Wednesday, March 4, 2015
How do you feel about matrices?
Today while we were calculating some things some people raised some bona fide questions about the matrix approach. I don't want to sweep that under the rug. Let's discuss here how you feel about matrix formulation of quantum physics (for electrons in a lattice). Or anything else that is on your mind.
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If I recall correctly, it wasn't a matter of guessing, it was more a matter of seeing that those terms are negligible compared to \( I \) , which itself is pretty small compared to \( E_0 \) .
ReplyDeleteThe motivation for why we placed \( I \) in the "second diagonals" was because we were trying to find a matrix that described the energy of the whole lattice by way of position vectors. That's not true, they were actually wavevectors, but seeing as how the absolute magnitude is highest at the "position", we used simple position vectors to help visualize what was going on in the matrix.
Anyway, after "locating" the energies in the matrix, we used the matrix as an operator to act upon a wavefunction and determine how it diffused across the lattice over time.
After Chris exposed us to the wonders of python, we decided to apply this formalism to study an entirely different system: we would no longer have a single electron in a 1000-well lattice, but TWO electrons in a 2-well "lattice". This proved to be a very different problem, since instead of having 1000 sites with only 1 energy state available, we now had 2 sites with 6 energy states available (actually 4, since the triplet states have the same energy), but you get the picture. Ipso facto, "more is different", and we ran into some very counter-intuitive physics.
Even though my linear algebra skills are lacking, I find the matrix approach really intuitive in a lot of ways. I like how you can just find ground states and gain really fundamental insights into the properties of the crystal without even doing a derivative or much messy algebra at all.
ReplyDeleteI really like working with the Hamiltonian in matrix form and how it gives you a chance to visualize how the different spin states influence each other. George and Aron's discussion regarding similarities between the matrix of the 2 electron, 2 sites problem and that of 1 electron and 1000 sites is interesting but also makes me kind of confused!
ReplyDeleteHi Nellie, what parts are confusing? I think my word choice might not be the best, since it turned out that George and I were on more or less the same page.
DeleteMaybe this part might be confusing? "1000 sites with only 1 energy state available, we now had 2 sites with 6 energy states available"
DeleteActually in each case we start with a single atomic level. In the non-interacting case with 1000 sites that one level morphs into 1000 one-electron eigenstates. In the interacting 2-site case that one level morphs into 6 two-electron eigenstates.
"The reason why all the other terms except the ones near the diagonal are zero is because in the derivation of E=E0+2Icos(qa) we decided that all terms except I11 where very small.
ReplyDeleteExactly, that's what I meant to point out.
"I think that in the 1000-well lattice there are 1000 different energy eigenvalues"
I meant to say that there is one energy state per site, any of which are identical (because of translational invariance).
Haha we are in complete agreement, it was just a problem with my wording. Yes, the matrices for the 1000-well and double electron systems were guided by the same motivations and reasoning, hence they have the same formalism.
I wonder what a 2-electron 1000-well system matrix would look like?
I'm personally fond of the matrix formulation, however I'm noticing I'm slightly lost as to how it works in the two well-two electron system. With the infinite lattice and one electron the matrix made intuitive sense (each column of the matrix represented a well, each row represented how that well interacted with another well). With our new two electron system I don't have an intuition for what the elements correspond to.
ReplyDeleteTwo electron states are subtle and more difficult to get a feel for. Spending time thinking about that is definitely worth while!
DeleteIn the case of the matrix for one-electron states and 1000 sites, we get 1000 energy eigenstates. We understand that in general that is N eigenstates, where N is the number of sites.
ReplyDeleteFor the case of two-electron states and 2 sites we get six energy eigenstates (i.e., the dimension of our Hilbert space is 6).
Does anyone want to guess what that 6 generalizes to? That is, how does it depend on the number of sites and number of electrons?
Another interesting thing to think about is the limit in which U goes to infinity, i.e., U/I >>>1 for fixed I. (e.g., set I = 1 or 2 eV and let U get big).
ReplyDeleteI feel pretty comfortable with the Heisenberg approach. In the context of understanding two electron states in 2 wells, it provides a strong visual and logical representation of the energy states. I would personally like to understand the nature of the tunneling parameter t and how it relates to the kinetic energy.
ReplyDeletegood points. we call it I though. How is it related to KE?? let's think about that!
DeleteI like the matrix formulation a lot, and I feel as though I understand it for the most part. The 1000 site hamiltonian time dependence problem was very intuitive to me. The two-electron system is a little bit more confusing to me, but I feel as though I can understand the physics that we get out of it decently well. Personally, I am having some trouble regarding our basis and with parsing the physical meanings of vectors that correspond to a superposition of different states; for instance, (1001) and (0110) seem as though they should be almost identical, as their combination would yield something with weights of both electrons on both sites; however, one of these vectors would give something with an interaction term, and one wouldn't. Currently, I think about this in terms of the degeneracy of the energies; i.e., (0110) is a sum of two states BECAUSE they have the same energy, and do not necessarily correspond to an actual state that has two electrons on a single site. But then, I am somewhat at a loss as to how i can physically think of the lowest energy state, which is a combination of all the unlike spin states. It seems as though they could not create a lower energy state unless the four states are physically in some sort of interaction. Perhaps this confusion is because I haven't taken Quantum yet and there is a simple way to think about this.
ReplyDelete"Perhaps this confusion is because I haven't taken Quantum yet "
DeleteI don't think so. I think we need to work this out now.
"and there is a simple way to think about this."
I don't think so. no simple way.
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ReplyDeleteI think maybe we need to spend some time thinking about the nature of our original basis states for the 2 electron system.
ReplyDelete