Are You Ready for Your Qualifier Exams?

[eNtRopY]

Here's the idea. We can help each other study for our qualifier exams by posting qualifer-type questions along with the solutions in this thread. I think this thread will also help those studying for the physics GRE... or even those just wanting to get good at solving exam-type physics problems fast.

eNtRopY

 

[quantumdude]

Sure, I've got 5 books full of qualifer problems, plus many, many old exams from my school (Rensselaer Polytechnic Institute).

 

[chroot]

I'm taking the Physics GRE on April 12... I've been studying my ass off for months now.

- Warren

 

[nbo10]

This is a good idea, I'll post some in the next few days when I have some free time. What kinda level were you talking about pre-graduate?

JMD

 

[hhegab]

I am waiting!

Hi all,
Well, I have waited for a month now and I had no post about this GRE Test.
Well, as I am sure that I will find help from ppl here in this forum I have the following questions:

1-How may I prepare for the GRE physics? and
2-What books should I read? and
3-Are there any useful materials on the net that I can download and use?

hhegab

 

[nbo10]

The best way to prepare is to solve problems, lots of them. My GRE had lots of QM on it but one my friends had mostly classicaly physics, so they vary the content on each. I don't recommend buying the Green Physics GRE book, it was not helpful at all. Just solve problems, and understand the concepts behind the solution.

JMD

 

[quantumdude]

Jeez, I forgot all about this thread. Sorry!

As for the GRE, I say get Princeton Review stuff. I used to work for them (taught SAT, GMAT, and GRE courses--not subject courses though).

I will post 5 qualifier questions tomorrow, I promise.

 

[Alexander]

Pardon my ignorance, but what the hell is GRE everybody talkin' about?

 

[hhegab]

GRE

GRE stands for Graduate Record Examinations, this is a standardized test requirred to join any american unversity to complete your graduate eduaction.
www.gre.org is their web site.

hhegab

 

[quantumdude]

OK kiddies, here’s Round 1. At my school, the exam is divided into 5 sections:

1. Classical Mechanics
2. Classical Electrodynamics
3. Special Relativity
4. Quantum Mechanics
5. Thermodynamics/Statistical Mechanics

We’ll start off easy to warm up.

1. (Classical Mechanics) Using Lagrangian mechanics, derive and solve the equation of motion for a simple pendulum of mass m and length L for small oscillations.

2. (Classical Electrodynamics) Consider a spherical capacitor of outer radius a and inner radius b.
a.) Derive the capacitance.
b.) What is the capacitance of a single conducting sphere?

3. (Special Relativity) Consider a reference frame in which an electric field E is observed, and B=0. Show that there is no reference frame S’ in which E’=0 and B’ is nonzero.

4. (Quantum Mechanics)
a.) Show that a spin-1/2 state is not invariant under 2π rotations and that a 4π rotation is required to return to the initial state.
b.) A necessary condition for a state to be physical is that it differ by no more than a phase factor e^-iθ upon 2π rotations. Consider the super position of fermion number states ψ given by

ψ=(1/2^1/2)(|0>+|1>)

Show that ψ is not physical.

5. (Thermodynamics/Statistical Mechanics)
In an ideal gas of molecules of mass M at temperature T, the probability for a molecule to have a velocity v is given by the Maxwell-Boltzmann distribution

P(v)=Cexp(-Mv²/k_BT)

a.) What is the root mean square speed?
b.) What is the most probable speed?
c.) What is the average speed?

You may find the following information helpful.

(sorry, I don't know how to make matrices in this forum)
F^μμ=0
F⁰ⁱ=-Fⁱ⁰=-E_i/c
F¹²=-F²¹=-B_z
F¹³=-F³¹=B_y
F²³=-F³²=-B_x

The metric tensor is g₀₀=1, g_ii=-1 (i=1,2,3)

 

[chroot]

Tom,

The Physics GRE is in fact made up of 100 short questions with five multiple choice answers. Each problem should be solvable in a couple of minutes by a knowledgeable candidate. In fact, nearly a third or so of the questions can be answered immediately simply by inspection.

In short, the kind of questions you posted, while excellent (I plan on doing them when I get home), will never appear on a GRE exam.

- Warren

 

[quantumdude]

Originally posted by chroot
In short, the kind of questions you posted, while excellent (I plan on doing them when I get home), will never appear on a GRE exam.

