Calculus for delta function based on wave function

[sandf]

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>

Homework Equations

NO

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

 

[nrqed]

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>

Homework Equations

NO

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.