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thepopasmurf
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I'm trying to follow a derivation of the ground state energy of Helium using perturbation theory. I've made it through most of the derivation but I'm stuck at the following integral
Find the value of C where
[itex]C=\frac{1}{(4\pi)^2}\int d\mathbf{z}_1d\mathbf{z}_2\frac{\exp(-(z_1+z_2))}{|\mathbf{z}_1-\mathbf{z}_2|}[/itex]
To give context to this equation, the vectors z_1 and z_2 are dimensionless rescalings of the position vectors (r_1 r_2) of the two electrons.
[itex]\mathbf{r}_1=\frac{a}{2}\mathbf{z}_1[/itex]
[itex]\mathbf{r}_2=\frac{a}{2}\mathbf{z}_2[/itex]
where a is the Bohr radius.
The integral is to be taken all over space.
Don't think there are any
integrating wrt to z_1 first:
[itex]\frac{1}{(4\pi)^2} \int z_1^2 \sin^2\theta_1 dz_1 d\theta_1 d\phi_1 \frac{\exp{z_1+z_2}}{\sqrt{z_{1}^2 + z_{2}^2 - 2z_1 z_2 \cos\theta_{12}}}d\mathbf{z}_2[/itex]
I basically don't know where to go from here. My first idea to simplify was to make the integral relative and set the z_2 vector to always be the z-axis of the coordinate frame (thus eliminating an angle) but I still couldn't do the integral itself.
Thanks,
Homework Statement
Find the value of C where
[itex]C=\frac{1}{(4\pi)^2}\int d\mathbf{z}_1d\mathbf{z}_2\frac{\exp(-(z_1+z_2))}{|\mathbf{z}_1-\mathbf{z}_2|}[/itex]
To give context to this equation, the vectors z_1 and z_2 are dimensionless rescalings of the position vectors (r_1 r_2) of the two electrons.
[itex]\mathbf{r}_1=\frac{a}{2}\mathbf{z}_1[/itex]
[itex]\mathbf{r}_2=\frac{a}{2}\mathbf{z}_2[/itex]
where a is the Bohr radius.
The integral is to be taken all over space.
Homework Equations
Don't think there are any
The Attempt at a Solution
integrating wrt to z_1 first:
[itex]\frac{1}{(4\pi)^2} \int z_1^2 \sin^2\theta_1 dz_1 d\theta_1 d\phi_1 \frac{\exp{z_1+z_2}}{\sqrt{z_{1}^2 + z_{2}^2 - 2z_1 z_2 \cos\theta_{12}}}d\mathbf{z}_2[/itex]
I basically don't know where to go from here. My first idea to simplify was to make the integral relative and set the z_2 vector to always be the z-axis of the coordinate frame (thus eliminating an angle) but I still couldn't do the integral itself.
Thanks,