Undergraduate-level explanation of Dirac Equation?

[lizzie96]

I am interested in learning about how the Dirac Equation was derived, how it allowed special relativity and QM to be unified, and how it predicted the existence of animatter. The explanations I have found so far are too advanced for me mathematically, and I was wondering if anybody could recommend a textbook or other source that explains the Dirac Equation at a basic undergraduate level.
Thank you for any help.

 

[edguy99]

I am interested in learning about how the Dirac Equation was derived, how it allowed special relativity and QM to be unified, and how it predicted the existence of animatter. The explanations I have found so far are too advanced for me mathematically, and I was wondering if anybody could recommend a textbook or other source that explains the Dirac Equation at a basic undergraduate level.
Thank you for any help.

Nothing like the master himself explaining it in this 1975 video:

http://www.youtube.com/watch?v=vwYs8tTLZ24&list=PLjNexov924eRr3L8aCirRyVCRN5rGi29W

 

[Jorriss]

I am interested in learning about how the Dirac Equation was derived, how it allowed special relativity and QM to be unified, and how it predicted the existence of animatter. The explanations I have found so far are too advanced for me mathematically, and I was wondering if anybody could recommend a textbook or other source that explains the Dirac Equation at a basic undergraduate level.
Thank you for any help.

The easiest book that I know of that covers it is Sakurai's Modern Quantum Mechanics.

That being said, the last chapter on the klein-gordon equation and dirac equation is not terribly advanced and I don't *think* an undergraduate who's taken quantum mechanics would have any issues with it.

 

[DimReg]

There's probably some errors in this, but all the important ideas are there and I don't have time to make sure its exactly right. You might want to go over it and test each step by yourself, since that's pretty informative.

Basically, you start with the klein gordon equation (in c=hbar = 1 units) $(\partial^2 + m^2 ) \phi = 0$

You get this equation by starting with the relativistic invariant and replacing E with the energy operator and p with the momentum operator. Because this is second order, you can't reproduce the result that probability is conserved like with the schrodinger equation (it won't hold it's normalization). So Dirac set out to find a first order equation, which would "square" to the klein gordon equation, so that it would both be relativistic and preserve probability.

To state the problem mathematically, you have some operator D, such that:

$D \psi = 0$ and $D^2 \psi = (\partial^2 + m^2) \psi = 0$

So you try to write down a first order operator for D, and solve for coefficients:

$D = (A\partial - Bm), D^2 = (A\partial - Bm)(A\partial - Bm) = A^2\partial^2 + m^2B^2 - (AB + BA)\partial = \partial^2 + m^2$

So A^2 and B^2 have to be one, but that doesn't allow AB + BA to be zero, as long as A and B are complex numbers. Dirac's great idea is that A and B can be matrices, and by finding a set of matrices with those properties you have derived the Dirac Equation.

That's roughly what the derivation is, and how it's linked to special relativity (because it's closely related to the Klein Gordon equation). Anti matter is predicted from solutions of the Dirac equation, but I don't remember how to derive them. Maybe someone else can help you out.

 

[WannabeNewton]

Chapter 1 of Srednicki's QFT text has a great exposition of exactly what you ask for and this particular chapter is not advanced mathematically for an undergraduate (at least in principle), in my opinion anyways.

 

[Jolb]

There's probably some errors in this, but all the important ideas are there and I don't have time to make sure its exactly right. You might want to go over it and test each step by yourself, since that's pretty informative.

Basically, you start with the klein gordon equation (in c=hbar = 1 units) $$(\partial^2 + m^2 ) \phi = 0$$

You get this equation by starting with the relativistic invariant and replacing E with the energy operator and p with the momentum operator. Because this is second order, you can't reproduce the result that probability is conserved like with the schrodinger equation (it won't hold it's normalization). So Dirac set out to find a first order equation, which would "square" to the klein gordon equation, so that it would both be relativistic and preserve probability.

To state the problem mathematically, you have some operator D, such that:

$D \psi = 0$ and $D^2 \psi = (\partial^2 + m^2) \psi = 0$

So you try to write down a first order operator for D, and solve for coefficients:

$D = (A\partial - Bm), D^2 = (A\partial - Bm)(A\partial - Bm) = A^2\partial^2 + m^2B^2 - (AB + BA)\partial = \partial^2 + m^2$

So A^2 and B^2 have to be one, but that doesn't allow AB + BA to be zero, as long as A and B are complex numbers. Dirac's great idea is that A and B can be matrices, and by finding a set of matrices with those properties you have derived the Dirac Equation.

That's roughly what the derivation is, and how it's linked to special relativity (because it's closely related to the Klein Gordon equation). Anti matter is predicted from solutions of the Dirac equation, but I don't remember how to derive them. Maybe someone else can help you out.

That's actually a pretty nice explanation. [You dropped an m in the algebra but the conclusions are still right.] Here's what you're missing: you leave off at trying to find a set of matrices with those properties. It turns out the simplest set of such matrices are 4x4 matrices (called the Dirac matrices or gamma matrices), and they are built out of the Pauli spin matrices. These matrices must act on $\psi$, so $\psi$ (which describes the fermion) must be a four-component object. Two of these components are associated with fermions with negative charge (electrons) and the other two with fermions of positive charge (positrons). The two components for each correspond to a spin up and spin down component.

With the appropriate notation, you could pretty much identify
$$\psi=\begin{pmatrix} \mathrm{spin \ up \ electron \ component} \\ \mathrm{spin \ down \ electron \ component} \\ \mathrm{spin \ up \ positron \ component} \\ \mathrm {spin \ down \ positron \ component} \end{pmatrix}$$
You need to do a little work to actually see how each component actually reflects those [you need to introduce an electromagnetic interaction term into the Hamiltonian and see how the components line up], and I would agree that Sakurai's "Modern Quantum Mechanics" is a good reference, but I think Shankar's "Principles of Quantum Mechanics" is a little better. If you're more comfortable with relativity/field theory than with quantum mechanics, then maybe check out Zee's "Quantum Field Theory in a Nutshell".

Paul Dirac was notoriously a man of few words. Dick Feynman told the story that when he first met Dirac at a conference, Dirac said after a long silence, "I have an equation; do you have one too?"

(From Zee's Quantum Field Theory in a Nutshell, 2nd edition, p. 105)

 

[DimReg]

I think another good undergraduate level text is Griffiths particle physics book. There's a lot you can learn from that book, so it's a good idea to read it. Particularly, I've found it to be the best book for describing how to actually read feynman diagrams I have ever used (out of 3 qft books and 2 particles physics books). Just take the shortest route you can to chapter 7, where the dirac equation is derived and explained, like I have done above (but in more detail).

A personal favorite, but at a higher level of detail than Griffiths text, is Halzen and Martin's Quarks and Leptons. These two books are good for getting you into the topics you seem interested in, without dropping you right into field theory (actually, they are field theory books but they focus of the physics rather than the formalism).

 

[gadong]

I too had a similar interest as an undergraduate. I bought a book by R.E. Moss (Advanced Molecular Quantum Mechanics) at the end of my first year which satisfied my curiosity. It is a beautiful book, surely out of print by now, but probably available in university libraries. The intended readership is physical chemists with an interest in spectroscopy, so all topics (including vector and matrix notation) are addressed from an elementary level.