Problem with Talmi Transformation

In summary, the conversation discusses the calculation of the Talmi transformation of two 3d harmonic oscillator wavefunctions in spherical coordinates with the quantum numbers n1=0, l1=1, n2=0, l2=1. The formula being used is Psi(r1)*Psi(r2)=Ʃ<n1 l1 n2 l2 λ|n l N L λ><l1 m1 l2 m2|λ 0><l m L M|λ 0>*Psi(r)*Psi(R), where r=(r1-r2)/√2 and R=(r1+r2)/√2. The conversation also mentions a Mathematica notebook with a test case and an extra term
  • #1
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I'm trying to calculate the Talmi transformation of two 3d harmonic oscillator wavefunctions in spherical coordinates with the quantum numbers: n1=0, l1=1, n2=0, l2=1.

For some reason, every time I do it I end up with extra terms.

The formula I am using (from several textbooks) is: Psi(r1)*Psi(r2)=Ʃ<n1 l1 n2 l2 λ
|n l N L λ><l1 m1 l2 m2|λ 0><l m L M|λ 0>*Psi(r)*Psi(R)

where r=(r1-r2)/√2; R=(r1+r2)/√2

I have the 0 as my value for μ, and this is guaranteed to always be true. Is this the right way to do this transformation?

I have attached a Mathematica notebook with my test case.
 

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  • #2
The answer I'm getting is:Psi(r1)*Psi(r2)=2*Ʃ<n1 l1 n2 l2 λ|n l N L λ><l1 m1 l2 m2|λ 0><l m L M|λ 0>*Psi(r)*Psi(R)+Ʃ<n1 l1 n2 l2 λ|n l N L λ><l1 m1 l2 m2|λ 0>*Psi(r)*Psi(R).The extra terms appear to be coming from the second sum, and I am not sure why they are there. Can anyone help me out?
 

FAQ: Problem with Talmi Transformation

What is the Talmi Transformation?

The Talmi Transformation is a mathematical model used in nuclear physics to describe the behavior of nuclear energy levels. It was first proposed by Yoav Talmi in 1952.

What is the problem with the Talmi Transformation?

The problem with the Talmi Transformation is that it does not accurately predict the energy levels of all nuclei. It is particularly inaccurate for nuclei with a large number of protons or neutrons.

Why is the Talmi Transformation still used if it has problems?

Despite its limitations, the Talmi Transformation is still used because it provides a simple and intuitive understanding of nuclear energy levels. It is also useful for making initial predictions before more sophisticated models are used.

Are there any proposed solutions to the problem with the Talmi Transformation?

Yes, there have been several proposed solutions to improve the accuracy of the Talmi Transformation. These include modifications to the original model and the development of new models based on different mathematical approaches.

How does the Talmi Transformation impact nuclear physics research?

The Talmi Transformation is an important tool in nuclear physics research, as it provides a starting point for understanding the energy levels of nuclei. However, its limitations mean that it must be used in conjunction with other models and experimental data to fully understand nuclear behavior.

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