Avoided Crossing in a One-Dimensional Asymmetric Quantum Well

Authors: Dou Du, Taylor James Baird and Giovanni Pizzi

Original notebook: asymmetricwell.ipynb

In this notebook we demonstrate the phenomenon of avoided crossing by solving the Shrödinger equation of a one-dimensional asymmetric quantum well.

Goals

Familiarize yourself with the phenomenon of avoided crossing.
Understand the mathematical origin of avoided crossing (i.e. how the behaviour arises from the description of a given system through a dependence on some set of parameters).
Relate the observation of avoided crossing in this toy model to its manifestation in realistic molecular and material systems.

Background theory

Tasks and exercises

For which value of $\mu$ are the two lowest eigenvalues closest to each other? Will the order of the states change by changing the $\mu$ parameter? And why?

Solution
In the figure, the blue and red lines show the lowest and second-lowest eigenvalues respectively. The subplot on the bottom shows the two lowest eigenvalues as a function of the parameter $\mu$ . One can see that these two eigenvalues are closest to each other at $\mu = 0$ . For the entire range of plotted $\mu$ values, the red line is always higher than the blue line, i.e. the eigenvalues change continuously. However, the nature of the state changes as we change the value of mu! For values $\mu \ll 0$ , the lowest eigenstate is localized in the right well, while for $\mu \gg 0$ is is localized in the left well. For values of $\mu$ close to zero, the two states mix and we get a symmetric and an antisymmetric solution. The $\mu x$ term can be considered as a perturbation to the original double well potential, which results in the avoided crossing. Please check the detailed information in the background theory section.
How about other states? Is there any changing of energy order of the states?

Solution
By tuning the $\mu$ slider we observe that the values of the eigenvalues change continuously with the size of the perturbation. Also in this case, when the value of $\mu$ would tend to make two eigenvalues identical, an anticrossing effect takes place (e.g. between the second and third state at $\mu \sim 0.085$ ), and we see the appearance of a symmetric and an antisymmetric eigenstate.
What type of molecular system could be described by the asymmetric double well model we consider here?

Solution
As demonstrated in the notebook, illustrating a double quantum well system, this type of potential gives a reasonable description of diatomic molecules. Adding in asymmetry simply accounts for the two atoms comprising the molecule being different (a heteronuclear molecule). This toy model therefore illustrates the fact that, in diatomic systems, which are parameterized by a single value - namely the separation of the two nuclei - the eigenvalues of the system never cross. One must consider polyatomic molecules consisting of three or more atoms before crossing of electronic energy levels is observed (see Wikipedia).

Interactive visualization

Avoided Crossing in 1D Quantum Well

(click on a state to select it)

μ-0.10Zoom factor2

ShowWavefunctionProbability density

$\mu$ : the potential parameter determining the symmetry of the double well potential.
Zoom factor: the zoom factor of the eigenfunctions.

Legend

(How to use the interactive visualization)

Interactive figures

There are two subplots shown above. In the uppermost subplot, the wide figure on the left shows the well potential alongside the eigenfunctions, $\psi$ , of different states (or their corresponding probability density, the square modulus $|\psi|^2$ ). The narrow figure on the right shows the corresponding eigenvalues.

The subplot on the bottom shows how the three lowest eigenvalues change with the modification of the potential parameter, $\mu$ .

Controls

There is a slider to adjust the $\mu$ parameter of the potential. The zoom factor slider aids in the inspection of the wavefunctions by multiplying them by a constant (this is purely for visualization purposes). One can highlight the square modulus of a specific eigenfunction, $\psi$ , and its eigenvalue, by clicking on the plot. All the other states shall be hidden from the plot. By clicking the Show all button, one can see all states again if they were previously hidden by clicking on one of them to highlight it. Additionally, there is a radio button which enables one to choose whether to plot the wavefunctions $\psi$ , or the probability densities $|\psi|^2$ .