The physics department at Cornell University conducts an oral qualifying exam for its PhD candidates. The exam tests basic undergrad knowledge of physics and is meant to rectify any gaps, if present, in the core physics subjects.
The exam is held in two parts.
Part 1 - Quantum Mechanics and Statistical Mechanics - 30 Minutes
Part 2 - Electromagnetism and Classical Mechanics - 30 Minutes
Here is the rough transcript of my oral PhD qualifying exam at Cornell University:
Part 1: Quantum and Statistical Mechanics
Two profs took the exam while I stood in front of a blackboard
There was already a potential energy figure on the board.
P1: (Points me to an already drawn potential energy figure on the blackboard) We will be asking you questions related to this potential well now.
P1: If some classically distributed particles are in this potential well, what would be the expectation value of the position of the particles?
Me: So they are under the Maxwell-Boltzmann distribution?
P2: Yes, correct.
Me: Umm, I guess if I know the functional form of the potential, then I can mathematically calculate the expectation value of position under the Maxwell-Boltzmann statistics. (I proceed to write the full expression for calculation of the expectation value of position)
P1: No no, you don't need to do a calculation here. Just qualitatively, point on the board where you would expect the expectation value to lie.
Me: Let me take a moment to think.
P2: You can explain your thinking out loud.
Me: I can see that the potential is skewed towards the right side. So naturally particles would be more probable in taking the positions towards the right. So I would say that the expectation value of position would be somewhere away from the minima of the potential towards the right side since those positions have more weight of probability.
P1: That's correct.
P1; Now, say you are at the minima of the potential well. Could you taylor expand the potential around at that point?
Me: Sure. (Start writing on the board)
P1: As a first approximation, what would be the shape of this potential?
Me: The first derivative would vanish since it is a minima and hence it would be a quadratic potential.
P1: Right. For this potential, can you find the expectation value of the classically distributed particles? How would you do it mathematically?
Me: (I start writing down the equation I was going to write before.)
P1: Would the momentum integral contribute?
Me: No, it would factor out and only the position integral needs to be done.
(I proceed setting up the gaussian position integral. I start saying how I can do the Gaussian integral by making a substitution in the integral)
P1: Can you another way in which you can get the answer of the integral quickly without calculating?
Me: (I think for a while). Ah! Yes. It's an odd integrand with symmetric limits from negative infinity to infinity. It is zero.
P1: Right. So you showed that the position expectation value for this harmonic approximation around the minima is zero. Can you think of a physical phenomena modelled by this potential considering the two expectation values you found?
(I was fortunate enough that I could quickly remember something I had already read about anharmonic potentials and thermal expansion)
Me: Yes, it is thermal expansion of a material.
P1: Correct. Do you know the name of this potential?
Me: Sorry, I don't know
P1: It is the Lenard Jones potential.
Now P2 chimes in and the test shifts to quantum mechanics.
P2: Can you see the diagram drawn on the other board?
P2: Consider that these are two quantum energy levels very far apart from each other. So they have no coupling with each other. And these are the only two levels. What would the hamiltonian matrix look like?
Me: It would be a 2X2 diagonal matrix. I write down the matrix with energies E1 and E2.
P2: Now if I bring these levels closer, they can get coupled. How would you change the hamiltonian?
Me: By adding off-diagonal terms.
P2: Add some off-diagonal term delta. Can you find the eigenenergies?
Me: Yes. I simply diagonalize the matrix and write the two eigenenergies.
P2: On a plot of energy vs difference of original two energies (E1-E2), what would the diagram of the energy levels look like?
Me: (I look at the energies and start figuring out how to plot them with respect to delta.) I am thinking about limiting cases. With E1=E2, the difference in energies is just delta. And with (E1-E2) very large, the separation between the levels is large.
I think and quickly remember the concept of avoided level crossing and drew the levels where their separation first decreases to a minimum (when E1 = E2) and then increases again.
P2: That seems correct. You know the atoms are of a size which is roughly a few angstroms, right?
P2: What would be the order of magnitude of the atomic electron energies?
Me: I can estimate with the uncertainty principle the order of magnitude of energy of the electron confined within a few angstroms.
P2: Can you tell us what it comes out to be without calculating?
Me: Yeah, it is a few eV.
P2: The SI unit of voltage was defined as 1 V and the electron energy in 1 V would be 1 eV. Why is there this coincidence that the atomic energies are also of this order of magnitude even though at that time there was no quantum mechanical understanding of atomic energies?
Me: (Think for a while) I guess.... thermionic emission was pretty commonly known that you get current when you heat metals. The stopping voltage measured there would be the atomic energies which would be a few eV and thus voltage was defined as 1 V which is close to that.
