## PHY456H1F: Quantum Mechanics II. Lecture 25 (Taught by Prof J.E. Sipe). Born approximation.

Posted by peeterjoot on December 7, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Born approximation.

READING: section 20 [1]

We’ve been arguing that we can write the stationary equation

with

Introduce Green function

Suppose that I can find , then

It turns out that finding the Green’s function is not so hard. Note the following, for , we have

(where a zero subscript is used to mark the case). We know this Green’s function from electrostatics, and conclude that

For we can easily show that

This is correct for all because it also gives the right limit as . This argument was first given by Lorentz. We can now write our particular solution

This is of no immediate help since we don’t know and that is embedded in .

Now look at this for

We get

where

If the scattering is weak we have the *Born approximation*

or

Should we wish to make a further approximation, we can take the wave function resulting from application of the Born approximation, and use that a second time. This gives us the “Born again” approximation of

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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