What is Avoided crossing or Level repulsion

[sugeet]

I was reading about 'Polaritons" and Happened to come across this principle called " Avoided Crossing Principle", it is also called "Level Repulsion". They say (wikipedia) that Polaritons are expressions of this quantum phenomena!

Can anyone throw some light on this? I really did not understand much!

 

[DrDu]

Whenever you have two energy levels $E_1$, $E_2$ which come close to each other, you also have a matrix element V between the corresponding states which is not vanishing. You may combine this into a 2x2 matrix hamiltonian
$\begin{pmatrix} E_1 & V \\ V & E_2 \end{pmatrix}$
If the two energies depend on some parameter x, e.g. in an approximately linear way, there may be a point where they are degenerate. Nevertheless the eigenvalues of the hamiltonian will not become degenerate but will have a hyperbolic shape where the minimal energetic distance is 2V. This is called an avoided crossing. In the (avoided) crossing region, the two states (e.g. one photonic and the other electronic excitation) become strongly mixed. This is then called a polariton.

 

[sugeet]

Thank you very much! Really grateful to you! However I would like you to elaborate and also give a qualitative explanation.

 

[DrDu]

Well $E_1$ could be the energy necessary to create an exciton dependent on its crystal momentum k (which takes the role of the variable x). $E_2(k)$ can be taken as the energy of a photon in the medium assuming some background value of refractive index n.

 

[Zarqon]

I've seen people mentioning avoided crossings many times, but never taken the time to understand them, myabe I will now.

Whenever you have two energy levels $E_1$, $E_2$ which come close to each other, you also have a matrix element V between the corresponding states which is not vanishing. You may combine this into a 2x2 matrix hamiltonian
$\begin{pmatrix} E_1 & V \\ V & E_2 \end{pmatrix}$
If the two energies depend on some parameter x, e.g. in an approximately linear way, there may be a point where they are degenerate. Nevertheless the eigenvalues of the hamiltonian will not become degenerate but will have a hyperbolic shape where the minimal energetic distance is 2V. This is called an avoided crossing. In the (avoided) crossing region, the two states (e.g. one photonic and the other electronic excitation) become strongly mixed. This is then called a polariton.

A couple of specific questions:

In your example, where does V come from? What is V physically and why doesn't it vanish?

I guess a simple case is having two spin levels, whose distance depends linearly on the applied magnetic field. However, they can usually become degenerate without problems (at zero field). What separates this case from a case that does have an avoided crossing instead?

Is it correct to say that at the point of the avoided crossing, the original eigenstates are no longer good, and the get the new energy values one should diagonalize your E/V matrix above?

 

[DrDu]

Ok, let's take something concrete, i.e. a crystal of dye molecules, e.g. naphtalene (strongly absorbing in the UV). A single electronic exitation in one molecule can get transferred to neighbouring molecules due to Foerster energy transfer which falls of like $1/r^6$ where r is the distance between two molecules. Hence the excitonic states will get broadened into a band which can be indexed by a k vector. As long as we neglect the absorption of light by a molecule, we can describe the propagation of the light inside the crystal using an approximately constant index of refraction $n_0=(n_l+n_u)/2$, where $n_{l/u}$ is the index of refraction well below/above the resonance. The dispersion of the light is then approximately constant $\omega(k)=kc/n_0$.
If the possibility of absorption and emission, possibily only virtually, is taken into account the picture changes. n becomes a stongly k dependent function so that there may be various k's for a given frequency. This corresponds to level repulsion and to a strong mixing of the electronic and photonic degree of freedom.

 

[Zarqon]

Thanks for your replies DrDu, but I have some issues with also involving the already tricky subjects of energy transfer and excitons into this. It makes it difficult to isolate the distinct mechanism responsible for the avoided crossing.

I did do some reading on this subject on my own, and found a passage in the atomic physics book by Bransden and Joachain. They state that for diatomic molecules "two electronic energy surfaces having the same symmetry cannot cross". Since we are talking about electrons, I was thinking that this implies that the non-crossing rule is just an extension of the Pauli exclusion principle? Does this sound reasonable?

 

[f95toli]

No, the term "avoided crossing" is used in many different quantum systems; not only atoms and other systems where the Pauli exclusion principle applies.

