The Northern California Physics GRE Bootcamp

Transcription

1 The Northern California Physics GRE Bootcamp Held at UC Davis, August 13-14, 2016 Damien Martin

2 Big picture (your exam) (24) percentile (6) <20 percentile

3 Big picture ( ) (35) percentile (18) <20 percentile

4 * Multiple passes through the exam * Dimensional analysis (which answers make sense?) Other hint -- look at exponentials, sines, cosines,... * Expansions, in particular (1+x)^n = 1 + n x +... * Limiting cases (e.g. make parameters go to 0 or infinity) * Special cases (e.g. looking at circles) * Powers of ten estimation * Know scales of things [wavelength / freq of visible light, binding energies of nuclei, mass ratios of common particles (up to muon, pion),..., mass of stars, mass of galaxies,...] Okay to specialize on scales Big tips and tricks

5 Big tips and tricks -- material * Know your first year general physics really well - Newtonian mechanics in particular * Know your modern physics general physics really well (usually Sophmore class) * Worth going through Griffiths: Intro to electromagnetism Griffiths: Intro to quantum mechanics (Concentrate on harmonic osc, infinite square well, spin systems, expectation values) Schroeder: Thermal physics * Look at the archive of monthly problems in The Physics Teacher (if you have access to a university library)

6 Evaluate whether question is special or first year

7 Specific tips and tricks -- material Bare bones thermal: Know the definition of partition function, probability of a state, how to find an expectation value Z = X state i hxi = X exp( E i )= X Prob(State i) = 1 Z e state i X i P (X i )= 1 Z energies j X E i state i g j exp( E j ) X i exp( E i ) Bare bones quantum: Know the modern physics course, know how to find probability of a state, know how to find expectation values, know the special systems, know spin-addition rules special systems: particle in box, harmonic oscillator, two spin 1/2 particles

8 Specific tips and tricks -- material Bare bones (advanced) classical: Know how to find the Hamiltonian, Hamilton s equations, Lagrangian, and the Euler Lagrange equations for a particle in a gravitational field, charged particle in a uniform electric field, and a pendulum. Don t just memorize the results, if you can do these three systems you will be aware of the pattern. Remember: GRE questions are typically short -- cannot get you to do any crazy calculations!

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10 Dimensions here This quantity is dimensionless Call x = hv kt C =3kN A x 2 e x (e x 1) apple 2 =3kN a x 2 1+x +... ((1 + x ) 2 =3kN a x 2 apple 1 x =3kN a +...

11 A distant galaxy is observed to have its H-beta line shifted to a wavelength of 480nm from its laboratory value of 434nm. Which is the best approximation to the velocity of the galaxy? (Note: 480/434 ~ 1.1) a) 0.01c b) 0.05c c) 0.1c d) 0.32c e) 0.5c

12 A distant galaxy is observed to have its H-beta line shifted to a wavelength of 480nm from its laboratory value of 434nm. Which is the best approximation to the velocity of the galaxy? (Note: 480/434 ~ 1.1) a) 0.01c b) 0.05c c) 0.1c d) 0.32c e) 0.5c obs = obs emit = v r c + v emit c v s 1+(v/c) 1 (v/c) p (1 + (v/c)) 2 obs 1 c emit

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14 Need fraction of area covered by sensor 100 cm A sensor = (4 cm) 2 = 16 cm 2 A sphere =4 (100 cm 2 )= cm 2 A sensor A sphere = 16 4 =

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16 max friction = centrifugal force [ so µn = µmg = m! 2 r r = µg! 2! = 2 T! = t (don t use!) = s s 1 )! 2 10 s 2 therefore r µ m 0.3 m

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18 Things to know (they always seem to come up) 1) Elastic collision formula

19 Things to know (they always seem to come up) 1) Elastic collision formula v 0 v 1f v 2f A B A B Before collision After collision v 1f = m 1 m 2 m 1 + m 2 v 0 v 2f = 2m 1 m 1 + m 2 v 0

