What does the equation for the radial wave function look like if we assume that the electron is an unchargedparticle

The radial wave function for an electron in a hydrogen atom is given by the equation:

R(r) = R_nl(r) = sqrt[(2/n)^3 * (n-l-1)!/(2n*(n+l)!) ] * (2r/n)^l * L^(2l+1)_{n-l-1}(2r/n) * e^(-r/n)

where:

• R(r) is the radial wave function
• n is the principal quantum number
• l is the azimuthal quantum number
• r is the distance of the electron from the nucleus
• L^(2l+1)_{n-l-1}(2r/n) is the associated Laguerre polynomial
• e^(-r/n) is the exponential decay factor

However, if we assume the electron is an uncharged particle, it would not be influenced by the electric field of the nucleus. This means that the potential energy term in the Schrödinger equation, which is responsible for the interaction between the electron and the nucleus, would be zero.

In this case, the radial wave function would be the solution to the Schrödinger equation without the potential energy term. This would be a free particle wave function, which is a plane wave solution:

ψ(r) = A * e^(ikr)

where:

• ψ(r) is the wave function
• A is the normalization constant
• k is the wave number
• r is the position of the particle

This is a much simpler function, but it does not describe a realistic electron in an atom because it does not account for the electron-nucleus interaction.