For an example, see e.g.
http://www.sciencemag.org/content/300/5625/1548.full

Such are also seen in e.g. semiconductor devices etc.

Hence. avoid crossings are can be potentially be observed in any "general" two-state system where there is some coupling. The physicsl "implementation" of these systems is irrelevant.

The physics that DrDu described in the first response apply to ALL such quantum systems.

 

[DrDu]

The non-crossing rule has nothing to do with Pauli principle. It is rather a statement about the eigenvalues of a 2x2 matrix hamiltonian dependent on one parameter which was first formulated by von Neumann and Wigner.
There is an interesting generalization to hamiltonians depending on more than one parameter. There you find that the potential energy surfaces can only touch on a sub-manifold of dimension n-2 if n is the number of parameters. E.g. for n=2, surfaces can only touch in isolated points, the so-called conical intersections. These intersection points are of immense importance in photochemical reactions and underly also the Jahn-Teller effect.

 

[Zarqon]

Hence. avoid crossings are can be potentially be observed in any "general" two-state system where there is some coupling. The physicsl "implementation" of these systems is irrelevant.

The physics that DrDu described in the first response apply to ALL such quantum systems.

But that doesn't seem to be the case. From the book I quoted above it says clearly that the avoided crossing only happens when the two levels have "exactly the same symmetry". Any different symmtery, like parity or multiplicity, and it doesn't happen. This is not unlike the situation with the Pauli exclusion principe which also only causes repulsion in the case where the two fermions have "exactly the same quantum numbers". (Note, I'm not convinced it actually is Pauli, I was simply trying to steer to discussion onto more fundamental effects, because starting with complicated systems with a ton of different effects makes it really hard to isolate the mechanism)

In addition, in the books where I have found it mentioned so far, they always introduce this effect when they reach diatomic molecules, never with single atoms. Also from my experience working with single atoms and ions I have never seen this effect at all. So this leads me to believe that there is something fundamental appearing when you start working with diatomic molecules, as the simplest system that has this effect.


And DrDu, I saw your math above, and I agree that this is what happens if you have such a matrix, however, I still don't understand where 'V' comes in. In light of the diatomic systems being the simplest possible system to observe this in, what would V be here?

 

[f95toli]

Did you look at the paper I linked to above?
This is clearly a case where an avoided crossing is observed in a system where two two-level systems are coupled together via some interaction (in the paper just a capacitor). The reason I linked to that paper is that it is -in my view- quite a nice demonstation of a real implentation of a typical "toy Hamiltonian" where an avoided crossing can be observed.

This is an good example of situation where one should NOT think too much about specific systems such as atoms etc, it is much better to focus on the underlying physics which will be the same in many different situations.

 

[DrClaude]

From the book I quoted above it says clearly that the avoided crossing only happens when the two levels have "exactly the same symmetry".

That's because there is no coupling between the states ($V=0$) if they don't have the same symmetry.

 

[DrDu]

And DrDu, I saw your math above, and I agree that this is what happens if you have such a matrix, however, I still don't understand where 'V' comes in. In light of the diatomic systems being the simplest possible system to observe this in, what would V be here?

A typical situation in diatomic molecules is an avoided crossing between an ionic and a covalent state.
Let $|1\rangle=|1s_A (1)1s_B(2)+1s_B(1)1s_A(2)\rangle$ be the a purely covalent basis function for a H2 molecule with electrons labeled 1 and 2 and atoms labeled A and B, respectively, with electronic spins paired as a singlet.
An ionic state of the same symmetry would be $|2\rangle=|1s_A(1) 1s_A(2) +1s_B(1) 1s_B(2)\rangle$.
Now you can expand H into that basis, e.g., $V(R)=H_{12}=\langle 2|H(R)|1\rangle$.

For a general proof you determine the eigenfunctions of H(R) for some definite point $R=R_0$:
$H(R_0)\psi_i(R_0)=E_i(R_0)\psi_i(R_0)$.
These eigenfunctions for a fixed $R_0$ are called a "crude adiabatic basis".
For other distances $V(R)=\langle \psi_j (R_0)|H(R)|\psi_i(R_0)\rangle$ will be nonvansishing.
I remember Landau Lifshitz quantum mechanics to contain a nice discussion of the theorem.