Hey not really sure where to start on this problem. I know you either use four-momentum or classical conservation of energy and momentum. Any ideas?
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Quantitatively or qualitatively? A photon cannot produce an electron-positron pair in free space because both energy and momentum cannot be conserved simultaneously. Say the photon has energy E=hf, while the combined energy of the pair E = 2γmc², where m is the mass of the electron. Now the momentum of the photon is p=hf/c and the momentum of the pair is p=2γmv. To satisfy conservation of energy:
hf = 2γmc²
Conservation of momentum:
hf/c = 2γmv
hf = 2γmvc
hf = 2γmc² = 2γmvc if and only if v=c, which is impossible.
hf = 2γmc²
Conservation of momentum:
hf/c = 2γmv
hf = 2γmvc
hf = 2γmc² = 2γmvc if and only if v=c, which is impossible.