Question

A method for determining the momentum of electrons by utilizing the distribution function of electrons in the ground state of hydrogen atoms is provided. Find the value

Answer 1

Hydrogen Atom Ground State

is a decreasing exponential function of a single variable r

the probability per unit volume of finding the electron at the point (r,θ,φ) is equal to the square of the normalized wavefunction

probability density is normalized such that

some ways ρ(r) does not provide the best description of the electron distribution, since the region around r = 0, where the wavefunction has its largest values, is a relatively small fraction of the volume accessible to the electron.

the radial distribution function

which represents the probability density within the entire shell of radius r, normalized such that

The general solution for

The angular momentum quantum number ` is by convention designated by a code: s for l = 0, p for l = 1, d for l = 2, f for l = 3, g for l = 4, and so on. The first four letters come from an old classification scheme for atomic spectral lines: sharp, principal, diffuse and fundamental

The solutions of the hydrogenic Schrodinger equation in spherical polar coordinates can now be written

n = 1, 2 .. . ` = 0, 1 .. .n − 1 m = 0, ±1, ±2 . .. ± ` where are the spherical harmonics