- #1

Drew Carey

- 10

- 0

Hi all!

I was reading up on the Klein paradox in Itzykson & Zuber's Quantum Field Theory (but I think this is a pretty standard part that's probably present in most QFT textbooks) and on page 62 they have a pretty straight forward solution to the Dirac equation with a step potential.

I've noticed something odd though - they're solution is based upon solving for the two regions (V=0 and V>0) and requiring continuity at the origin. They do not however require continuity of the first derivative, and in fact their solution's first derivative isn't continuous.

When solving Schrodinger's equation with a step potential it is always customary to enforce continuity of the first derivative as the equation contains a second space derivative.

I've tried to solve the Dirac problem myself with this added constraint, but is seems unsolvable (over constrained. Though I may have some mistake in the algebra). I've searched online for other solutions to this problem, and they all seem to ignore continuity of the first derivative, without even mentioning it.

What am I missing? How can we give up on continuity of the first derivative? Especially in light of the fact that the Dirac equation is supposed to be equal to Schrodinger's equation in a non-relativistic limit these two constraints need to be satisfied by the Dirac equation solution as well, no?

Thanks in advance!

I was reading up on the Klein paradox in Itzykson & Zuber's Quantum Field Theory (but I think this is a pretty standard part that's probably present in most QFT textbooks) and on page 62 they have a pretty straight forward solution to the Dirac equation with a step potential.

I've noticed something odd though - they're solution is based upon solving for the two regions (V=0 and V>0) and requiring continuity at the origin. They do not however require continuity of the first derivative, and in fact their solution's first derivative isn't continuous.

When solving Schrodinger's equation with a step potential it is always customary to enforce continuity of the first derivative as the equation contains a second space derivative.

I've tried to solve the Dirac problem myself with this added constraint, but is seems unsolvable (over constrained. Though I may have some mistake in the algebra). I've searched online for other solutions to this problem, and they all seem to ignore continuity of the first derivative, without even mentioning it.

What am I missing? How can we give up on continuity of the first derivative? Especially in light of the fact that the Dirac equation is supposed to be equal to Schrodinger's equation in a non-relativistic limit these two constraints need to be satisfied by the Dirac equation solution as well, no?

Thanks in advance!

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