Oh, I know. These are the kinds of questions that appeared on my qualifying exam. These would be among the "easy" ones (not worth many points, but embarassing if you don't get them!).

I have 5 books full of qualifier questions, as well as 5 or 6 complete exams from my school. I'll try to post an entire exam this weekend.

 

[chroot]

Originally posted by Tom
These would be among the "easy" ones (not worth many points, but embarassing if you don't get them!)

Terrifying. :(

- Warren

 

[Greg Bernhardt]

Princeton Review

I would second this recommendation. I got the princeton review for mathematics and it's really helpful.

 

[chroot]

Originally posted by Greg Bernhardt
I would second this recommendation. I got the princeton review for mathematics and it's really helpful.

I don't think there is one for the GRE physics exam.

- Warren

 

[Adam]

Good grief Tom. Those questions serve to show me how much I have yet to learn about all this stuff.

 

[Adam]

I think I'll just buy basic textbooks in these fields to read through some day. There is the possibility that I may do some physics and chemistry units (maybe some robotics as well) after I complete my current degree, so it can't hurt to get a head-start.

 

[quantumdude]

Here's the first one

Originally posted by Tom
1. (Classical Mechanics) Using Lagrangian mechanics, derive and solve the equation of motion for a simple pendulum of mass m and length L for small oscillations.

Solution:
There is one degree of freedom, which I take to be the angular displacement θ. Let θ=0,V=0 at the bottom of the swing.

L=T-V
T=(1/2)m(Lθ')²
V=mgL(1-cosθ)

So the Lagrangian is:

L=(1/2)m(Lθ)²-mgL(1-cosθ)

Plug into Lagrange's equation:

(d/dt)(∂L/∂θ')-∂L/∂θ=0
(d/dt)(mL²θ')+mgLsinθ=0
mL²θ''+mgLsinθ=0

Now for the bit about small oscillations:

sin&theta;~&theta; for &theta;<<1

mL²&theta;''+mgL&theta;=0

&theta;''+(g/L)&theta;=0

let &omega;²=g/L

&theta;''+&omega;²&theta;=0
&omega;(t)=Acos(&omega;t)+Bsin(&omega;t)

Now I did not give you any initial conditions (my oversight), so let's say that the pendulum is released from rest at &theta;(0)=&theta;₀.

&theta;'(0)=0 ==>B=0
&theta;(0)=&theta;₀ ==> A=&theta;₀

I know that you already know the result, and could probably do it with Newtonian mechanics (or possibly by simply writing the solution down!), but specifically asking you to solve it with Lagrangian dynamics is something that would be fair to ask. It was asked of me on my (Goldstein-based) Advanced Mechanics midterm.

 

[poorcollegegirl]

Homework

Hello Tom,

It is me-Ramona. Are you still in town?
If so, can I meet you tonight?

 

[quantumdude]

Check your Private Message box.

 

[quantumdude]

Solution #2

Originally posted by Tom
2. (Classical Electrodynamics) Consider a spherical capacitor of outer radius a and inner radius b.
a.) Derive the capacitance.

This one is worked out in Halliday, Resnick and Walker, but it is also a fair qualifier question. It appeared on a qualifier one year at my school, and it appeared on my (Jackson-based) Classical Electrodynamics midterm. It's one of those "catch you with your pants down" type questions, because you're expecting to find long, dirty problems involving roots of Bessel functions and whatnot.

Start from Gauss' law:

&int;E^.dS=q/&epsilon;₀

The Gaussian surface here is a sphere concentric with the two plates and with radius a<r<b. Since the entire charge in enclosed, the field is that of a point charge, so:

E(r)=(4&pi;&epsilon;₀)^-1q/r²

Now for the potential difference.

&Delta;V=-&int;_CE^.ds

We use as a path C a straight line from one shell (r=a) to the other shell (r=b). Take the inner plate as positive and the outer plate as negative (this choice does not affect the result for capacitance, which depends only on geometry). Since E is everywhere antiparallel to ds, the integral reduces to:

&Delta;V=+(4&pi;&epsilon;₀)^-1)&int;_a^bdr/r²
&Delta;V=q/(4&pi;&epsilon;₀)(1/a-1/b)

Solving for capacitance C=q/&Delta;V we get:

C=(4&pi;&epsilon;₀)/(1/a-1/b)

b.) What is the capacitance of a single conducting sphere?

