Physics Graduate Prelim exam

Size: px

Start display at page:

Download "Physics Graduate Prelim exam"

Brianna Leonard
5 years ago
Views:

1 Physics Grdute Prelim exm Fll 2008 Instructions: This exm hs 3 sections: Mechnics, EM nd Quntum. There re 3 problems in ech section You re required to solve 2 from ech section. Show ll work. This exm is closed book. No texts of ny kind llowed. You re llowed to bring single sheet of formule. You cn use clcultor.

2 Mechnics You re required to solve ny 2 out of 3 Problem 1 Dmped oscilltors re described by the following eqution: m & x + bx& + kx = 0 ) Use digrms nd less thn 50 words to explin how one rrives t this eqution b) Show the trnsformtion to the following eqution: 2 & x + 2β& x + ω0 x = 0 c) Describe mthemticlly β nd ω 0. Wht do β nd ω 0 physiclly represent? d) Wht three cses of oscilltions re generlly discussed? e) Sketch the behvior of ech of those three cses s function of time f) We often include term on the right of the eqution. Wht does f(t) represent? 2 & x + 2β& x + ω0 x = f ( t) g) Assume tht f ( t) = cos( ωt) nd tht β is much less thn ω 0. Without going through huge derivtion, how would you expect the oscilltor to behve fter reching the stedy stte for ω<ω 0? ( grphicl description is fine) h) How would you expect the oscilltor to behve fter reching stedy stte for ω>ω 0? (gin, grphicl description is fine) Problem The Lgrngin for mechnicl system is L = q& + bq where q is generlized coordinte nd nd b re constnts. The eqution of motion for the system is: b 2 ) q & = q 2b b) q 3 q & = 2b c) q 3 q& = 2b d) q 3 q & = + b 3 e) q & = q Note: in order to get full credit you hve to show work.

3 Problem 3 r F = byxˆ + bxyˆ r F = byxˆ + bxyˆ Above re representtions of two force fields nd their corresponding mthemticl representtions. Arrow points on the grphs represent the mgnitude nd directions of the force evluted t the center of ech rrow. ) Test ech of the bove forces (either grphiclly or mthemticlly) to see if they re conservtive. b) Write the potentil energy function for the cse(s) where the force is conservtive. r c) Consider the cse of force given by F = yxˆ + bxyˆ where b re both positive numbers. Is this conservtive force? d) Nme three things tht re specil bout conservtive forces?

4 Electromgnetism You re to solve ny 2 out of 3 1) A prllel plte cpcitor is composed of two metl circulr discs of dimeter D, seprted by distnce in vcuum (where D>>). A resistor of resistnce R connects the two pltes. At t=0 the cpcitor is suddenly given chrge of Q 0. Neglecting ny edge effects write down: ) An expression for the cpcitnce of the cpcitor in terms of it s dimensions (2 pts) b) An expression for the time dependnt chrge on the pltes (4 pts) c) An expression for the time dependnt electric field between the pltes nd indicte its direction on the digrm below. (5 pts) D R d) Write down three of the following electrodynmics equtions defining your constnts. Explin their use nd context with digrm. (9 pts) 1) The Biot Svrt Lw 2) Ampere s Lw 3) Kirchhoff s rule for current junctions 4) Frdy s Lw of Induction 5) The Poynting vector

2) Describe the opertion of the trnsformer with digrm nd explin why they re used in long distnce power trnsmission.

The rottionl xis is dimeter of the ring.

Assume the energy loss goes into Joule heting in the ring. The conductivity of copper is 5.5 x 10 7 s 1, the density of copper is 8.

5 2) Describe the opertion of the trnsformer with digrm nd explin why they re used in long distnce power trnsmission. (5 pts) A thin copper ring of rdius 10cm rottes bout n xis perpendiculr to uniform mgnetic field H of 0.02T. The rottionl xis is dimeter of the ring. (15pts) H If the initil ngulr frequency of rottion is ω 0, clculte the time it tkes for the frequency to decrese to ω 0 /e. Assume the energy loss goes into Joule heting in the ring. The conductivity of copper is 5.5 x 10 7 s 1, the density of copper is 8.9g/cm 3 CLUE: I = 1 2 Mr2 for thin ring 3) A sphere of rdius R is uniformly chrged with chrge density ρ. Inside it is sphericl cvity of rdius R whose center is t distnce from the center of the sphere. Find the electric field vector (i) inside the cvity, (ii) outside the cvity but inside the sphere nd (iii) outside the sphere. R R

6 Quntum hrmonic oscilltor describes prticle in n hrmonic potentil with the Hmiltonin H = p2 2m mω 2 x 2. For quntum prticle, the coordinte opertor x nd the momentum opertor p obeys the commuttion reltion Define the following opertors 1. derive the commuttion reltion [, + ] 2. express the Hmiltonin H in terms of nd + 3. if wve function ψ(x) is n eigenstte of the hrmonic oscilltor Hmitonin with Hψ(x) = Eψ(x), show tht + ψ(x) is lso n eigenstte with eigenenergy. 4. The ground stte wve function of the oscilltor is derive Δx 2 nd Δp 2 nd show tht they stisfy uncertinty principle. Consider prticle in one dimensionl infinite squre well. The length of the well is L. 1. clculte the first three eigensttes of the infinite squre well. 2. with n initil wve function ψ(x,0) =ψ 1 (x) ψ 2(x), write down the time evolution ψ(x,t)

7 3. in (b), time-independent perturbtion is pplied with V (x) = V 0,L /2 < x L 0,otherwise use perturbtion theory to clculte the new ground stte energy, only considering the first three sttes. In double well potentil, prticle exhibit quntum tunneling between the wells. Here, we ssume only one locl stte is considered in ech of the potentil well. The sttes re lbeled s L nd R, with the energy E L nd E R respectively. The quntum tunneling rte is given s t 0 nd the tunneling Hmiltonin is. 1. for the two level system of L nd R including the tunnel Hmitonin, solve the eignesttes nd eigenenergies λ ±, (eigensttes cn be expressed in terms of λ ±, no need to simplify) 2. for E L = E R, write down the the energy difference between the new eigensttes. let the initil stte be L. Write down the time evolution of the stte. 3. ssume t 0 << E L E R, use perturbtion theory to write down the second order modifiction of the energy E L.

Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions

Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis