Any tricks to nail down a few Quantum questions?

uhurulol
Posts: 79
Joined: Wed Aug 27, 2014 12:38 am

Any tricks to nail down a few Quantum questions?

Postby uhurulol » Tue Oct 21, 2014 10:11 pm

The PGRE is just a few days away for me, and I've boned up on all the physics that my coursework has covered thus far.

I, however, have never had a QM course. Nor linear algebra. I don't know much about either, but the past few days I've spent learning a little bit about them.

I intend to take QM and Linear next semester, but for now, is there any tricks I can memorize to get answers for some of the basic QM questions? I understand that QM and Stat Mech take up a good portion of the test, and it'd be nice to at least be able to get a few of the questions if they're some of the easier ones.

Thanks, and good luck test-takers!

djh101
Posts: 97
Joined: Tue Jul 15, 2014 4:08 pm

Re: Any tricks to nail down a few Quantum questions?

Postby djh101 » Wed Oct 22, 2014 3:34 am

Hmm, it's getting to be a little late, otherwise I'd recommend skimming through the introduction of a quantum book. Hopefully you've at least read a chapter on quantum in the modern physics section of an introductory physics textbook? A few things to note:
-Wavefunctions (psi) will always be continuous. If ETS gives you a graph of a wavefunction with discrete jumps, it's wrong.
-Wavefunctions do not diverge
-Particle in a box wavefunctions sort of resemble damped oscillators. In a classically forbidden region (i.e. you have to cross a barrier with more potential energy than the energy of your particle), the graph will look like an overdamped oscillator while an energy with no potential will look like an undamped oscillator. This is called quantum tunneling and can only be avoided with an infinite well.
-Wavefunctions should be normalized. In other words, the integral over all space of the psi times psi* should be 1.
-The probability of finding your particle between a and b is the integral over that region of space of psi times psi*.
-Expectation value of variable x is the integral of psi* times x times psi.
-Remember symmetry. If your wavefunction is symmetric about 0, then the average value of x should be 0.
-A Hermitian matrix is equal to its conjugate transpose and has all real eigenvalues. An eigenvalue a is a scalar such that Av = av for matrix A and vector v. If A is 2x2, you can find the eigenvalues by solving a^2-Ta+D=0, where T is the trace and D is the determinant of A.
-Heisenberg's uncertainty principle applies to energy and time as well as position and momentum.
-Hopefully you have taken an introductory chemistry course that talked about wavefunctions (remember the orbital probability distributions?). s orbitals have wavefunctions symmetric about rotation. psi should be 0 at the nodes. If you are given a wavefunction for p that is not 0 at the origin, it is wrong.
-The commutator [A,B] is equal to AB-BA (matrices/operators do not generally commute). If two operators commute, then they can be measured simultaneously. x and p do not commute, thus the uncertainty principle. Oh, and a matrix is just a representation of a more general entity called an operator.
-A basis is any linearly independent set of vectors that span an entire space. Linearly independent vectors are vectors that cannot be written as linear combinations of each other (a linear combination of v and w is av+bw for any scalars a and b). To span a space means that any vector in that space (which might be R, R^3, or something more abstract like the space made up of all possible wavefunctions for your Hamiltonian). x, y, and z unit vectors, for instance, make up a basis for R^3.
-A basis is orthonormal if all the basis vectors are orthogonal (inner/dot product = 1) and normalized (length is 1).
-If you see the word "perturbation theory", the first order correction to psi will be a linear combination of your basis psi functions. In other words, your first order perturbation correction isn't going to perturb your psi out of your original psi space.
-The eigenvalue for Lz (angular momentum, z component) is m h/2pi, quantum number m.
-The eigenvalue for L^2 (angular momentum squared) is l(l+1), quantum number l.
-[Ly,Lz] = i h/2pi Lx.
-[L^2,Lx] = 0
-An operator called the propagator can be applied to your wavefunction at t=0 to find the wavefunction at t= not 0 (its time evolution). U is unitary, which means its conjugate transpose is equal to its inverse.

Crash course in quantum mechanics off the top of my head. Knowing the qualitative properties of the wavefunction, at least, you should learn. Quantitative things like probability and expectation value, even better. Go learn what an eigenvalue is and how to find it (hopefully you at least know about matrices, determinants, and vectors). And definitely keep your eyes open for symmetry. Also be sure to note on your statement of purpose that you took the physics GRE before taking quantum, since your score will take a big hit from that.

And for stat mech, Google Widom and skim through the first chapter so that you know what a partition function is and how to get an expectation value from that. It's pretty much the same as wavefunctions except discrete, so you have a sum instead of an integral.
Last edited by djh101 on Wed Oct 22, 2014 1:50 pm, edited 1 time in total.

slowdweller
Posts: 62
Joined: Wed Oct 15, 2014 8:47 am

Re: Any tricks to nail down a few Quantum questions?

Postby slowdweller » Wed Oct 22, 2014 8:12 am

djh101 wrote:-[Ly,Lz] = 2pi i h Lx.


It should be 1/2pi, since it's hbar.

djh101
Posts: 97
Joined: Tue Jul 15, 2014 4:08 pm

Re: Any tricks to nail down a few Quantum questions?

Postby djh101 » Wed Oct 22, 2014 1:49 pm

Ah, sorry about that. Thanks for the catch.

uhurulol
Posts: 79
Joined: Wed Aug 27, 2014 12:38 am

Re: Any tricks to nail down a few Quantum questions?

Postby uhurulol » Wed Oct 22, 2014 7:56 pm

Apologies for the dumb, but I know psi is a wavefunction, is psi* the derivative? Not sure what the notation means.

Catria
Posts: 353
Joined: Fri Oct 26, 2012 4:14 pm

Re: Any tricks to nail down a few Quantum questions?

Postby Catria » Wed Oct 22, 2014 9:31 pm

uhurulol wrote:Apologies for the dumb, but I know psi is a wavefunction, is psi* the derivative? Not sure what the notation means.


Given psi a complex-valued wavefunction, psi* is its complex conjugate...




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