# Solving Hydrogen Wavefunctions with Coulomb Potential

• gosuly
In summary, to find out for what values of the orbital quantum number and magnetic quantum number the given wave function is an eigenfunction of the Hamiltonian, you will need to know the dependence of R and Y on the l and m quantum numbers, specifically in relation to the Coulomb potential.
gosuly
Hello I have a question if someone could help I would be grateful.

We have a wave function $$\psi(r,\theta,\varphi) = R_{5,4}Y_{l,m} + R_{5,3}Y_{3,0} + R_{5,2}Y_{2,1}$$ which is describing a hydrogen atom.

I have to find out for what values of the orbital quantum number and magnetic quantum number the above wave function is an eigenfunction of the Hamiltonian.

How can I find out if i only know that the atom is in Coulomb potential? Do i have to know the dependence of R and Y of l and m quantum numbers?

Thank You !

Yes, you will need to know the dependence of R and Y on the l and m quantum numbers in order to determine which values of l and m yield an eigenfunction of the Hamiltonian. In particular, you need to know how these functions depend on the potential, which is given by the Coulomb potential in this case.

## 1. What is the Coulomb potential?

The Coulomb potential is an electrostatic potential that describes the interaction between two electrically charged particles. It is given by the equation V(r) = k(Q1Q2)/r, where k is the Coulomb constant, Q1 and Q2 are the charges of the particles, and r is the distance between them.

## 2. Why is it important to solve hydrogen wavefunctions with Coulomb potential?

Solving hydrogen wavefunctions with Coulomb potential allows us to accurately predict the behavior and properties of the hydrogen atom. This is important in understanding the fundamental principles of quantum mechanics and in various applications such as spectroscopy and materials science.

## 3. What is the Schrödinger equation and how does it relate to solving hydrogen wavefunctions with Coulomb potential?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system. It relates to solving hydrogen wavefunctions with Coulomb potential as it allows us to determine the energy levels and wavefunctions of the hydrogen atom by taking into account the Coulomb potential.

## 4. Can the Coulomb potential be solved analytically?

Yes, the Coulomb potential can be solved analytically using the Schrödinger equation. This results in a set of solutions known as the hydrogen wavefunctions, which describe the different energy levels and spatial distributions of the electron in the hydrogen atom.

## 5. Are there any limitations to solving hydrogen wavefunctions with Coulomb potential?

While solving hydrogen wavefunctions with Coulomb potential is a useful tool, it does have some limitations. For example, it does not take into account the effects of relativity and electron-electron interactions, which may become important for heavier atoms. Additionally, the solutions for more complex systems may not have simple analytical solutions and may require numerical methods.

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