Name Solutions to Test 2 October 14, 2015

Size: px

Start display at page:

Download "Name Solutions to Test 2 October 14, 2015"

Elijah Ryan
5 years ago
Views:

1 Name Solutios to Test October 4, 05 This test cosists of three parts. Please ote that i parts II ad III, you ca skip oe questio of those offered. The equatios below may be helpful with some problems. Costats 4 h J s 5 h 4.60 ev s J s 6 =6.580 ev s kb.8070 J/K 5 kb ev/k k N m / C 9 e.60 0 C 9 ev.60 0 J ke c 7 9 Part I: Multiple Choice [0 poits] For each questio, choose the best aswer ( poits each). Oe of the ucertaity relatios says that A) Et B) Et C) t E D) t E E) oe of these Hydroge-Like Atoms Hydroge Spectrum 4 kez c Z E 9.7 m m.60 ev Z E Black Bodies 4 kt B Wave Reduced Mass U Relatioships 5c mm max T m K m M k Compto Effect Rutherford Scatterig f h kqq T b cot cos mc mv h.46 0 m Ze k mc R E. Why was there a problem i statistical mechaics whe studyig a thermal distributio of electromagetic waves? A) The eergy i each mode could be aythig, so there was o idea how to figure out how much there was B) Log wavelegths had most of the eergy, leadig to the ifrared catastrophe C) Short wavelegths had most of the eergy, leadig to the ultraviolet catastrophe D) The predictios could ot explai the photoelectric effect E) There was already ample evidece that light was made of particles (photos). Whe you have a wave packet, the speed of the etire packet is govered by the velocity, while the speed of idividual waves withi it are govered by the velocity. A) Group, wave B) Wave, group C) Group, phase D) Phase, group E) Phase, wave

2 4. Which hypothesis is attributed to debroglie? A) Electros have wave-like properties B) Electros must satisfy Schrödiger s equatio C) Electros orbit the ucleus o circular paths D) Electros have agular mometum that is a multiple of E) Electros satisfy a ucertaity relatio o their positio ad mometum 5. If 4 i, what is? A) 7 B) 5 C) 7 + 4i D) 5 + 4i E) Noe of these 6. If f xy, Ax y, the f x A) Ax B) Ax C) A(x + y) D) A(x + y) E) Ay 7. I which of the followig ways is a atom very much like the solar system? A) The mai force ivolved is electromagetism i both cases B) The mai force ivolved is gravity i both cases C) The force betwee two of the orbitig objects is egligible compared to the force betwee oe of the orbitig objects ad the cetral object D) The agular mometum for a sigle electro or plaet is usually a small multiple of E) The cetral object (Su or ucleus) is much smaller tha the size of the whole object (solar system or atom) 8. A dispersio relatio is a relatio betwee which two quatities? A) ad k B) ad f C) ad T D) ad k E) f ad T 9. Rutherford scattered alpha particles off of gold atoms, ad discovered, to his surprise, that A) Most of them wet straight through, suggestig the atom was mostly empty space B) The gold atoms were ofte kocked out, suggestig they were loosely boud C) The electros did little to stop them, showig that electros were light D) The alpha particles sometimes scattered at large agles, suggestig there was a lot of mass ad charge cocetrated i the ucleus E) The gold atoms would ofte be trasmuted to other metals, like lead 0. Wave effects i quatum mechaics are ot oticeable i the everyday world because A) Quatum mechaics does t apply to composite objects, like all everyday objects B) Quatum effects oly describe wave packets, which do t apply to particles C) Quatum effects are so slow that you ca t observe them D) The wavelegth is eormous compared to the size of everyday objects E) The wavelegth is tiy compared to the size of everyday objects

3 Part II: Short aswer [0 poits] Choose two of the followig questios ad give a short aswer (- seteces) (0 poits each).. Explai, i the photoelectric effect, why there is a miimum frequecy which suffices to free a electro from a metal. Whe you are above that frequecy, give a formula for the maximum eergy with which a electro ca be ejected. The photos have eergy E = hf. There is a miimum amout of work (called the work fuctio) for ay give metal that must be iput i order to free the electro. If hf <, the o electros ca be freed. If hf >, the electros ca be freed, ad those electros ca have at most a eergy of Emax = hf.. Sice a electro is attracted to the ucleus, it should have a miimum eergy whe it is exactly at the ucleus ad has zero velocity. Explai why this ca t happe i quatum mechaics. You should have at least oe mathematical relatio i your aswer. The miimum eergy would correspod to positio x = 0 ad mometum p = 0. However, specifyig both of these simultaeously is impossible for quatum waves, because the x p. product of the ucertaities are cotraied by the iequality. Whe electros are accelerated through a thi gas, ad the voltage is gradually icreased, at first the curret will icrease, the at certai threshold velocities, the curret will suddely drop ad the gas begis to glow. Explai qualitatively what is happeig. This is a descriptio of the Frack-Hertz experimet. Because atoms have oly certai levels that are eergies, there is a miimum amout of eergy that ca be absorbed by the atoms. Hece whe electros with less tha this eergy collide, they must collide elastically, maitaiig their speed. But whe we reach this threshold eergy, the atoms suddely ca absorb the eergy, slowig dow the electros ad ultimately decreasig the curret. But the atom is ow i a excited state, ad it emits a photo (it glows) whe it falls back to its groud state.

