I want to write the kinetic energy operator as a matrix within a finite element approach for electrons moving in a crystal with some effective mass that is a function of position.(adsbygoogle = window.adsbygoogle || []).push({});

Now usually we have:

K = -ħ^{2}/2m d^{2}/dx^{2}

such that the second order derivative of a wavefunction maybe written as:

d^{2}/dx^{2}= 2/(x_{i+1}-x_{i-1})* (ψ_{i+1}-ψ_{i})/(x_{i+1}-x_{i}) - (ψ_{i}-ψ_{i-1})/(x_{i}-x_{i-1}))

But for electrons moving in a crystal where the effective mass depends on the spatial coordinate, then the kinetic energy operator is:

K = -ħ^{2}/2 d/dx(1/m*(x) d/dx)

How can I write this in a finite element approach? Do I just put 1/m_{i}in front of ψ_{i}etc.? I tried that but it gives some funny results that do not seem physical. In the problem I am solving the effective mass makes a large jump from one grid point to the next, so maybe this could cause some problems?

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# I Kinetic energy with effective mass

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