Physics 201 Final Exam December

Size: px

Start display at page:

Download "Physics 201 Final Exam December"

Sybil Cross
5 years ago
Views:

1 Physics 01 Fial Exam December Name (please prit): This test is admiistered uder the rules ad regulatios of the hoor system of the College of William & Mary. Sigature: Fial score:

2 Problem 1 (5 poits) It should be obvious that to be able to deliver all the gifts o Christmas ight, Sata eeds to move fast! Let s assume that he ca accelerate ad decelerate istatly, ad moves with a costat speed of v =0.8c. a) If Sata leaves the North Pole at exactly midight o December 5, what time will the clock o Christmas Islad (real thig!) show, whe he arrives there, if the distace betwee the North Pole ad Christmas Islad is 10,000km? b) What will Sata s wrist watch show at that momet? c) Whe Sata is d=1000km away from a reflective sky scraper, Rudolph s ose emits a bright short flash, which travels toward the mirror ad the is reflected back to the sleigh. What is the total travel time of the pulse as measured by observers o the groud ad by Sata? If you are ot ito Sata, here is the last part of the problem i a borig physics laguage: A observer i a rocket moves toward a mirror at speed v =0.8c relative to the referece frame labeled S (see figure). The mirror is statioary with respect to S. A light pulse is emitted by the rocket travels toward the mirror ad is reflected back to the rocket. The frot of the rocket is a distace d=1000km from the mirror (as measured by observers i S) at the momet the light pulse leaves the rocket. What is the total travel time of the pulse as measured by observers i the S frame ad the frot of the rocket? calculators oly for simple arithmetical operatios.

3 Problem (5 poits) (a) What is the eergy of the secod excited (=3) electro state for He + io (ucleus Z=)? (b) What is the wavelegth of a photo emitted whe the electro trasitios to the groud state? (c) This radiatio is used to precisely measure the velocity of a quasar 3C-9 that is movig away from the Earth at a speed of about v=0.8c. At what wavelegth the photo, emitted i part (b) will be detected by a Earth astroomer? calculators oly for simple arithmetical operatios.

4 Problem 3 (5 poits) (a) A physicist claims that i a experimet he measured the magitude of agular mometum of a electro i a hydroge atom to be 90 h. Is it possible? If yes, how may possible values of the z-compoet of the orbital agular mometum Lz that electro ca have? (b) What is the miimum possible priciple quatum umber for this state? (c) What is the eergy of the electro i that state? (d) What is the frequecy of a photo required to completely ioize this state (i.e. to excite the electro ito a uboud state E 0)? calculators oly for simple arithmetical operatios.

5 Problem 4 (5 poits) (a) Fid first five eergy levels of a particle with mass m, cofied iside a three-dimesioal rigid box size L L L (i.e., U(x,y,z)=0 for 0<x <L, 0<y,z<L, ad U(x,y,z)= elsewhere). What are the degeeracies of these eergy states? (b) Let us use such 3D rigid box as a crude model for a asymmetric quatum dot, cotaiig 10 electros. What is its groud-state eergy? (c) I a alterative uiverse, electros are spi-1 particles. What would be the groud state eergy for a 10-electro quatum dot i that case? calculators oly for simple arithmetical operatios.

6 Problem 5 (30 poits) U0 E 0 L L x I order to fid wave-fuctios ad eergies for statioary boud states of a particle with mass m i a rigid potetial well (see figure), we eed to solve a time-idepedet Schrodiger equatio with potetial: +, x < 0, x > L U ( x ) = 0, 0 < x < L. U L < x < L, 0 (a) At what iterval of x the probability of detectio for a particle with eergy E < U0 is o-zero? (b) Sketch a wave-fuctio of =6 boud state i this potetial (assumig E6 < U0). Make sure you sketch clearly shows the behavior of the wave fuctios i differet regios, feel free to add otes clarifyig the features you are drawig. (c) Write the geeral form for the wave fuctio ψ1(x) for all values of x. Make sure to defie ay eergyrelated coefficiets (like k ad α ). (d) What costrait(s), if ay, ca be imposed o the coefficiets by cosiderig the boudary coditios? Elimiate as may coefficiets as possible, ad give equatios for the rest. You do t eed to solve these equatios! calculators oly for simple arithmetical operatios.