20 Things to know (they always seem to come up) 1) Elastic collision formula v 0 v 1f v 2f A B A B Before collision After collision v 1f = m 1 m 2 m 1 + m 2 v 0 v 2f = 2m 1 m 1 + m 2 v 0 2) The limiting behavior of capacitors and inductors in DC acts like (e.g. high pass filter question) (while uncharged) (while fully charged)

21 3) Virial theorem (and the quick way to get it) m is reduced mass! (To get levels for e.g. positronium, same formula but use reduced mass for that system)

22 3) Virial theorem (and the quick way to get it) F (r) =Ar +n ) V = mv 2 r A 1+n r1+n = F (r) 1+n r = F (r) ) 1 2 mv2 = 1 2 F (r)r m is reduced mass! (To get levels for e.g. positronium, same formula but use reduced mass for that system)

23 3) Virial theorem (and the quick way to get it) F (r) =Ar +n ) V = mv 2 r A 1+n r1+n = F (r) 1+n r = F (r) ) 1 2 mv2 = 1 2 F (r)r hkei = 1+n 2 hv i m is reduced mass! (To get levels for e.g. positronium, same formula but use reduced mass for that system)

24 3) Virial theorem (and the quick way to get it) F (r) =Ar +n ) V = mv 2 r A 1+n r1+n = F (r) 1+n r = F (r) ) 1 2 mv2 = 1 2 F (r)r hkei = 1+n 2 hv i 4) The Bohr formula (or know how to get it quickly) E = Z2 (ke 2 ) 2 m 2~ 2 n 2 m is reduced mass! (To get levels for e.g. positronium, same formula but use reduced mass for that system)

25 3) Virial theorem (and the quick way to get it) F (r) =Ar +n ) V = mv 2 r A 1+n r1+n = F (r) 1+n r = F (r) ) 1 2 mv2 = 1 2 F (r)r hkei = 1+n 2 hv i 4) The Bohr formula (or know how to get it quickly) E = Z2 (ke 2 ) 2 m 2~ 2 n 2 m is reduced mass! (To get levels for e.g. positronium, same formula but use reduced mass for that system) 5) Combining masses, springs, capacitors, resistors k 1 k 2 k 1 Can you find kequiv? k 2 Frequency of oscillation? m m Know reduced mass!

26 Quick Bohr (semi-classical) derivation Electron traveling in a circle:

27 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 r = k(ze)(e) r 2

28 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: r = k(ze)(e) r 2

29 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: r = k(ze)(e) r 2 L = pr = mvr = n~

30 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: r = k(ze)(e) r 2 L = pr = mvr = n~ Put together to find r (Bohr radius!)

31 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: 1 r = k(ze)(e) r 2 L = pr = mvr = n~ Put together to find r (Bohr radius!) r = kze2 mv 2 r 2 = kzme2 (mvr) 2 = kzme2 n 2 ~ 2

32 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: 1 r = k(ze)(e) r 2 L = pr = mvr = n~ Put together to find r (Bohr radius!) Potential energy: r = kze2 mv 2 r 2 = kzme2 (mvr) 2 = kzme2 n 2 ~ 2

33 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: 1 r = k(ze)(e) r 2 L = pr = mvr = n~ Put together to find r (Bohr radius!) r = kze2 mv 2 r 2 = kzme2 (mvr) 2 = kzme2 n 2 ~ 2 Potential energy: PE = kze2 r = k2 Z 2 me 4 n 2 ~ 2

34 Quick Bohr (semi-classical) derivation Electron traveling in a circle: mv 2 Angular momentum is quantized: 1 r = k(ze)(e) r 2 L = pr = mvr = n~ Put together to find r (Bohr radius!) r = kze2 mv 2 r 2 = kzme2 (mvr) 2 = kzme2 n 2 ~ 2 Potential energy: PE = kze2 r = k2 Z 2 me 4 n 2 ~ 2 Virial thm: <E>= <PE>/2

35 Apply elastic collision equations!

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37 ? 1? 1 t 1/2 = t + t or = t 1/2 t + 1 t

38 ? 1? 1 t 1/2 = t + t or = t 1/2 t + 1 t 1 = 1 t 1/2 t + 1 t = = 1 6 ( ) = ) t 1/2 = = = 14.4 min

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40 f high K large!?? f high K small!?? Reduced mass

41 What is the emission energy from a photon going from n = 3 to n = 1 in positronium (one electron and one positron orbiting one another)?

