Question

Show that the solution to the Schrodinger equation for n=1, l=0, m=0 yield the Bohr radius. Hint: find the most probable value for r.

Answer 1

The Schrödinger equation for single-electron Coulomb systems in spherical coordinates is

This type of equation is an example of a partial differential equation.

the Schrödinger equation is set up starting from the classical energy, which we said takes the form

which we can write as

where

The term is actually dependent only on and , so it is purely angular. Given the separability of the energy into radial and angular terms, the wave function can be decomposed into a product of the form

Solution of the angular part for the function yields the allowed values of the angular momentum and the -component . The functions are then characterized by the integers and , and are denoted . They are known as spherical harmonics. Here we present just a few of them for a few values of .

for , there is just one value of , , and, therefore, one spherical harmonic, which turns out to be a simple constant:

The remaining function is characterized by the integers and , as this function satisfies the radial part of the Schrödinger equation, also known as the radial Schrödinger equation:

The radial parts of the wave functions that emerge are given by (for the first few values of = 1 and =0 ):

where is the Bohr radius

The full wave functions are then composed of products of the radial and angular parts as

The probability of finding the electron in a small volume element of space around the point is

			The electron in a hydrogen atom () is in the state with quantum numbers , and . What is the probability that a measurement of the electron's position will yield a value ? The wave function is
			.