- #1

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A particle moving in a potential V is described by the Klein-Gordon equation:

[tex] \left[-(E-V)^2 -\nabla^2 + m^2 \right] \psi = 0 [/tex]

Consider the limit where the potential is weak and the energy is low:

[tex] |V| << m \;;\; |\epsilon| << m \;;\; \epsilon = E - m [/tex]

Show that in this limit the KG equation can be approximated by the Schrodinger equation:

[tex] \left[-\nabla^2 + 2mV \right] \psi = 2m\epsilon \psi [/tex]

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It seems that a taylor expansion is required or other approximation is needed to get the factors of 2 but I've tried several now and can't seem to make the answer.

Any help much appreciated.