Background Theory: Avoided Crossing in One Dimensional Asymmetric Quantum Well#

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Source code: https://github.com/osscar-org/quantum-mechanics/blob/master/notebook/quantum-mechanics/theory/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 the 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 symmetry 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:
\[\begin{split}H = \begin{pmatrix} E_1 & 0 \\ 0 & E_2 \end{pmatrix} \quad (1)\end{split}\]

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

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

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 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.