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eNtRopY

eNtRopY

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eNtRopY

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Tom Mattson

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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.

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JMD

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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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JMD

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Tom Mattson

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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.

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Alexander

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

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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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Tom Mattson

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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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Tom Mattson

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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!).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.

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.

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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!)

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

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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.

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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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Tom Mattson

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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.

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poorcollegegirl

Hello Tom,

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

If so, can I meet you tonight?

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Tom Mattson

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Check your Private Message box.

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Tom Mattson

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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.Originally posted by Tom

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

a.) Derive the capacitance.

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

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

C=4πε

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Tom Mattson

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Note that FOriginally 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.

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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Tom Mattson

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

|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)

First, as to the part in red, I meant to sayb.) 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.

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.

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Tom Mattson

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

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

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

This is the speed at which the probability distribution is maximized.b.) What is the most probable speed?

dP(v)/dv=(2Cv)exp(-Mv

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

Setting dP(v)/dv=0 yields:

v

The average speed is:c.) What is the average speed?

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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Tom Mattson

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