This is the same as if the outer sphere were not there. We can translate this into mathematics by taking the limit of C as b-->[oo] to get:

C=4&pi;&epsilon;₀a

 

[quantumdude]

Three's a charm!

Originally posted by Tom
3. (Special Relativity) Consider a reference frame in which an electric field E is observed, and B=0. Show that there is no reference frame S’ in which E’=0 and B’ is nonzero.

Note that F^&mu;&nu;F_&mu;&nu; is a Lorentz scalar. You can find F_&mu;&nu; from F^&mu;&nu; by:

F_&mu;&nu;=g_&mu;&sigma;g_&nu;&tau;F^&sigma;&tau;

You will find that the electric field components change sign while the magnetic field components do not.

The inner product of the two tensors is then:

F^&mu;&nu;F_&mu;&nu;=2(|B|²-|E|²/c²)

Note that this is a Lorentz scalar, which is an invariant. That means that F^&mu;&nu;F_&mu;&nu;=F'^&mu;&nu;F'_&mu;&nu;.

In frame S, we have nonzero E and B=0. That means that F^&mu;&nu;F_&mu;&nu; is less than zero[/color]. So, we cannot have nonzero B' and E'=0 because that would mean that F'^&mu;&nu;F'_&mu;&nu; is greater than zero.

This is a problem from Jacskon Chapter 11.

 

[quantumdude]

Numero quatro

Originally posted by Tom
4. (Quantum Mechanics)
a.) Show that a spin-1/2 state is not invariant under 2&pi; rotations and that a 4&pi; rotation is required to return to the initial state.

An arbitrary spin-1/2 state is (expanded in S_z eigenstates):

|&alpha;>=c₊|+>+c_-|->

The rotation operator is:

D(n;&phi;)=exp(-iS^.n&phi;/hbar)

For a rotation of 2&pi; radians about the z-axis, D becomes:

D(k;2&pi;)=exp(-2&pi;iS_z/hbar)

Apply this to |&alpha;>:

D(k;2&pi;)|&alpha;>=c₊exp(-2&pi;iS_z/hbar)|+>+c_-exp(-2&pi;iS_z/hbar)|->
D(k;2&pi;)|&alpha;>=c₊exp(-i&pi;)+c_-exp(+i&pi;)
D(k;2&pi;)|&alpha;>=-c₊|+>-c_-|->
D(k;2&pi;)|&alpha;>=-|&alpha;>

So, we do not come back to the initial state after a 2&pi; rotation. It is easy to see how a 4&pi; rotation does return us to the initial state. (That wouldn't be accepted on a qualifier, but I have already passed mine! :p)

b.) A necessary condition for a state to be physical is that it differ by no more than a phase factor e^-i&theta; upon 2&pi; rotations. Consider the super position of fermion number states[/color] &psi; given by

&psi;=(1/2^1/2)(|0>+|1>)

Show that &psi; is not physical.

First, as to the part in red[/color], I meant to say spin-1/2 number states[/color]--sorry.

As for the vacuum state |0>, it is unaffected by any rotation at all, so we know it will be invariant under the action of D(n;&phi;) for any n or &phi;. As for the single-particle state, we just worked out in part a.) that it will suffer a change in sign.

Choosing n=k for definiteness, we have:

D(k;2&pi;)|&psi;>=(1/2)^1/2(D(k;2&pi;)|0>+Dk;2&pi;)|1>)
D(k;2&pi;)|&psi;>=(1/2)^1/2(|0>-|1>)

Since the resulting state is not related to the initial state |&psi;> via a simple phase change, |&psi;> is unphysical.

 

[quantumdude]

And 5 makes a handful!

Originally posted by Tom
5. (Thermodynamics/Statistical Mechanics)
In an ideal gas of molecules of mass M at temperature T, the probability for a molecule to have a velocity v is given by the Maxwell-Boltzmann distribution

P(v)=Cv²[/color]exp(-Mv²/k_BT)

Sorry to anyone who tried this problem. I screwed up in writing the Boltzmann distribution: the v²[/color] was missing.

The problem itself is just 3 exercises in basic definitions.