7 Problem 6 (30 poits) A ustable ++ particle decays ito two lighter particles, a positively-charged pio ad a proto +. The rest mass of a pio is mπc =140 MeV ad its measured mometum pπc = 70MeV i the positive x directio. The rest mass of a proto is mpc =940 MeV ad mometum ppc = 350MeV i the positive y directio. a) What is the mometum of the origial ++ particle? b) What is the total eergy of the origial ++ particle? c) What is the mass of the ++ particle measured at its rest frame? Bous (+5 pts): the origial particle ++ is a spi 3/ particle ad cosists of three up quarks. Use Pauli exclusio priciple to argue that quarks must have some additioal quatum umber (like a color ) besides electrical charge (+/3e for a up quark) ad spi (1/ for a up quark). Why we eed three colors? calculators oly for simple arithmetical operatios.

8 Problem 7 (5 poits) A hypothetical fissio reactio is give by? + + +? The bidig eergy per ucleo of the elemets X, Y ad Z are 7.4 MeV, 8 MeV ad 8. MeV correspodigly. (a) What is the umber of protos i Z? (b) How may eutros is produced? (c) What is the eergy released i the reactio? calculators oly for simple arithmetical operatios.

9 Problem 8 (15 poits) The four absorptio lies of logest wavelegth i lithium iodide gas are foud at λ=1.13, 0.564, 0.376, ad 0.8 cm (ad o other lies i betwee). Fid correspodig photo eergies. Use the provided level diagram (ot to scale!) to idetify the rotatioal trasitios ivolved i emissio of each spectral lie. l=4 l=3 l= l=1 l=0 calculators oly for simple arithmetical operatios.

10 Relevat equatios you may or may ot eed: Loretz trasformatios Relativistic ivariat x = γ( x vt) y = y; z = z v t = γ t x c u v x u = x u v 1 x c u y, z u = y z u v γ 1 x c ( c t x ; Doppler effect: s = ) r r p = γmv Relativistic eergy ad mometum: Photoelectric effect: Wave-particle duality Ucertaity priciple hf = ϕ + K Geeral Schrödiger equatio: max E = γmc ; γ = 1 ( v ), c h π ω πf E = h ω = hf, p = hk = ; k = = = λ λ c c h h p x ; E t ψ ih t 1 1+ v / c fobs = fsource (approachig observer) 1 v / c r r 4 v pc E = p c + m c ; = c E ( x,t) h ψ( x,t) = m Statioary (time-idepedet) 1D Schrödiger equatio: ( x t) = ψ ( x) ψ, e ie t h ( x,t) + U( x ) ψ x h d ψ ( x) + U ( x) ψ m dx hω= hf= πhc/λ= E E ii fi d ψ Geeral solutios for the equatio + ψ = 0 k are A cos( kx ) + B si( kx ) or Ae dx d ψ x x Geeral solutio for the equatio α ψ = 0 is Ae α α + Be dx h π E = ml E = h ω ; ψ πx si L L 1-D ifiite square well: ( ) Simple harmoic oscillator ( + 1 ) Z me ( ke ) Hydroge-like io eergy levels: Agular mometum values: L L z x = ER Z E = = h = h l( l + 1); l = 0( s ), 1( p ), ( d ), 3( f = hm m = l, l + 1,.. 0..l 1,l Magitude of the agular mometum vector is L = h l( l + 1) Spi values: E S S z h = l I = s( s ) = h h + 1 m s ( l + 1) + ν + hω s = ; for electro: 1 1 rot vib ; selectio rules l = ± 1 ; = ± 1 Nuclear bidig eergy = + ν )... 1; ( x) = E ψ ( x) ikx + Be ; ikx Some useful costats c= m/s h=πh=π J s h= ev s hc=140 ev m hc=197 ev m me=0.51 MeV/c mp= MeV/c m= MeV/c E R = m e ( ke ) h =13.6eV

PHYS-3301 Lecture 7. CHAPTER 4 Structure of the Atom. Rutherford Scattering. Sep. 18, 2018

PHYS-3301 Lecture 7. CHAPTER 4 Structure of the Atom. Rutherford Scattering. Sep. 18, 2018 CHAPTER 4 Structure of the Atom PHYS-3301 Lecture 7 4.1 The Atomic Models of Thomso ad Rutherford 4.2 Rutherford Scatterig 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydroge Atom 4.5 Successes