Calculus for delta function based on wave function

In summary, the conversation discusses the calculation of <ψ|δ(r12)|ψ> and <ψ|δ(r1)|ψ> for a two-electron wavefunction, where A, B, and C are coefficients. The integrals are calculated using the given equations and it is determined that the only mistake made was not setting r12 = r2 in the last step.
  • #1
sandf
21
0

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
 
Last edited:
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  • #2
sandf said:

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.
 

1. What is the delta function in calculus?

The delta function is a concept in calculus that represents an infinitely tall and thin spike at a specific point on a graph. It is used to model a point mass or impulse in physics and engineering problems.

2. How is the delta function related to the wave function in quantum mechanics?

In quantum mechanics, the delta function is used as the basis for the wave function, which describes the probability of finding a particle at a certain position. The delta function acts as a "sampling" function for the wave function, allowing us to calculate probabilities at specific points.

3. What is the significance of using the delta function in calculus?

The delta function allows us to model and solve certain problems in physics and engineering that involve point masses or impulses. It also plays a crucial role in quantum mechanics, where it helps us understand the probabilistic nature of particles.

4. How is calculus used to derive the delta function from a wave function?

Calculus is used to integrate the wave function over a small interval around the point of interest. As the interval gets smaller and smaller, the integral approaches the value of the delta function at that point, giving us a way to calculate its value.

5. Can the delta function be used in other areas besides calculus and quantum mechanics?

Yes, the delta function has applications in various fields, including signal processing, fluid mechanics, and probability theory. It is a powerful tool for solving problems that involve sudden changes or impulses.

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