Normalised Radial Coulomb Wave Function

In summary, the conversation is about finding the non-normalised wave function of an electron near a Hydrogen nucleus using Mathematica. The result is converted from Whittaker functions to generalised Laguerre polynomials. However, when multiplied by the normalisation constant, it is still not normalised. The conversation then goes on to discuss potential solutions and the importance of including the angular part in the calculation for the probabilistic interpretation. The conversation ends with a suggestion to integrate the absolute square of the whole wave function over certain values to find the probability of finding the electron within a specific radius.
  • #1
tomdodd4598
138
13
Hey there,

I used Mathematica to find the (non-normalised) wave function of an electron in the vicinity of a Hydrogen nucleus, and converted the answer from one involving Whittaker functions to one involving generalised Laguerre polynomials. My result is shown below:

LXINrrz.png


This agrees with the documented non-normalised wave function. However, when I multiply by the documented normalisation constant, to get the following wave function:

BFd01RQ.png


It is still not normalised - the integral from 0 to infinity of the absolute square is still not equal to 1.

Does anyone know what I'm doing wrong, and/or how to fix the wave function? Thanks in advance.
 
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  • #2
Which reference did you take for the unnormalized bound-state wavefunction?
 
  • #3
dextercioby said:
Which reference did you take for the unnormalized bound-state wavefunction?

This is the page where I found the function with a Laguerre polynomial (I've rearranged a few terms and added the constants, but otherwise it's the same): http://ampl.github.io/amplgsl/coulomb.html

I think I know where I'm going wrong, however. It seems that the function is only normalised once this radial part is multiplied by the angular part. I guess I could find the angular part, but I was wandering whether there was a way to just normalise this function on its own, so that I can calculate the probability of finding an electron however far from the nucleus, without worrying about the angular coordinates.
 
  • #4
Of course you can find a constant only from the normalization of the radial part alone, but it's not the one you for the probabilistic interpretation. The unnormalized radial distribution can be used to graph it (critical points, inflection, zeros), but the probability is gotten only from the full wavefunction (which includes the angular part and its own normalization constant).
 
  • #5
dextercioby said:
Of course you can find a constant only from the normalization of the radial part alone, but it's not the one you for the probabilistic interpretation. The unnormalized radial distribution can be used to graph it (critical points, inflection, zeros), but the probability is gotten only from the full wavefunction (which includes the angular part and its own normalization constant).

Ah, ok. So if I find the angular function(s), could I integrate the absolute square of the whole wave function over θ from 0 to 2π, over ϕ from -π/2 to π/2, then between two values of radius to find the probability of finding the electron between those radii?
 
  • #6
It's actually in a spherical shell of finite width, because, by the way you've chosen them, the 2 angles vary maximally therefore covering a whole sphere.
 

1. What is a Normalised Radial Coulomb Wave Function?

A Normalised Radial Coulomb Wave Function is a mathematical function that describes the probability of finding an electron in a specific position around a positively charged nucleus. It is used in quantum mechanics to model the behavior of electrons in atoms and molecules.

2. How is the Normalised Radial Coulomb Wave Function calculated?

The Normalised Radial Coulomb Wave Function is calculated using the Schrödinger equation, which is a fundamental equation in quantum mechanics. It takes into account the charge of the nucleus, the mass of the electron, and the distance between them to determine the probability of finding an electron at a given position.

3. What are the applications of the Normalised Radial Coulomb Wave Function?

The Normalised Radial Coulomb Wave Function is used extensively in atomic and molecular physics to understand the behavior of electrons. It is also used in fields such as quantum chemistry, solid-state physics, and nuclear physics to model the electronic structure and energy levels of atoms and molecules.

4. What is the significance of normalization in the Normalised Radial Coulomb Wave Function?

Normalization ensures that the probability of finding an electron within a certain region around the nucleus is equal to 1. This is important because it allows for the accurate prediction of the electron's behavior and helps in understanding the electronic structure of atoms and molecules.

5. Are there any limitations to the Normalised Radial Coulomb Wave Function?

While the Normalised Radial Coulomb Wave Function is widely used and has been successful in explaining many phenomena, it does have limitations. It does not accurately describe the behavior of electrons in high-energy situations, such as those found in particle accelerators. Additionally, it does not take into account the effects of relativity, which can be significant for heavy elements.

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