Tight binding model - David Miller

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Lecture 24a of Quantum Mechanics for Scientists and Engineers
Part of Lecture 24 Tight binding and variational models
Text reference: Section 6.5, QMSE
QMSE - D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge)

[@abhishekcp2024]

Thank you for the wonderful explanation

[@nnmaster4271]

Thank you! Most clear explanation I've ever found.

[@nnmaster4271]

The voice just like Morgan Freeman

[@Frank-ie8dh]

If you put the two wells next to each other such that an electron spilled equally into both wells, and then you put in energy to separate the two wells without making a measurement, what would the wavefunction electrons look like. What would be the energy levels of these new separated wavefunctions?

[@davidmillermusic]

So, to be specific, let's say we are looking at the problem of an electron in a coupled well, in the lowest state. That lowest state has equal probability of finding the electron in each well. If we now consider problems in which the barrier is progressively thicker (but with the same wells), the electron is still in a state with equal probability of being in each well. The energy of this state is, however, a bit higher, and eventually asymptotes to the energy of an electron in an isolated well. So, pulling the wells apart (without making a measurement) leads to an increase in the energy of the electron in this lowest state. That means that there is a kind of attractive force that tends to pull the wells closer (because the electron will have lower energy if the wells are closer), and that is partly the explanation for covalent bonding in chemistry.

[@Frank-ie8dh]

@@davidmillermusic Suppose we progressively increased the energy between the well barriers like you said. Would the peaks of the wave function remain in phase as the energy barrier is raised? For example, in the lowest state in your video, the two peaks line up in the well and they will both rotate in the complex plane (if we added an imaginary axis coming out of the screen) around the line they are drawn above, as time goes on (according to the time dependent SE). Would they still do the same as the energy barrier is raised, or would the phases somehow completely dissociate since the wave function is “split in half” now. To me, it makes more sense that the math would say that their phases remain “linked together” since there seems to be no well defined point where the wave functions in each well suddenly dissociate. However, the implication of this is that if two electrons each started out in each well, and the barrier is lowered then raised again, the two electrons suddenly have locked phases with each other by necessity right? That’s really weird. By the way, thank you for your explanations. I’m just a college student trying to teach myself QM.

[@davidmillermusic]

@@Frank-ie8dh You are right that (if we could figure a way to raise the barrier without "measuring" the system, which I think is possible in principle), the electron wave function phase would remain linked on the two sides. The kind of sudden dissociation you describe would be more characteristic of a "measurement" on the electron, such as asking which well it was in. One subtle point in this discussion is that these states we are discussing are the states of one electron, not two. (The states of two electrons are much more complicated!) So we are saying that, unless we make some "measurement", then in the kind of barrier-raising process you describe, the electron remains in both wells at once - an example of the kind of superposition that people often think is a strange aspect of quantum mechanics even if it is actually quite normal (such as electrons in molecules being spread over the entire molecule).

[@teddy0819tw]

Is there a reason for the lower energy state being symmetrical and higher energy state being antisymmetric in the tight binding model? I failed to understand the difference between this and a two-electron system (the lower state being the singlet state due the the exchange interaction reduces overlapping probability in the barrier that's the preferable configuration for electrons)...

[@puneettripathi740]

Thank you very much sir. The lecture is precious.

[@rigoluna1491]

This was a nice explanation however, why did we arbitrairly assume that in each well, our wave function was only on the lowest atomic state as opposed to a linear combinations of its atomic orbitals?

[@davidmillerquantum]

That is part of the whole idea of the tight binding approximation, and it is an approximation. We start from the simplest atomic state, and try to construct a first model presuming we are only weakly coupling those atoms. It is just a first model to expose some of the basic behavior.

[@rigoluna1491]

@David Miller Quantum Mechanics Courses I see, thanks for responding so fast. Does that imply that there are models/approximations that use more orbitals for better approximations? I am reading a paper on quantum chemistry simulations and felt this video helped

[@davidmillerquantum]

@@rigoluna1491 There are many different models for handling the complicated problems of molecules and crystals, and even for just the quantum mechanics of atoms with many electrons. Essentially all of these "many particle" problems are practically unsolvable in any exact sense; this is not a problem just of quantum mechanics - it would be the case even for "classical" many-body problems. As a result, approximation methods are practically essential. One, called "linear combinations of atom orbitals" or LCAO, is along the lines of what you are suggesting, for example. Because of these substantial difficulties, simple models like tight-binding approximations are often used to give a first guide to the kinds of things that may also come up in more complex approximations and calculations.

[@rigoluna1491]

@@davidmillerquantum Thank you, I will look into LCAO then.