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eNtRopY

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eNtRopY

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quantumdude

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chroot

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

- Warren

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nbo10

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JMD

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hhegab

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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

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nbo10

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JMD

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quantumdude

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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.

- #8

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

- #9

hhegab

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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

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quantumdude

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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)

2. (Classical Electrodynamics) Consider a spherical capacitor of outer radius

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

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

ψ=(1/2

Show that ψ is not

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

P(v)=Cexp(-Mv

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

F

F

F

F

The metric tensor is g

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chroot

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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

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quantumdude

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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.

- #13

chroot

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Terrifying. :(Originally posted by Tom

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

- Warren

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Princeton Review

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

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chroot

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I don't think there is one for the GRE physics exam.Originally posted by Greg Bernhardt

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

- Warren

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Adam

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

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Adam

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quantumdude

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Originally posted by Tom

1. (Classical Mechanics)Using Lagrangian mechanics, derive and solve the equation of motion for a simple pendulum of massmand lengthLfor small oscillations.

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')

V=mgL(1-cosθ)

So the Lagrangian is:

L=(1/2)m(Lθ)

Plug into Lagrange's equation:

(d/dt)(∂L/∂θ')-∂L/∂θ=0

(d/dt)(mL

mL

Now for the bit about small oscillations:

sinθ~θ for θ<<1

mL

θ''+(g/L)θ=0

let ω

θ''+ω

ω(t)=Acos(ωt)+Bsin(ωt)

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

θ'(0)=0 ==>B=0

θ(0)=θ

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.

- #19

Hello Tom,

It is me-Ramona. Are you still in town?

If so, can I meet you tonight?

- #20

quantumdude

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Check your Private Message box.

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quantumdude

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Originally posted by Tom

2. (Classical Electrodynamics) Consider a spherical capacitor of outer radiusaand inner radiusb.

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:

∫

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)=(4πε

Now for the potential difference.

ΔV=-∫

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

ΔV=+(4πε

ΔV=q/(4πε

Solving for capacitance C=q/ΔV we get:

C=(4πε

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=4πε

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quantumdude

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Originally posted by Tom

3. (Special Relativity) Consider a reference frame in which an electric fieldEis observed, andB=0. Show that there is no reference frame S’ in whichE’=0andB’ is nonzero.

Note that F

F

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

Note that this is a Lorentz scalar, which is an invariant. That means that F

In frame S, we have nonzero

This is a problem from Jacskon Chapter 11.

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quantumdude

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Originally posted by Tom

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.

An arbitrary spin-1/2 state is (expanded in S

|α>=c

The rotation operator is:

D(

For a rotation of 2π radians about the z-axis, D becomes:

D(

Apply this to |α>:

D(

D(

D(

D(

So, we do not come back to the initial state after a 2π rotation. It is easy to see how a 4π 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 bephysicalis that it differ by no more than a phase factor e^{-iθ}upon 2π rotations. Consider the super position offermion number statesψ given by

ψ=(1/2^{1/2})(|0>+|1>)

Show that ψ is notphysical.

First, as to the part in red, I meant to say

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(

Choosing

D(

D(

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

- #24

quantumdude

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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 velocityvis given by the Maxwell-Boltzmann distribution

P(v)=Cv^{2}exp(-Mv^{2}/k_{B}T)

Sorry to anyone who tried this problem. I screwed up in writing the Boltzmann distribution: the v

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

I

I

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

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

v

Recognizing that the integral above is I

v

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(-Mv

dP(v)/dv=(2C)(v-2Mv

Setting dP(v)/dv=0 yields:

v

c.) What is the average speed?

The average speed is:

v

v

where the integral is taken from 0 to [oo]. Recognizing that the integral above is I

v

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

edit: fixed subscript bracket

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quantumdude

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quantumdude

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Originally posted by Tom

Hint: Look at the limits of integration.

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quantumdude

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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 ∞, but look at what we are integrating: speed.

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

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

- #28

quantumdude

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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.

(I'm skipping the stuff about "use only one side of the answer sheet, etc.)

Answer a total of 8 problems with

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.

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)=(k

where k

a.) What are the normal frequencies in this case?

b.) Write down the equation for the normal modes.

c.) Discuss the limit ε-->0.

[4+4+2]

Problem 3:

A particle of mass m moves in a constant gravitational field

ρ=αz,

φ=βz.

Here α and β 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

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]

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 Φ(R,θ)=V

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 μ=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 π/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]

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

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/c

[6+4]

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- #29

quantumdude

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Answer a total of 8 problems with

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.

Problem 1:

Consider a physical system with a three-dimensional state space, spanned by the orthonormal basis formed by the three kets |u

H=(hbar)ωM

and M is a (3x3)matrix with elements (sorry I can't make it pretty in this forum):

M

All other elements =0

A particular state vector at t=0 is given by

|ψ(t=0)>=(1/2)

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 |φ

a.) Determine the eigenvectors of the annihilation operator a in terms of the eigenvectors of the Hamiltonian. (Denote the eigenvalue of a to be α (a complex number) and define a=(mω/2hbar)

b.) When the system is in an eigenstate of a with an eigenvalue of α, 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

J

a.) Show that

J

where N is a normalization constant.

b.) Show that

J

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:

H

H

H

Assume that ΔE<<|E

a.) What is the leading order correction to the energy eigenvalues (of the Hamiltonian for which ΔE=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 ΔE-->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 V

a.) Use the time-dependent Schrodinger equation to calculate the loss of flux due to absorption.

b.) Show that the cross section per atom for absorption is

σ=2V

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

[8+2]

Problem 7:

Consider the van der Waal's equation of state

P=k

where k

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}=ε

a.) Calculate the partition function Z and the thermal energy E.

b.) Discuss your calculated results with respect to the specific heat c

[(3+3)+4]

Problem 9:

Consider a system of N

Σ

where n

a.) determine the average total magnetization

<M>=<Σ

as a function of β=1/k

b.) similarly, determine <(δM)

δM=M-<M>,

and compare your result with the susceptibility given by

χ=(∂<M>/∂H)

c.) examine the behavior of <M> and <(δM)

[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

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)=N

[7+3]

edit: fixed subscript bracket

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- #30

chroot

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

- Warren

- #31

quantumdude

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To get into grad school, you take the GRE General and Subject test.

There is no exam for the MS degree.

The PhD program proceeds in the following steps:

1. First year courses.

2. Advanced/breadth courses.

3. Choose research advisor and ease into group.

4. Qualifying exam by end of second year. (that's the subject of this thread)

5. Choose thesis and start writing proposal.

6. Candidacy exam (oral exam w/ panel of 5 profs, present proposal, get reamed with questions).

7. Complete thesis.

8. Defend thesis.

9. Get blind drunk.

I am still working on #7, and eagerly anticipating #8, and already practicing for #9.

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