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In reading through old course material, I found the assignment (my translation):

Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.

My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.

How to solve this task?

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@Qmechanic: I don't know if this counts as homework as it's part of a course that was given several years ago? The tag wiki isn't very helpful. On the other hand, I'm grateful for hints as opposed to answers. – Andrey Mar 28 '12 at 12:41
I didn't add the homework tag. If you think it doesn't apply, please edit it out again. – Qmechanic♦ Mar 28 '12 at 12:56
Yes, the homework tag does apply here. The tag wiki is meant to explain that the tag doesn't just apply to actual homework questions, but rather any question of an educational nature - that is, any question where the real goal is to learn something more general than just the answer to the problem. – David Zaslavsky♦ Mar 28 '12 at 22:14

3 Answers

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Another way of solving such problems is to go to another reference frame, where you obviously don't have enough energy.

For example you've got a $5 MeV$ photon, so you think that there is plenty of energy to make $e^-e^+$ pair. Now you make a boost along the direction of the photon momentum with $v=0.99\,c$ and you get a $0.35 MeV$ photon. That is not enough even for one electron.

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Neat trick, thank you! – Andrey Mar 28 '12 at 20:58
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Check if momentum can be conserved. That ought to do the trick.

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One needs both momentum conservation and energy conservation to get a contradiction! (Or 4-momentum conservation) – Arnold Neumaier Mar 28 '12 at 12:24
@ArnoldNeumaier: Yes, but this is a homework question. As such, we only give hints. It looks like he's already covered energy anyway. – Manishearth♦ Mar 28 '12 at 12:37
Ah, I got it. Thanks! – Andrey Mar 28 '12 at 12:40
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Other way to see why this is impossible is to look at inverse process: why annihilating positron and electron can't give up only one photon? Imagine these two particles at rest near each other (or look at center-of-mass system). They will annihilate giving 1MeV of energy, but single photon can't pick this energy up by itself because it would also have E/c of momentum and starting setup, the two charged particles, had non. You need two photons that move in opposite directions.

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