I will be making use of the following mathematical information. At my school, each examinee is given a complimentary copy of Schaum's Mathematical Handbook to keep. It helps dull the pain if you fail. Not.

For I_n(a)=&int;uⁿexp(-au²)du, where the integration is from 0 to [oo], we have:

I₃(a)=1/2a²
I₄(a)=(3/8a²)(&pi;/a)^1/2

a.) What is the root mean square speed?

This is exactly what the name implies: the square root of the mean of the square of the speed.

v_rms=(&int;v²F(v}dv)^1/2

where the integral is taken from 0 to [oo].

v_rms=(C&int;v⁴exp(-Mv²/k_BT))^1/2

Recognizing that the integral above is I₄(a) with a=M/k_BT, we have:

v_rms=((3C(k_BT)²/8M²)(k_BT&pi;/M)^1/2)^1/2

I am not going to simplify that any further. :p

b.) What is the most probable speed?

This is the speed at which the probability distribution is maximized.

dP(v)/dv=(2Cv)exp(-Mv²/k_BT)+(Cv²)(-2Mv/k_BT)exp(-Mv²/k_BT)
dP(v)/dv=(2C)(v-2Mv³/k_BT)exp(-Mv²/k_BT)

Setting dP(v)/dv=0 yields:

v_mp=(k_BT/M)^1/2

c.) What is the average speed?

The average speed is:

v_avg=&int;vP(v)dv
v_avg=C&int;v³exp(-Mv²/k_BT)dv

where the integral is taken from 0 to [oo]. Recognizing that the integral above is I₃(a) with a=M/k_BT, we have:

v_avg=C(k_BT)²/2M²

OK, anyone else want to put any up? If not, I'll put some more up in a few days.

edit: fixed subscript bracket

 

[quantumdude]

I just thought of something. The above solution to #5 is correct, but anyone who can see anything fishy with it gets bonus points.

 

[quantumdude]

Originally posted by Tom
I just thought of something. The above solution to #5 is correct, but anyone who can see anything fishy with it gets bonus points.

Hint: Look at the limits of integration.

 

[quantumdude]

Answer

Originally posted by Tom
I just thought of something. The above solution to #5 is correct, but anyone who can see anything fishy with it gets bonus points.

OK, since no one answered here, I'll spill the beans. The limits of integration go from 0 to &infin;, but look at what we are integrating: speed[/color].

The upper limit of speed in the integral is v=_________.
The upper limit of speed in special relativity is v=_________.

 

[quantumdude]

Spring 1996 Part I (RPI)

Feel free to do all of the problems, but I am going to post the directions here anyway so you can get an idea of what is needed to pass. Your school could (and probably does) have different requirements, so check up on it.

Part I: Classical Mechanics, Classical Electrodynamics and Special Relativity

INSTRUCTIONS FOR SUBMITTING ANSWER SHEETS FOR QUALIFYING EXAMINATION
(I'm skipping the stuff about "use only one side of the answer sheet, etc.)

Answer a total of 8 problems with at least 3 problems in mechanics, 3 in E&M and 1 in special relativity. Do not hand in more than 8 problems.

To pass the exam, 6 problems must be completed satisfactorily (with a score of 6/10 or better) with at least 2 in mechanics, 2 in E&M and 1 in special relativity. The point value of each part of each problem is stated below the problem.

A. Classical Mechanics
Problem 1:
a.)Explain under what circumstances there can be conservation of linear or angular momentum in a dynamical system.
b.)What is the role of the constraint in the dynamics of a smiple pendulum? Construct the Lagrangian for the pendulum and illustrate the constraint therein.

[(2+2)+(3+3)]

Problem 2:
A particle of mass m moves in the (x,y) plane. The motion is in a potential U(x,y) such that

U(x,y)=(k₁/2)x²+(k₂/2)y²-&epsilon;(k₁k₂)^1/2xy

where k₁, k₂, and &epsilon; are constants.

a.) What are the normal frequencies in this case?
b.) Write down the equation for the normal modes.
c.) Discuss the limit &epsilon;-->0.

[4+4+2]

Problem 3:
A particle of mass m moves in a constant gravitational field g=gk. The motion of the particle is confined to a spiral. The spiral is described in cylindrical coordinates by

&rho;=&alpha;z,
&phi;=&beta;z.

