The Northern California Physics GRE...
Transcript of The Northern California Physics GRE...
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The Northern California Physics GRE Bootcamp
Held at UC Davis, August 11-12, 2018
Damien Martin
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* 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
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* Know your “first year” general physics really well - Newtonian mechanics in particular
* Know your “modern physics” general physics really well (usually Sophomore 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)
Big tips and tricks -- material
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Evaluate whether question is “special”
or “first year”
15
17
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Bare bones thermal:Know the definition of partition function, probability of a state, how to find an expectation value
Specific tips and tricks -- material
Z =X
state i
exp(��Ei) =X
energies j
gj exp(��Ej)
Prob(State i) =1
Ze��Ei
hXi =X
state i
XiP (Xi) =1
Z
X
state i
Xi exp(��Ei)
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
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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.
Specific tips and tricks -- material
Don’t just memorize the results, if you cando these three systems you will be awareof the pattern.
Remember: GRE questions are typically short -- cannot get you to do any crazy calculations!
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This quantity is dimensionlessDimensions here
Call x = hvkT
C = 3kNAx2 ex
(ex � 1)2
= 3kNax2
1 + x + . . .
((1 + x + . . .� 1)2
�
= 3kNax2
1x2
+ . . .
�
= 3kNa + . . .
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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) 0.01cb) 0.05cc) 0.1cd) 0.32ce) 0.5c
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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) 0.01cb) 0.05cc) 0.1cd) 0.32ce) 0.5c
�obs = �emit
rc + v
c� v
�obs
�emit=
s1 + (v/c)1� (v/c)
⇡p
(1 + (v/c))2
v ⇡✓
�obs
�emit� 1
◆c
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27
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Need fraction of area covered by sensor
Asensor = ⇡(4 cm)2 = 16⇡ cm2
100 cm
Asphere = 4⇡(100 cm2) = 4⇡ ⇥ 104 cm2
Asensor
Asphere=
16⇡
4⇡ ⇥ 104= 4⇥ 10�4
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max friction = “centrifugal” force [http://xkcd.com/123/]
µN = µmg = m!2r r =µg
!2
! =2⇡
T(don’t use!)
! =�✓
�t=
2⇡ ⇥ 33.3
60 s⇠ ⇡ s�1 ) !2 ⇠ 10 s�2
so
thereforer ⇠ µ m ⇠ 0.3 m
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Things to know (they always seem to come up)
1) Elastic collision formula
A B A B
Before collision After collision
v0 v1f v2f
v1f =m1 �m2
m1 + m2v0
v2f =2m1
m1 + m2v0
2) The limiting behavior of capacitors and inductors in DC
acts like (while uncharged)
(while fully charged)
(e.g. high pass filter question)
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3) Virial theorem (and the quick way to get it)
F (r) = Ar+n ) V =A
1 + nr1+n =
F (r)1 + n
r
mv2
r= F (r)) 1
2mv2 =
12F (r)r
hKEi =1 + n
2hV i
5) Combining masses, springs, capacitors, resistors
mm
k1k1k2
k2
Can you find kequiv ?Frequency of oscillation?
Know reduced mass!
4) The Bohr formula (or know how to get it quickly)
E = �Z2(ke2)2m2~2n2
m is reduced mass!(To get levels for e.g. positronium,same formula but use reduced massfor that system)
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Quick Bohr (semi-classical) derivation
Electron traveling in a circle:
mv2
r=
k(Ze)(e)
r2
L = pr = mvr = n~Angular momentum is quantized:
1
r=
kZe2
mv2r2=
kZme2
(mvr)2=
kZme2
n2~2
Put together to find r (Bohr radius!)
PE = �kZe2
r= �k2Z2me4
n2~2
Potential energy:
Virial thm: < E >= � < PE > /2
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Apply elastic collision equations!
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t1/2?= t� + t� or
1
t1/2
?=
1
t�+
1
t�
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t1/2?= t� + t� or
1
t1/2
?=
1
t�+
1
t�
1
t1/2=
1
t�+
1
t�=
1
24+
1
36
=1
6(1
4+
1
6)
=1
6
10
24
) t1/2 =24⇥ 6
10=
144
10= 14.4 min
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fhighK large�! ??
fhighK small�! ??
Reduced mass
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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)?
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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: En =�13.6 eV
n2 Z2
But 13.6 is proportional to the reduced mass m = melectron in Hydrogen
In positronium m = me/2, so we have to halve the 13.6
En = �6.8 eVn2
, (positronium energy levels)
Ephoton = E3 � E1 = 6.8 eV✓
112� 1
32
◆= 6.8 eV⇥ 8
9⇡ 6 eV
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mm
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
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mm
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
f1
f2=
!1
!2=
ske↵,1/m
ke↵,2/m=
ske↵,1
ke↵,2=
s2k
k/2= 2
(Hint: can pretend k1 and k2 are not
the same to take limits to determine formula for ke↵)
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A particle sits in a periodic potential
V (x) = d sin(kx)
What is its oscillation frequency about the minimum?
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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 minimumwe have:
V (y) = Vmin + 0y +12
d2V
dy2|min y2 + . . .
Force isF = �dV
dy= �d2V
dy2|miny + . . .
0 (because min) Just a number, not a function
SHM with “spring constant” k = d2V/dy2evaluated at min!
spring constant = �dk2 sin(kx) = +dk2 evaluated at min
f = 2⇡
rspring const.
m= 2⇡
rdk2
m
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6) Making problems look like a harmonic oscillator
!2 =(d2V/dx2)|min
m
7) Remember spectroscopic notation (ugh)2s+1(orbital angular momentum symbol)j
and the selection rules for an electric dipole
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8) Know the pattern of spherical harmonics Y m`
Too detailed ...... !
(` = 0)
(` = 1) (` = 2)
(But if you can remember these, congratulations)
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8) Know the pattern of spherical harmonics
Y m`
Y m` contains ' dependence of the form eim�
Y m` contains ` dependence of the form sin` ✓, sin`�1 ✓ cos ✓, . . .
(i.e. can write as ` sines or cosines mulitpled, or as sin(`✓), cos(`✓).)
m – magnetic quantum number (�`,�` + 1, . . . , `)` – orbital quantum number (0, 1, 2, . . .)
Compare these rules to the spherical harmonics listed one slide ago.