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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Kinetic energy with effective mass

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**