Concerning spherical potential wells

[moriheru]

Can one work with spherical potential wells as square wells with an infinite amount of steppotentials of infinitly small size , thus integrating or summing the steppotentials? Would be great bunch of work, treating all the steppotentials and the different energys of the particle I mean for E>V and E<V and so on?

 

[mfb]

What do you mean with "summing the step potentials"? You can get a potential function by summing over multiple other functions - but you cannot get the solution to the corresponding Eigenvalue problem this way.

 

[moriheru]

Sorry this was badly formulated. I meant instead of treating the spherical potential well as a spherical potential well treating it as a bunch of step potentials ie. a aproximation to the spherical potential

 

[mfb]

I don't see how that would help in any way, but a good approximation to a true potential will give good approximations for everything else.

 

[jfizzix]

It's not that terribly difficult to solve the infinite spherical well (U= infinity for r>R, and zero otherwise) outright.

First, one would rewrite Schrödinger's equation in spherical coordinates.

The angular variables separate out, so that the energy eigenfunctions are radial functions (of n and L) times spherical harmonics. (of L and Lz)

The radial equation is a bit more challenging, but one can show that the radial solutions (which depend on n and L) are spherical Bessel functions.

Just as with the ordinary particle-in-a-box, one can use the boundary conditions to fix the scale of these spherical Bessel functions, and obtain the energy eigenvalues. Here, the energy eigenvalues don't go up as the square of positive integers, but as the square of the positive zeroes of the spherical Bessel function of order L, where L is the angular momentum quantum number.

Hope this helps:)

 

[moriheru]

It's not that terribly difficult to solve the infinite spherical well (U= infinity for r>R, and zero otherwise) outright.

First, one would rewrite Schrödinger's equation in spherical coordinates.

The angular variables separate out, so that the energy eigenfunctions are radial functions (of n and L) times spherical harmonics. (of L and Lz)

The radial equation is a bit more challenging, but one can show that the radial solutions (which depend on n and L) are spherical Bessel functions.

Just as with the ordinary particle-in-a-box, one can use the boundary conditions to fix the scale of these spherical Bessel functions, and obtain the energy eigenvalues. Here, the energy eigenvalues don't go up as the square of positive integers, but as the square of the positive zeroes of the spherical Bessel function of order L, where L is the angular momentum quantum number.

Hope this helps:)

helps lots thanks...yet why replace the radial part with the sign of the density matrix?

 

[jfizzix]

I think I missed something... what do you mean by the sign of the density matrix? I was just talking about how one could solve the infinite spherical well.

 

[moriheru]

Doesn't matter I meant when one is solving the radial part of the Schrödinger equation in spherical coordinates one replaces kr with rho the sign of the density matrix(I had just forgotten the name of the sign). What about Neumann functions they are mixed in to?!

 

[jfizzix]

I believe Bessel functions and Neumann functions are like sines and cosines as far as both being solutions to their own particular equations (sines and cosines for the simple harmonic oscillator equation). You can represent the solutions as a sum of both Bessel and Neumann functions, but your boundary conditions will fix the constants in that sum.