Normalization of the radial part of the spherical harmonics

[Taz]

TL;DR

How can i normalized the radial part of the Harmonic oscillator .

Im trying to solve the equation 62.7 of this numerical on mathematica. Whenever i try to normalized the function it shows function diverges. As the Bessel function contains trigonometry term so it diverges. I don't know how to solve the integral. Can i use the hydrogen atom wavefunction in exp form? But the question is that hydrogen atom equation is solved by laugurre polynomial and my problem contains legendar polynomial

 

[Haborix]

You start by talking about the radial part of harmonic oscillator wavefunctions. Then you bring up using hydrogen radial wavefunctions. Finally, the picture you show us appears to be about particles confined within a sphere. We are going to need a little more clarity about what you're trying to do. Can you type up the exact integral you're trying to compute and the functions being used in that integral?

[edit] @BvU wants the details, too!

 

[BvU]

Hello Taz, !

I am confused:

• your title is somewhat contradictory: Spherical harmonics don't have a radial part as far as I remember
• your summary mentions a harmonic oscillator (can be treated as 3 independent 1D HO) and
• your (badly photographed and barely legible) screen shot is about a particle in a sphere (so no integration beyond the radius is ever at hand).

Which is it to be ?

[edit]Ah, @Haborix noticed, too !

 

[Taz]

Sorry for the inconvenience,.Its the sphere enclosed inside the infinite potential .Once we normalized the wavefunction the spherical harmonic(angular part) solved by kronecker delta but the radial part becomes difficult to solve as it contains bessel function (containing trigonometry terms) . So the wavefunction diverges due to limit from zero to infinity... so, in which way i can normalized this radial part shown in equation 62.7 . The image is poor i will try to replace it but lamda is the radial part and J shows bessel function.I hope its more clear now

 

[BvU]

limit from zero to infinity

Still don't understand. Why not integrate up to the radius ( i.e. a zero point of the Bessel function $j$ -- the $y$ fall off as non-physical ) ?

image is poor i will try to replace it

I'll wait for a PDF or a link to a PDF

 

[Haborix]

I think I may see the problem. Equation 62.7 in the book, taken literally, doesn't make sense. Because you are studying a particle confined to a sphere of radius $R$, the radial wavefunction must go to zero at $R$ and be zero for all $r>R$. It is fair to say $\int_0^\infty dr |u|^2=1$, but only with the understanding that $$u(r)=\begin{matrix}Cj_\ell(kr) && \text{for } r<R\\ 0 && \text{for } r\geq R \end{matrix}.$$ From this definition you can see the second integral of 62.7 is not correct.

 

[Taz]

I am attaching the problem again with my hand written sol which explain what is my target

 

[Taz]

I think I may see the problem. Equation 62.7 in the book, taken literally, doesn't make sense. Because you are studying a particle confined to a sphere of radius $R$, the radial wavefunction must go to zero at $R$ and be zero for all $r>R$. It is fair to say $\int_0^\infty dr |u|^2=1$, but only with the understanding that $$u(r)=\begin{matrix}Cj_\ell(kr) && \text{for } r<R\\ 0 && \text{for } r\geq R \end{matrix}.$$ From this definition you can see the second integral of 62.7 is not correct.

 

[Taz]

The 62. 7 equation is correct as its the radial part only the angular part is solved by kronecker delta. can you send me the link of equation you have written?as its confusing as i always thought that after normalization we can have 1 for same l,m and 0 for different l,m

 

[BvU]

I am attaching the problem again with my hand written sol which explain what is my targetView attachment 266427View attachment 266428 View attachment 266424View attachment 266425View attachment 266426

The screen shots are illegible -- check for yourself !

You are obviously trying to do the best you can, but perhaps this thread by @ZapperZ can help you to get better help, now and in the future:
https://www.physicsforums.com/threa...our-instructor-during-remote-learning.991016/

 

[jtbell]

Because you're new here, you may not know about our LaTeX implementation:

https://www.physicsforums.com/help/latexhelp/

 

[Taz]

The screen shots are illegible -- check for yourself !

You are obviously trying to do the best you can, but perhaps this thread by @ZapperZ can help you to get better help, now and in the future:
https://www.physicsforums.com/threa...our-instructor-during-remote-learning.991016/

Its a real shame that the images are not clear. I will put the clear one

 

[Taz]

Hello,
I am adding new photos .I want to solve the equation 62.7 of this problem. Its radial part diverges for zero to infinity limit. I don't know how they normalized that part. I want to do it in Mathematica

 

[Taz]

Because you're new here, you may not know about our LaTeX implementation:

https://www.physicsforums.com/help/latexhelp/

thankyou

 

[BvU]

A link goes quite a long way too ...

OK, so now we can read that 62.7 is in a context where $C_2$ is concluded to be zero. The argument goes on with $C_1$ for which you want to find the proper normalizing value. The author wisely avoids the normalization issue and focuses on the energy levels.

But you want to make life difficult (Bravo!) and dig in deeper. If we drill to the core, your self-assigned mission is to integrate $$\chi_l^2(r) = {\pi k r\over2} |C_1|^2 J_{l+{1\over 2}}^2 (kr) \qquad\qquad (62.3) $$from zero to any of the points where $$ J_{l+{1\over 2}} (kR) = 0 \qquad\qquad\qquad\qquad\qquad (62.10) $$ and no further as @Haborix and I have tried to point out: beyond that, the potential function is infinite and the wave function is zero.

Should be a breeze for Matlab .



[edit] oh, no: for mathematica. Don't know that program at all
[edit2] but google is everyone's friend and it looks promising

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