Schrodinger/Dirac equation

• jamie.j1989
In summary, the use of the Dirac equation instead of the Schrodinger equation in graphene is due to its emergence as an excellent approximation at low energy. The fermi velocity of electrons in graphene is not relativistic, but it becomes relevant at 0K. The "speed of light" in the Dirac equation is not the actual speed of light, but rather ~10^6 m/s.
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

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

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.

jamie.j1989 said:
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.

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:
bhobba

What is the Schrodinger/Dirac equation?

The Schrodinger/Dirac equation is a mathematical equation that describes the behavior of quantum particles, such as electrons, in a quantum system. It combines the principles of quantum mechanics and special relativity to accurately predict the behavior of these particles.

Who developed the Schrodinger/Dirac equation?

The Schrodinger/Dirac equation was first proposed by Austrian physicist Erwin Schrodinger in 1926, and later expanded upon by British physicist Paul Dirac in 1928. Both scientists made significant contributions to the field of quantum mechanics and their work laid the foundation for the development of the Schrodinger/Dirac equation.

What is the significance of the Schrodinger/Dirac equation?

The Schrodinger/Dirac equation is a fundamental equation in quantum mechanics and has been instrumental in our understanding of the behavior of subatomic particles. It has been used to accurately predict the behavior of particles in a variety of physical systems and has been verified through numerous experiments.

How does the Schrodinger/Dirac equation differ from the original Schrodinger equation?

The original Schrodinger equation only takes into account non-relativistic effects and is therefore limited in its application to particles moving at speeds much slower than the speed of light. The Schrodinger/Dirac equation, on the other hand, takes into account the principles of special relativity and can be used to accurately describe the behavior of particles moving at any speed.

What are some real-world applications of the Schrodinger/Dirac equation?

The Schrodinger/Dirac equation has numerous applications in fields such as quantum physics, chemistry, and materials science. It is used to understand the behavior of electrons in atoms and molecules, as well as in the development of new materials and technologies, such as transistors and quantum computers.

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