Here &alpha; and &beta; are constants.

a.) Obtain an expression for the Lagrangian function describing this motion.
b.) Get an expression for the corresponding Hamiltonian.
c.) Write down Hamilton's equations of motion.

[3+4+3]

Problem 4:
A particle of mass m and charge q moves in an electric field E=Ei.

a.)Write down the time-dependent Hamilton-Jacobi equation describing the nonrelativistic motion of this particle.
b.) Write down the time-independent Hamilton-Jacobi equation.
c.) Obtain an explicit expression for Hamilton's principal function.
d.) Obtain the time dependence of x and p.

[3+1+3+3]

B. Classical Electrodynamics
Problem 5:
a.) In what particular applications of the integral form of Gauss' law in electrostatics is invoking symmetry of the problem useful?
b.) Why is the dielectric constant of a crystalline medium, in general, a tensor?
c.) If you want an approximate formula for the vector potential of a localized current distribution, show that a multipole expansion is helpful. Also, compute the magnetic dipole moment of the current loop.

[2+2+(2+4)]

Problem 6:
A spherical surface of radius R is held at an electrostatic potential &Phi;(R,&theta;)=V₀cos&theta;, where V₀ is a constant and &theta; is the polar angle.

a.) Find the potential inside and outside the sphere.
b.) Determine the surface charge density on the sphere.

[4+6]

Problem 7:
A magnetic dipole, m, is placed a distance d from a superconducting plane surface (the magnetic permeability is &mu;=0). The dipole is oriented perpendicularly to the surface.

a.)Find the position and orientation of the image dipole.
b.) Calculate the force exerted by the surface on the dipole.

[3+7]

Problem 8:
A plane wave is incident on a metal plate at an angle of &pi;/4.

a.) Write the boundary conditions for the electric and magnetic fields at the boundary.
b.) Determine the time-averaged electric energy density in the space for both polarizations.

[4+6]

C. Special Relativity
Problem 9:
A particle of mass m has a total energy that is twice as large as its rest energy. It collides with an identical particle at rest. The particles stick together after the collision and form a new particle. What are the velocity and rest mass of the new composite particle?

[8+2]

Problem 10:
a.) What is the acceleration of a charged particle in a uniform electric field E? Assume the particle moves along a straight line parallel to the electric field. A relativistic treatment is required. Determine the speed and position of the particle.
b.) The maximum speed attained at the Stanford Linear Accelerator by the accelerated electrons is 0.99999999967c (there are nine 9's there). Find the kinetic energy for an electron moving with this speed. Assume the electron mass to be 0.5 MeV/c².

[6+4]

 

[quantumdude]

Spring 1996 Part II (RPI)

Part II: Quantum Mechanics, Thermodynamics and Statistical Mechanics

INSTRUCTIONS FOR SUBMITTING ANSWER SHEETS FOR QUALIFYING EXAMINATION

Answer a total of 8 problems with at least 5 problems in quantum mechanics and 3 problems in thermodynamics and statistical mechaincs. Do not hand in more than 8 problems.

To pass the exam, 6 problems must be completed satisfactorily (with a score of 6/10 or better) with at least 4 in quantum mechanics and 2 in thermodynamics and statistical mechanics. The point value of each part of each problem is stated below the problem.

A. Quantum Mechanics
Problem 1:
Consider a physical system with a three-dimensional state space, spanned by the orthonormal basis formed by the three kets |u₁>, |u₂>, and |u₃>. The Hamiltonian operator has the form

H=(hbar)&omega;M

and M is a (3x3)matrix with elements (sorry I can't make it pretty in this forum):
M₁₃=M₃₁=1
All other elements =0

A particular state vector at t=0 is given by

|&psi;(t=0)>=(1/2)^1/2|u₁>+(i/2)|u₂>+(1/2)|u₃>.

a.) If a single measurement of the energy is made at t=0, what possible values of the energy can be found, and with what probabilities? What is the expectation value of H?
b.) Calculate the state vector at time t later. If the energy is measured at time t, what energy values can be obtained and with what probabilities?

