeleration of gravity. Find te group veloity of oean waves Te pase veloity ay be expressed in ters of te wave nuber k π/λ as ω g v p, or ω gk or ω gk. k k Finding te group veloity by differentiating ω(k wit respet to k, dω g ω vg g v p. dk k k k Using ipliit differentiation in te forula for ω (k, dω ω ωvg g, dk g gk ω ω so tat vg v p, ω ωk ωk k te sae result. For tose ore ofortable wit alulus, te dispersion relation ay be expressed as ln( ω ln( k ln( g, dω dk ω fro wi, and vg v p. ω k k Ina University Departent of Pysis

7 Capter. Proble Solutions 9. Find te pase and group veloities of te de Broglie waves of an eletron wose kineti energy is 500 kev. K For a kineti energy of 500 kev, γ v / 5 Solving for v, v ( / γ ( / , and fro Equation (.6, v g v Te pase veloity is ten v p /v g.6.. (a Sow tat te pase veloity of te de Broglie waves of a partile of ass and de Broglie wavelengt λ is given by λ v p (b Copare te pase and group veloities of an eletron wose de Broglie wavelengt is exatly x 0 -. (a Two equivalent etods will be presented ere. Bot will assue te validity of Equation (.6, in tat v g v. First: Express te wavelengt x in ters of v g, Ina University λ p v γ g v g v g. Departent of Pysis

8 Ina University Departent of Pysis Multiplying by v g, squaring and solving for v g gives. / ( ( v g λ λ Taking te square root and using Equation (., v p /v g, gives te desired result. Seond: Consider te partile energy in ters of v p lv g ; ( ( ( (. / ( v p E p λ γ Dividing by ( leads to so tat, /( λ v p, / ( ( ( /( v p λ λ λ λ wi is an equivalent stateent of te desired result. It sould be noted tat in te first etod presented above ould be used to find λ in ters of v p diretly, and in te seond etod te energy ould be found in ters of v g. Te final result is, or ourse, te sae.

9 (b Using te result of part (a, 8 - ( 9. 0 kg(.0 0 /s(.0 0 v p , J s and v g /v p For a alulational sortut, write te result of part (a as - λ ( 5 0 ev(.00 0 v p ev In bot of te above answers, te stateent tat te de Broglie wavelengt is exatly 0 - eans tat te answers an be given to any desired preision.. Wat effet on te sattering angle in te Davisson-Gerer experient does inreasing te eletron energy ave? Inreasing te eletron energy inreases te eletron's oentu, and ene dereases te eletron's de Broglie wavelengt. Fro Equation (., a saller de Broglie wavelengt results in a saller sattering angle. Ina University Departent of Pysis

10 Capter. Proble Solutions 5. In Se..5 it was entioned tat te energy of an eletron entering a rystal inrease, wi redues its de Broglie wavelengt. Consider a bea of 54-eV eletrons direted at a nikel target. Te potential energy of an eletron tat enters te target anges by 6 ev. (a Copare te eletron speeds outside and inside te target. (b Copare te respetive de Broglie wavelengts. (a For te given energies, a nonrelativisti alulation is suffiient; v K ( 54 ev( J/eV 4. 6 /s 9. 0 kg outside te rystal, and (fro a siilar alulation, wit K 80 ev, v 5.0 x 0 6 /s inside te rystal (keeping an extra signifiant figure in bot alulations. (b Wit te speeds found in part (a, te de Brogile wavelengts are found fro J s λ p v 6 ( 9. 0 kg(4.6 0 /s or 0.67 n outside te rystal, wit a siilar alulation giving 0.7 n inside te rystal., Ina University Departent of Pysis

11 Capter. Proble Solutions 7. Obtain an expression for te energy levels (in MeV of a neutron onfined to a one-diensional box.00 x 0-4 wide. Wat is te neutron's iniu energy? (Te diaeter of an atoi nuleus is of tis order of agnitude. Fro Equation (.8, ( J s E n n n n. 8 0 J n 7-4 8L 8( kg(.00 0 Te iniu energy, orresponding to n, is 0.5 MeV 4 9. A proton in a one-diensional box as an energy of 400 kev in its first exited state. How wide is te box? Te first exited state orresponds to n in Equation (.8. Solving for te widt L, 0. 5 MeV. L n 8E 8( ( kg(400 0 J s ev( J/eV f. Ina University Departent of Pysis

12 Capter. Proble Solutions. Te atos in a solid possess a ertain iniu zero-point energy even at 0 K, wile no su restrition olds for te oleules in an ideal gas. Use te unertainty priniple to explain tese stateents. Ea ato in a solid is liited to a ertain definite region of spae - oterwise te assebly of atos would not be a solid. Te unertainty in position of ea ato is terefore finite, and its oentu and ene energy annot be zero. Te position of an ideal-gas oleule is not restrited, so te unertainty in its position is effetively infinite and its oentu and ene energy an be zero.. Te position and oentu of a.00-kev eletron are siultaneously deterined. If its position is loated to witin 0.00 n, wat is te perentage of unertainty in its oentu? Te perentage unertainty in te eletron's oentu will be at least p p 4πp x 4π x K 4π x ( K 4π( ( (5 0 ev ev(.00 0 ev. 0. %. Note tat in te above alulation, onversion of te ass of te eletron into its energy equivalent in eletronvolts is purely optional; onverting te kineti energy into joules and using 6.66 x 0-4 J s will of ourse give te sae perentage unertainty. Ina University Departent of Pysis

13 Ina University Capter. Proble Solutions 5. How aurately an te position of a proton wit v << be deterined witout giving it ore tan.00 kev of kineti energy? Te proton will need to ove a iniu distane v t v, 4π E were v an be taken to be K E v, so tat K v t 4π E π K π ( K π. 4 0 ( ev MeV(.00 0 ev p. (See note to te solution to Proble - above. Te result for te produt v t ay be reognized as v t /πp; tis is not inonsistent wit Equation (., x p /4π. In te urrent proble, E was taken to be te (axiu kineti energy of te proton. In su a situation, ( p p E p v p, wi is onsistent wit te previous result. Departent of Pysis

14 7. A arine radar operating at a frequeny of 9400 MHz eits groups of eletroagneti waves µs in duration. Te tie needed for te refletions of tese groups to return indiates te distane to a target. (a Find te lengt of ea group and te nuber of waves it ontains. (b Wat is te approxiate iniu bandwidt (tat is, spread of frequenies te radar reeiver ust be able to proess? (a Te lengt of ea group is t ( /s( s 4. Te nuber of waves in ea group is te pulse duration divided by te wave period, wi is te pulse duration ultiplied by te frequeny, 8 6 ( s( Hz 75 waves. (b Te bandwidt is te reiproal of te pulse duration, 8 - ( s. 5 MHz. Ina University Departent of Pysis

15 Ina University Departent of Pysis Capter. Proble Solutions. / / π ν C 9. Te frequeny of osillation of a aroni osillator of ass and spring onstant C is Te energy of te osillator is E p / Cx /, were p is its oentu wen its displaeent fro te equilibriu position is x. In lassial pysis te iniu energy of te osillator is E in 0. Use te unertainty priniple to find an expression for E in ters of x only and sow tat te iniu energy is atually E in ν/ by setting de/dx 0 and solving for E in. To use te unertainty priniple, ake te identifiation of p wit p and x wit x, so tat p / (4πx, and. ( 8 x C x x E E π Differentiating wit respet to x and setting, 0 E dx d 0, 4 Cx x π wi is solved for. C x π Substution of tis value into E(x gives. in 8 ν π π π π C C C C E