Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,548 answers , 19,352 comments
1,470 users with positive rep
409 active unimported users
More ...

How to obtain Dirac equation from Schrodinger equation and special relativity?

+ 2 like - 0 dislike
846 views

I'm reading the Wikipedia page for the Dirac equation:

The Dirac equation is superficially similar to the Schrödinger equation for a free massive particle:

A) $-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi.$

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energy are the space and time parts of a space-time vector, the 4-momentum, and they are related by the relativistically invariant relation

B) $\frac{E^2}{c^2} - p^2 = m^2c^2$

which says that the length of this vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,

C) $\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$

I am not sure how the equation A and B lead to equation C. It seems that it is related to substituting special relativity value into quantum mechanics operators, but I just keep failing to get a result...


This post imported from StackExchange Physics at 2014-03-24 03:51 (UCT), posted by SE-user Paul Reubens

Closed as per community consensus as the post is Undergraduate level questions are off-topic.
asked Oct 4, 2012 in Closed Questions by Paul Reubens (10 points) [ revision history ]
closed Apr 19, 2014 as per community consensus
Dirac himself talks about how he derived most of his ideas in this great lecture video he did in 1973. A little shaky, but very informative on the background of his ideas. rel="nofollow">youtube.com/…

This post imported from StackExchange Physics at 2014-03-24 03:51 (UCT), posted by SE-user AnimatedPhysics

2 Answers

+ 4 like - 0 dislike

First, C) isn't the Dirac Equation, it's the Klein-Gordon equation

Now, to your main question. A) comes from the classical equation for a free massive particle:

$\dfrac{p^2}{2m} = E$

by making the operator (operating on $\phi$) substitutions:

$p^2 \rightarrow - \hbar^2 \nabla^2$

$E \rightarrow i \hbar \dfrac{\partial}{\partial t}$

C) comes from B) by further recognizing that:

$E^2 \rightarrow -\hbar^2 \dfrac{\partial^2}{\partial t^2}$

This post imported from StackExchange Physics at 2014-03-24 03:51 (UCT), posted by SE-user Alfred Centauri
answered Oct 4, 2012 by Alfred Centauri (110 points) [ no revision ]
+ 0 like - 0 dislike

$$E^2 = p^2c_0^2 + m^2c_0^4$$ $$E^2\Psi=-\hbar^2\frac{\partial^2\Psi}{\partial t^2}$$ $$\left(p^2c_0^2+m^2c_0^4\right)\Psi=-\hbar^2\frac{\partial^2\Psi}{\partial t^2}$$ $$-c_0^2\hbar^2\nabla^2\Psi+m^2c_0^4\Psi=-\hbar^2\frac{\partial^2\Psi}{\partial t^2}$$ $$\frac{m^2c_0^2}{\hbar^2}\Psi=\left(\nabla^2+\frac{\partial^2}{\partial \left(ic_0t\right)^2}\right)\Psi$$ $$\left(\frac{mc_0}{\hbar}\right)^2\Psi=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\Psi$$

Where the signature of the metric tensor is taken to be $\left(-c_0^2,1,1,1\right)$. And there you have it, the fatally flawed Klein-Gordon Equation which cannot accomodate for potentials nor impose the norm-squared of the wavefunction to be non-negative which Schrodinger discarded, Dirac repaired, and the Higgs field was satisfied with!

answered Jun 18, 2013 by dimension10 (1,950 points) [ revision history ]




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...