Finite Square Well

There are two common approaches for solving systems of equations: linear and non-linear. For linear unsteady systems, it can be solved by Gaussian elimination and inverse matrix. However, for nonlinear equations, Newton’s method can let us approach the solution. This project demonstrates how Newton’s method can be applied to solve the nonlinear eigenvalue problem of a finite potential square well, where no theoritical solution exists.

Finite Square Well

In quantum mechanics, the energy eigenvalues of finite potential square well are given by transcendental equations. Therefore, these equations are highly nonlinear and must be solved numerically or graphically. In this project, I used single-variable Newton’s method to find the roots corresponding to the allowed energy eigenvalues of a bound electron in the well.

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System conditions

Consider an electron in a quantum square well potential of depth \(V=-13.6 eV\) and half-width \(a=20a_B\) where \(a_B\) is the Bohr radius.

Energy Equations

The energy eigenvalues, \(E>V\), are given by the transcendental equations. For the odd states, $$ \sqrt{-E} = \sqrt{E-V}\tan\left(\frac{a}{\hbar}\sqrt{2m(E-V)}\right) $$ , and the even states $$ \sqrt{-E} = -\sqrt{E-V}\cot\left(\frac{a}{\hbar}\sqrt{2m(E-V)}\right) $$ Since these equations cannot be solved analytically, we apply a numerical root-finding approach.

Initial Guess

The eigenenergies of an infinite square well with the same width are $$ E_n = \frac{n^2\pi^2 \hbar^2}{8ma^2} - 13.6. $$ These values serve as initial guesses for the Newton iteration process.

Simulation Result

By plotting the difference between the left- and right-hand sides of the transcendental equations, we visually identify the root locations where the plot intersects \(y=0\). These roots represent valid eigenenergies.