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I want to compute the eigenvalues and eigenstates of a quantum well with different effective masses of electron in the barrier and in the quantum well. As can be seen [1]: https://github.com/mholtrop/QMPython/blob/master/Finite%20Well%20Bound%20States.ipynb [1], the mass of the electron is constant, therefore the Hamiltonian can be easily diagonalized and eigenvalues/eigenstates can be evaluated.

However, in my situation, the mass of the electron takes different values in the barrier and the well. Could any one suggest how I could obtain the matrix which satisfies the effective mass condition, and hence can be used to compute eigenvalues and states using similar method.

$$a_i\psi_{i-1}+b_i\psi_{i}+c_i\psi_{i+1}=E\psi_{i}$$

$$a_{i+1}=c_i=\frac{-\hbar^2}{2m^*(\delta z)^2} \text{ and } b_i=\frac{\hbar^2}{m^*(\delta z)^2}+V_i$$

In the above equations, it is apparent that the Hamiltonian can be diagonalized as mentioned in the link above.

$$a_{i+1}=c_i=\frac{-\hbar^2}{2m_{i+\frac{1}{2}}^*(\delta z)^2} \text{ and } b_i=\frac{\hbar^2}{2(\delta z)^2}\bigg(\frac{1}{m^*_{i+\frac{1}{2}}}+\frac{1}{m^*_{i-\frac{1}{2}}}\bigg)+V_i$$

My question is: how can I diagonalize my Hamiltonian with $m^*$(z) being different in the barrier and the well using the finite difference.

The complete Schrodinger equation can be written as,

$$\frac{-\hbar}{2}\frac{\partial}{\partial z}\frac{1}{m^*(z)}\frac{\partial}{\partial z}\psi(z)+V(z)\psi(z)=E\psi(z)$$

$\qquad\qquad\qquad\qquad\qquad\quad$Quantum square-well diagram

Here, mass depends on z growth axis of quantum well (in picture its x instead of z) $m^*(z)$ in case of finite quantum well the value mass in barrier remains same in both regions 1 and 3 and is different in the well region 2.

I would like to know how can I use the same approach as mentioned [1] to solve the Schrodinger equation with a spatially dependent mass.

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    $\begingroup$ Would Computational Science be a better home for this question? $\endgroup$
    – Qmechanic
    Sep 29 '21 at 12:17
  • $\begingroup$ Thank you, I will do that. $\endgroup$
    – celerion
    Sep 29 '21 at 13:00
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    $\begingroup$ I don't understand the question. The way you write it down, you already have a matrix eigenvalue problem. You just need to combine all of your equations into a matrix and throw the matrix into an eigenvalue solver. I assume you have already done that for the case of same-mass electrons. It's not clear to me what is different about the case with different masses if you can numerically solve the case with same masses. $\endgroup$ Sep 29 '21 at 14:33
  • $\begingroup$ Can you write down the original differential equation for your problem? $\endgroup$
    – nicoguaro
    Sep 29 '21 at 16:39
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    $\begingroup$ @celerion when including equations, please use mathjax formatting rather than an image. Images aren't searchable and can't be interpreted by screen readers. $\endgroup$
    – Tyberius
    Sep 30 '21 at 13:50

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