# Schrodinger/Dirac equation

1. May 8, 2015

### jamie.j1989

Hi, I'm doing a project on graphene and don't really understand why we use the Dirac equation instead of the Schrodinger equation. The fermi velocity of electrons in graphene is not relativistic, I know the particles are considered as quasiparticles but don't see how this changes things. My only reasoning is that the effective mass is zero and the 1/m dependence in the Schrodinger equation is incompatible with this? Thanks.

Scrodinger equation

$$i\hbar\frac{\partial}{\partial{t}}\psi({\textbf{r},t})=\left[-\frac{\hbar^2}{2m}\nabla^2V(\textbf{r})+E\right]\psi({\textbf{r},t})$$

Dirac equation

$$\left[\gamma^{\mu}\partial_{\mu}+\frac{c}{\hbar}m\right]\psi(\textbf{r},t)=0$$

2. May 8, 2015

### Brage

Hi jamie.j1989, I am not sure what aspects of graphene you are investigating, but for example the conductance of graphene is very good, which translates to very fast moving electrons (definitely within the range where we see relativistic effects). Also the fermi velocity is only relevant at 0K, so for temperatures above that the velocities of the fastest electrons will exeed the fermi velocity.

3. May 8, 2015

### atyy

The fundamental equation for graphene is the Schroedinger equation. However, an equation with the form of the Dirac equation "emerges" from the Schroedinger equation as an excellent approximation at low energy. The "speed of light" in the equation with the form of the Dirac equation is not the speed of light. It is ~10^6 m/s.
http://arxiv.org/abs/cond-mat/0509330
http://www.physics.upenn.edu/~kane/pedagogical/295lec3.pdf

Last edited: May 8, 2015
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