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I'd be greatful for a bit of help on this question, can't seem to get the answer to pop out:
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.
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]
--------------
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.