P2: You can say that, but there is another straightforward reason.
I start thinking but P1 points out that they are out of time.
P2: Never mind. You see that batteries were made of chemical compounds and the electron energies there were a few eV. So you got voltages in the order of 1 V.
(Thinking how I missed that one completely.)
P1: Thank you for the exam. You would know the result in the evening.
Me: Did I do well? Would I pass?
P1: We can't really tell you that.
P2: Smiles and says you should be fine.
I thank them and exit the room.
The next part was the Classical Mechanics + Electromagnetism test with two different professors.
The setting was the same but the blackboard was clear.
P1: We would be talking about particle collisions and its simple analysis. Do you know the mass of a proton?
Me: About 1 GeV.
P1: Yes. Write it on the board. We would consider a very high energy cosmic ray anti proton (10^20 eV energy) colliding with a proton to produce something. What particles would you say can come out apart from photons?
Me: Since charge would be conserved, another particle-anti particle pair can come out, like electron and positron perhaps.
P1: Yes. Can you think of some other possibility like that, but not electron-positron pair?
Me: Umm, dimuon pair?
At this point there was a bit of a confusion. P1 said that there is a neutral muon too. I was confused and thought maybe recently something was discovered. Who knows! But then P2 chimes in and said P1 got confused by pions which has a neutral pion as well. Things got back to business then.
P1: Okay, so a muon plus and muon minus pair. How would you analyze this collision?
Me: I would just do momentum and energy conservation for the process.
P1: Alright. Do you think there is a convenient reference frame that makes these problems easier?
Me: (Knowing only one name) The center of mass frame.
P1: Correct. So we would do the problem in the center of mass frame. What's the advantage in doing it in this frame?
Me: The net momentum is zero before and after the collision in this frame and then you are only concerned with balancing the energy conservation.
P1: OK. We would try to find the mass of the muons. Can you write the energy conservation?
Me: (I write down the relativistic energy balance for the process and look towards P1)
P1: With just dimensional analysis, can you get the mass of the muon in terms of what you know?
Me: Umm, I don't quite know.
P1: Try to make a quantity with dimensions of mass.
Me: (I just use the relativistic energy formula to get dimensions of mass but that leaves me pretty much with the equation I had)
It got a bit weird there. Either I didn't get his question or I didn't know what to do
With time running out, P2 said that we can move on to the next part since not much time is left.
P1 stands up and writes something like mass of proton*mass of proton/energy on the board or something like that and said this is what I was alluding at.
I didn't pay much attention as P2 quickly took over.
P2: Draw a sphere with a charge Q uniformly spread on it.
Me: (Drew the sphere)
P2: If you very far from the sphere, what would be the electrostatic potential?
Me: It would be Q/(4*pi*epsilon*r)
P2: Is it correct or would there be any corrections?
Me: I think it is correct. In this case, there would be no higher order terms in the multipole expansion.
P2: Why is that?
Me: Umm, Newton proved the shell theorem for gravity which states that far away from a spherically symmetric body, potential is just like point mass potential. And the electrostatic problem directly maps to it.
P2: Alright. Now draw a cylinder of say radius R and same height R. Make the upper part uniformly positively charged and the lower one uniformly negatively charged. What would be the potential now?
Me: It is neutral overall, so the monopole term would be absent. The leading term would be the dipole potential.
P2: With the distance scales in the problem, what can you say would be a dimensionless parameter that can characterize the size of the dipole correction term?
Me: (I start writing down the dipole potential integral but realize that this won't help)
P2: You don't need to calculate. Think about the scales available in the problem.
Me: Oh, it would be R/r (r is the distance from the cylinder)
P2: Right. Would there be a quadrupole term?
Me: (I thought of writing down the quadrupole integral but realized that I probably don't need to calculate anything again) Umm, I think by symmetry it would be zero.
P2: Could you elaborate?
Me: I think it goes like ... either you have monopole, quadrupole terms or dipole and octupole terms. In the multipole expansion, you get first, third, fifth terms and so on or second, fourth, sixth terms and so on.
P2: Yeah, there is a word for that.
(The word didn't strike me at all at that moment. I don't know why)
P2: It's parity!
Me: Ah yes. Right.
P2: Can you draw a configuration with a quadrupole moment?
Me: (I proceed to draw a cylinder with top left having positive charge, top right negative, bottom left negative and bottom right as positive.
P2: Okay. I think that's enough for now.
I thank them and leave the room.
I guess I bombed the collision problem but did fine on the electromagnetism one.