Green's Functions & Density of States

[Master J]

After a fruitless search for a good undergraduate resource for Green's Functions (Economou's book is far too advanced for an intro course) , I hope someone here can clear this up.


So I have the Greens Functions (gf) for the time independent Schrodinger equation:

SUM |a><a| / (E - e(k)) where |a> form an orthonormal basis, e(k) is the particle energy. First off, for a free particle, E and e(k) are the same thing are they not? h^2.k^2/2m , so is the denominator not just zero, or is there a subtle meaning here?


Now, the more interesting part. Ok, so I have found some notes, but unfortunately they don't explain what is going on:

It takes the adove gf and adds an imaginary part to the denominator, and takes the limit that this imaginary parts goes to zero...then we get:


P ( 1 / E - e(k) ) + i(pi) DELTA( E - e(k))

where delta is the Dirac Delta function. What is this P? And how is this so? Is it just a mathematical relation??


And finally, I see that the density of states follows from this, but how is it motivated?? The result is just stated.

 

[weejee]

'P' means Cauchy principal value.

http://en.wikipedia.org/wiki/Cauchy_principal_value

 

[DrDu]

First off, for a free particle, E and e(k) are the same thing are they not? h^2.k^2/2m , so is the denominator not just zero, or is there a subtle meaning here?

No, e(k)=h^2.k^2/2m, but E is a variable which may take on any value, positive or negative (or even complex).
Think of a classical harmonic oscillator. There e(k) would correspond to the eigenfrequency of the oscillator omega_0 while E would correspond to an external driving perturbation which may oscillate with any frequency omega. The case E=e(k) would mean that the external perturbation is in resonance with some eigenfrequency of the system.

 

[Master J]

Thanks DrDu, I see that now. I see its similar to the way the gf reflects dispersion in the driven wave equation.

Any help with the next part? I don't know where or how the imaginary part is derived

 

[weejee]

$$\frac{1}{x+i\eta} = \frac{x}{x^{2}+\eta^{2}} - i\frac{\eta}{x^{2}+\eta^{2}}$$

Can you convince yourself that $$\frac{\eta}{x^{2}+\eta^{2}}$$ is pi times the delta function?

 

[Master J]

Sorry, no, I can't see that at all...

 

[Physics Monkey]

Try thinking about the limit as $$\eta \rightarrow 0$$. In particular, check what happens for $$x = 0$$ versus $$x \neq 0$$. Also, try computing the integral of $$\frac{\eta}{x^2+\eta^2}$$ over all x.