Archived Calculus for delta function based on wave function

sandf

1. Homework Statement

Two-electron Wavefunction: ψ(r1,r2,r12) = exp(-Ar1-Br2-Cr12), r12 = |r1-r2|
A, B, and C are coefficients

Calculate <ψ|δ(r12)|ψ> and <ψ|δ(r1)|ψ>
2. Homework Equations

NO
3. The Attempt at a Solution

<ψ|δ(r12)|ψ>

= ∫∫dv1dv2ψ2(r1,r2,r12)δ(r12)

=4π∫dr2ψ2(r2,r2,0)

<ψ|δ(r1)|ψ>

= ∫∫dv1dv2ψ2(r1,r2,r12)δ(r1)

=4π∫dr2ψ2(0,r2,-r2)

Could you tell me where I am wrong?

Best regards.
sandf

Last edited:
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nrqed

Homework Helper
Gold Member
1. Homework Statement

Two-electron Wavefunction: ψ(r1,r2,r12) = exp(-Ar1-Br2-Cr12), r12 = |r1-r2|
A, B, and C are coefficients

Calculate <ψ|δ(r12)|ψ> and <ψ|δ(r1)|ψ>
2. Homework Equations

NO
3. The Attempt at a Solution

<ψ|δ(r12)|ψ>

= ∫∫dv1dv2ψ2(r1,r2,r12)δ(r12)

=4π∫dr2ψ2(r2,r2,0)

<ψ|δ(r1)|ψ>

= ∫∫dv1dv2ψ2(r1,r2,r12)δ(r1)

=4π∫dr2ψ2(0,r2,-r2)

Could you tell me where I am wrong?

Best regards.
sandf
To be precise, in the last step, one should use that $r_{12} = |r_1-r_2|= r_2$ upon setting $r_1=0$. But other than that, everything is good, one simply has to calculate the integrals which are easy.

"Calculus for delta function based on wave function"

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