Background Theory: Avoided Crossing in One Dimensional Asymmetric Quantum Well

Original notebook: theory_asymmetricwell.ipynb

Introduction

In quantum physics and quantum chemistry, an avoided crossing is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable, and depending on N continuous real parameters, cannot become equal in value (“cross”) except on a manifold of N-2 dimensions. Please read the Wikipedia page on avoided crossing for more information.

In this notebook, we solve the Schrödinger equation for a 1D potential. The formula of the potential is:

\large V(x) = x^4 - 0.6x^2 + \mu x

where, $x$ is the position, and $\mu$ is the potential parameter (which can be tuned using a slider in the interactive notebook) that determines the asymmetry of the two quantum wells (i.e., the relative “depth” of the two quantum wells).

Two-state system

Consider a quantum system which has only two states E $_1$ and E $_2$ . It is a good model to study a variety of physical systems. The two-state Hamiltonian H can be formulated in matrix form:

H = \begin{pmatrix} E_1 & 0 \\ 0 & E_2 \end{pmatrix} \quad (1)

When a perturbation is introduced, the Hamiltonian of the perturbed system H’ is written as:

H' = H + P = \begin{pmatrix} E_1 & 0 \\ 0 & E_2 \end{pmatrix} + \begin{pmatrix} 0 & W \\ W^* & 0 \end{pmatrix} = \begin{pmatrix} E_1 & W \\ W^* & E_2 \end{pmatrix} \quad (2)

The new eigenvalues are calculated as:

E_+ = \frac{1}{2}(E_1+E_2) + \frac{1}{2}\sqrt{(E_1-E_2)^2+4|W|^2} \quad (4)

E_- = \frac{1}{2}(E_1+E_2) - \frac{1}{2}\sqrt{(E_1-E_2)^2+4|W|^2} \quad (5)

One can see that, by definition, E $_+$ is always bigger than E $_-$ . The energy difference $E_+ - E_-$ is $\sqrt{(E_1-E_2)^2+4|W|^2}$ . Hence, the perturbation prevents the two energy states from crossing. However, we stress that the eigenvectors depend on the value of $W$ ; they will be closer to the $(1,0)$ eigenvector (associated with the $E_1$ eigenvalue of the unperturbed Hamiltonian $H$ ) or to $(1,0)$ (associated with $E_2$ ) depending on the value of $W$ .