Friday, September 16, 2016

A basic quantum concept: energy level repulsion (avoided crossings)

When I learnt and later taught basic quantum mechanics I don't think the notion of energy level repulsion (or equivalently avoided crossings) was emphasised (or even discussed?).

Much later I encountered the idea in advanced topics in theoretical physics such as random matrix theory and in theoretical chemistry  (non-adiabatic transitions and conical intersections).

Yet level repulsion is a very simple phenomena that can be illustrated with just a two by two matrix describing two coupled quantum states, as nicely discussed on the Wikipedia page.

Last semester when I was teaching Solid State Physics I realised just how central and basic the phenomena is and that the students did not appreciate this.

Level repulsion is the origin of several key phenomena in chemistry and physics.

In solid state physics, it is the origin of the appearance of band gaps at the zone boundary and thus the all important distinction between metals and insulators.

Previously, I posted how Chemistry is quantum science because chemical bonding (the lowering of energy due to interacting atoms) arises due to the superposition principle. This could also be viewed as level repulsion.

Another key idea in chemistry is that of transition states and activation energies for chemical reactions. When one uses a diabatic state picture, particularly as emphasised by Shaik and Warshel, the transition state emerges naturally in terms of level repulsion.

The figure is taken from here.

Can you think of any other nice examples?


  1. To teach the concept, I much appreciate the example of two classical uncoupled oscillators. When we add a spring between them, the energies of the normal modes split (are no longer degenerate). I like this example because it is completely classical and demonstrates that level repulsion is not a "mysterious quantum mechanical effect".

    1. Thanks for the comment.
      I am glad you teach the subject.
      I agree that the classical analogue is important and useful.

  2. It's totally central, along with spin effects, to topological insulators, and their interacting corollaries. It would be interesting to know how many papers you've published with level repulsion as a core concept. I bet it's a few

  3. What still wracks my brain is how best to think about level repulsion in the context of dynamics. Strictly, level repulsion is an adiabatic phenomenon -- for the chemistry case, electronic level repulsion is observed if the BOA holds. When the nuclei move fast enough, though, the adiabatic states are not observable, but diabatic states may be. So, one would think that the repulsion depends on the time sampled in the crossing region. Levine wrote on this, and in many ways I'm still trying to wrap my brain around this article, years after I first tried to read it:

    1. Levine, R. D. & Kinsey, J. L. On the repulsion of energy eigenstates in the time domain. Proceedings of the National Academy of Sciences 88, 11133–11137 (1991).

  4. In crystals, the level splitting is caused by perfect back-reflection of an election wave, which can only happen when the electron wavelength is equal to the atomic spacing. When that happens, the forward- and backward-propagating waves form a standing wave, and the square of that standing wave (i.e. the electron density) can be either in-phase or out-of-phase with the positive ions in the lattice. These correspond to the low- and high-energy states (respectively) at the energy gap. This is all coming from Kittel. So, in this case, level splitting seems to be an interference effect, though I still don't understand why the states in the gap are forbidden.

    I think there must also be an explanation of level splitting in terms of modulation, since 2*cos(wt)*cos(Wt)=cos((W+w)t)+cos((W-w)t). In the case of the two oscillators with equal masses and spring constants, coupling them seems to have the effect of modulating their individual motions at the frequency corresponding to the resonance frequency of the coupling spring which, according to the equation, is equivalent to splitting.