[(3+2)+(2+2+1)]

Problem 2:
Consider a simple one-dimensional harmonic oscillator with the Hamiltonian H and its eigenvectors |&phi;_n> and eigenvalues (n+1/2)(hbar)&omega;, where n=0,1,2,3...

a.) Determine the eigenvectors of the annihilation operator a in terms of the eigenvectors of the Hamiltonian. (Denote the eigenvalue of a to be &alpha; (a complex number) and define a=(m&omega;/2hbar)^1/2x+ip/(2m(hbar)&omega;)^1/2, with symbols having the usual meaning).
b.) When the system is in an eigenstate of a with an eigenvalue of &alpha;, what is the probability of finding the system in its ground state?
c.) What is the expectation value of H and of the position operator x when the system is in an eigenstate of a?

[4+2+(2+2)]

Problem 3:
Describe the difference between

a.) adiabatic and sudden approximations.
b.) scalar and pseudoscalar observables in quantum mechanics.
c.) the laboratory system and the center-of-momentum system in two-body scattering.

[3+4+3]

4. The operators J₊ and J_- are defined as

J_&plusmn;=J_x&plusmn;iJ_y.

a.) Show that

J_-|j,m>=N|j,m-1>,

where N is a normalization constant.

b.) Show that

J²=J_z²+J₊J_--J_z.

c.) Obtain an explicit expression for the normalization constant N in terms of j and m.

[4+2+4]

Problem 5:
The matrix for the Hamiltonian H is given by:

H₁₁=E₁
H₂₂=E₂
H₁₂=H₂₁=&Delta;E.

Assume that &Delta;E<<|E₁-E₂|.

a.) What is the leading order correction to the energy eigenvalues (of the Hamiltonian for which &Delta;E=0)?
b.) Obtain the eigenfunctions in the lowest order of perturbation theory.
c.) Obtain an expression for the exact eigenvalues and show that, in the limit of &Delta;E-->0, they are consistent with perturbation theory results.

[4+4+2]

Problem 6:
A beam of particles with uniform velocity v enters an interaction region, where some of them are absorbed. This is represented by a complex potential V₁+iV₂ (V₁, V₂ are real).

a.) Use the time-dependent Schrödinger equation to calculate the loss of flux due to absorption.
b.) Show that the cross section per atom for absorption is

&sigma;=2V₂/((hbar)Nv)

where N is the number of absorbing atoms per unit volume.

[8+2]

B. Thermodynamics and Statistical Mechanics
Problem 7:
Consider the van der Waal's equation of state

P=k_BT/(v-b)-a/v²

where k_B, a and b are constants, T is the temperature, P is the pressure and v is the molar volume.

a.) Give a physical interpretation of the constants a and b.
b.) Determine the volume, temperature and pressure of the critical point in terms of the given constants.

[3+7]

Problem 8:
Assume that particles of mass m at temperature T are in the one-dimensional potential V(x}=&epsilon;₀|x/a|ⁿ, where &epsilon;₀ and a are parameters with the units of energy and length respectively, and n is a constant.

a.) Calculate the partition function Z and the thermal energy E.
b.) Discuss your calculated results with respect to the specific heat c_v of the above system.

[(3+3)+4]

Problem 9:
Consider a system of N distinguishable, non-interacting spins in a magnetic field H. Each spin has a magnetic moment of size &mu;, and each can point either parallel or antiparallel to the field. Thus, the energy of a particular state is

&Sigma;_i=1^N-n_i&mu;H, with n_i=&plusmn;1

where n_i&mu; is the magnetic moment in the direction of the field. For the above described system:

a.) determine the average total magnetization

<M>=<&Sigma;_i=1^N&mu;n_i>

as a function of &beta;=1/k_BT, H and N (the total number of spins);

b.) similarly, determine <(&delta;M)²>, where

&delta;M=M-<M>,

and compare your result with the susceptibility given by

&chi;=(&part;<M>/&part;H)_&beta;,N;

c.) examine the behavior of <M> and <(&delta;M)²> in the limit T-->0, and interpret your results.

[4+4+2]

Problem 10:
A nonideal gas has an equation of state P=P(n,T), where n is the particle density. The gas is at equilibrium in a uniform gravitational field g=-gk.

a.) Using hydrostatic arguments, obtain a differential equation for the density at height z.
b.) Show that for an ideal gas, the equation predicts the usual exponential density function n(Z)=N₀e^(-mgz/k_BT).

[7+3]

edit: fixed subscript bracket

 

[chroot]

Tom,

Are these qualifiers to get a PhD? Or to get a masters? Or to get into graduate school?

- Warren