4 Part III: Calculatio: [60 poits] Choose three of the followig four questios ad perform the idicated calculatios (0 poits each). 4. Oe importat evet i the early uiverse is called recombiatio, ad occurred whe the temperature of the uiverse was approximately T = 970 K. The uiverse was filled with almost perfect black body radiatio at this time. (a) Fid the eergy desity of the blackbody radiatio at this time i J/m. The eergy desity is give by.8070 J/K970 K 4 kt B U c J s.9980 m/s J/m. (b) Fid the wavelegth where the blackbody radiatio was strogest at this time, i m. We use Wie s Law, max T m K, to fid max T m K m m. 970K (c) Fid the eergy for a sigle photo correspodig to the wavelegth you foud i part (b). The eergy of a photo is give by E hf, where f ca be foud from f c, so J J s m/s hc E hf m.7 ev.600 J/eV J (d) Usig the eergy desity you foud i part (a) ad the eergy you foud i part (c), estimate the umber of photos per uit volume (i m - ) at this time. The eergy desity is equal to the eergy per photo times the umber desity of photos, so a simple estimate would yield U J/m 9 E.060 J m. There is a flaw i this argumet, i that we used the eergy of the peak wavelegth photo. The 7 actual umber is more like 5. 0 m istead.

5 5. A N +6 io has atomic umber Z = 7, oe electro, ad is i the = 7 state iitially. (a) What is the eergy of this state? The eergy is give by.60 evz.60 ev7 E.60 ev. 7 (b) A photo of eergy E =.056 ev is absorbed by this atom. What is the ew eergy of the atom? What is the value of ow? Sice a photo is absorbed, the eergy icreases to The fial state ca the be foud from E E E.60 ev ev ev f i Z.60 ev.60 ev ev E,.60 ev7 5, ev 5. (c) The same atom, still at = 7, istead emits a photo of eergy E =.056 ev. What is the ew eergy of the atom? What is the value of ow? This time the eergy is lost, so E E E.60 ev.056 ev ev The fial state ca the be foud from f i Z.60 ev.60 ev ev E,.60 ev7 5, ev 5. (d) What is the wavelegth of light correspodig to a eergy of E =.056 ev? The wavelegth ca be foud from f c, ad the frequecy from E hf, so ev s m/s c hc hc f hf E.056 ev m 94.9 m.

6 6. The 05 Nobel Prize i physics wet to experimeters who demostrated that oe of the eutrios has mass. Based o the measured mass, a log wavelegth eutrio would have a dispersio relatio of the form Ak where A 6650 m /s. (a) Fid a formula for the phase velocity ad the group velocity i terms of k. The phase velocity ad group velocity are give by v p Ak Ak, k k w k k vg Ak Ak vp. (b) A typical eutrio left over from the Big Bag might have k 85 m. For this wave, fid the wavelegth, the agular frequecy ad the frequecy f. Each of these quatities is easily computed from the relevat formulas: k 85 m m 7.8 mm, 9 Ak 6650 m /s 85 m s, s f s MHz. (c) Fid the group ad phase velocity for the eutrio from part (b) i m/s. We simply substitute ito the formulas from part (a) to yield v v p g Ak m /s 85 m m/s, 7 Ak 6650 m /s 85 m. 0 m/s. (d) Fid the eergy i ev of the eutrio you foud i part (b). We use the simple formula E hf ev s s.7 0 ev.7 ev.

7 7. The wave fuctio for a particle is give by (x) Nx a x a, a a x Na 0 elsewhere. x a Na where N ad a are positive costats. The wave is Na sketched at right. (a) Where i the allowed regio a x a is the particle most likely to be? Where is it impossible for it to be? You ca rely o the sketch if you wish. It is most likely to be where the wave fuctio has the largest magitude (positive or egative), so this is clearly at x a. It is impossible for it to be where the wave fuctio vaishes, which is at x = 0. (b) What is the ormalizatio costat N? We fid the ormalizatio costat by demadig that the probability that it is somewhere be give by. We therefore have a a x dx N x dx N x N a a N a 8 N a, a a N, a N a We could have chose N egative, but the problem says N is positive. (c) If the particle s positio is measured, what is the probability that it will have x > 0? This is simply give by a a x dx 0 N x dx 0 N x 0 N a a 9.%. a

PHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep.

PHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep. Aoucemet Course webpage http://www.phys.ttu.edu/~slee/3301/ PHYS-3301 Lecture 10 HW3 (due 10/4) Chapter 5 4, 8, 11, 15, 22, 27, 36, 40, 42 Sep. 27, 2018 Exam 1 (10/4) Chapters 3, 4, & 5 CHAPTER 5 Wave