42 What is the emission energy from a photon going from n = 3 to n = 1 in positronium (one electron and one positron orbiting one another)? Rule for Hydrogen like atoms: E n = 13.6 ev n Z 2 2 But 13.6 is proportional to the reduced mass m = m electron in Hydrogen In positronium m = m e /2, so we have to halve the 13.6 E n = E photon = E ev n 2, (positronium energy levels) 1 1 E 1 =6.8 ev =6.8 ev 8 9 6eV

43 m m Situation 1) Situation 2) Two different ways of connecting a mass m to two identical springs with spring constant k are shown above. If we denote the frequency of oscillation in situation 1 by f1 and the frequency of oscillation in situation 2 by f2 then f1/ f2 is: a) 4 b) 2 c) 1/2 d) 1/4 e) depends on m and / or k

44 m m Situation 1) Situation 2) Two different ways of connecting a mass m to two identical springs with spring constant k are shown above. If we denote the frequency of oscillation in situation 1 by f1 and the frequency of oscillation in situation 2 by f2 then f1/ f2 is: a) 4 b) 2 c) 1/2 d) 1/4 (Hint: can pretend k 1 and k 2 are not the same to take limits to determine formula for k e ) e) depends on m and / or k f 1 f 2 =! 1! 2 = s k e,1 /m k e,2 /m = s k e,1 k e,2 = s 2k k/2 =2

45 A particle sits in a periodic potential V (x) =d sin(kx) What is its oscillation frequency about the minimum?

46 A particle sits in a periodic potential V (x) =d sin(kx) What is its oscillation frequency about the minimum? Let y be the distance from the minimum. Expanding about the minimum we have: V (y) =V min +0y + 1 d 2 V 2 dy 2 min y (because min) Just a number, not a function Force is F = dv dy = d2 V dy 2 miny +... SHM with spring constant k = d 2 V/dy 2 evaluated at min! spring constant = dk 2 sin(kx) =+dk 2 evaluated at min f =2 r spring const. m =2 r dk 2 m

47 6) Making problems look like a harmonic oscillator! 2 = (d2 V/dx 2 ) min m 7) Remember spectroscopic notation (ugh) 2s+1 (orbital angular momentum symbol) j and the selection rules for an electric dipole

48 8) Know the pattern of spherical harmonics Y m ` (` = 0) (` = 1) (` = 2) Too detailed...! (But if you can remember these, congratulations)

49 8) Know the pattern of spherical harmonics Y m ` m magnetic quantum number ( `, ` +1,...,`) ` orbital quantum number (0, 1, 2,...) Y m ` contains ' dependence of the form e im Y m ` contains ` dependence of the form sin`, sin` 1 cos,... (i.e. can write as ` sines or cosines mulitpled, or as sin(` ), cos(` ).) Compare these rules to the spherical harmonics listed one slide ago.

50 Random mechanics problem: A ball and a block of mass m are moving at the same speed v. When they hit the ramp they both travel up it. The block slides up with (approximately) no friction, the ball experiences just enough friction to roll without slipping. Which goes higher? m v m v a) the ball goes higher b) the black goes higher c) they go the same height d) Impossible to tell from information given

51 Random mechanics problem: A ball and a block of mass m are moving at the same speed v. When they hit the ramp they both travel up it. The block slides up with (approximately) no friction, the ball experiences just enough friction to roll without slipping. Which goes higher? m v m v a) the ball goes higher b) the black goes higher c) they go the same height d) Impossible to tell from information given The ball has both translational kinetic energy (equal to that of the block) and rotational kinetic energy. Therefore KE ball,initial >KE block,initial The ball converts all this energy into potential energy, and therefore goes higher.