Radial Wave Function for Hydrogen

Hydrogen 1s Radial Probability

Probability of Finding Electron in a given Volume

( ) φdθdθsindrrφ,θ,rΨP 22

V∫=

( ) ( ) φdθdθsindrrφ,θ,rΨdrrP 22π

0θ

π2

0φ lnlmnl ∫ ∫= ==

( ) ( ) drrπ4φ,θ,rΨdrrP 22nl =

The Most Probable RadiusHydrogen Ground State

The radial probability density for the hydrogen ground state is obtained by multiplying the square of the wavefunctionby a spherical shell volume element.

It takes this comparatively simple form because the 1s state is spherically symmetric and no angular terms appear.

Dropping off the constant terms and taking the derivative with respect to r and setting it equal to zero gives the radius for maximum probability.

which gives

where

The most probable radius is the ground state radius obtained from the Bohr theory. The Schrodinger equation confirms the first Bohr radius as the most probable radius but goes further to describe in detail the profile of probability for the electron radius.

The Expectation Value for RadiusHydrogen Ground State

The average or "expectation value" of the radius for the electron in the ground state of hydrogen is obtained from the integral

This requires integration by parts. The solution is

All the terms containing r are zero, leaving

It may seem a bit surprising that the average value of r is 1.5 x the first Bohr radius, which is the most probable value. The extended tail of the probability density accounts for the average being greater than the most probable value.

Probability for a Range of RadiusHydrogen Ground State

Finding the probability that the electron in the hydrogen ground state will be found in the range r=b to r=c requires the integration of the radial probability density.

This requires integration by parts. The form of the solution is

Hydrogen 2s Radial Probability

Hydrogen 2p Radial Probability

Hydrogen 3s Radial Probability

Hydrogen 3p Radial Probability

Hydrogen 3d Radial Probability

The Principal Quantum Number

The principal quantum number or total quantum number n arises from the solution of the radial part of the Schrodinger equation for the hydrogen atom. The bound state energies of the electron in the hydrogen atom are given by

In the notation of the periodic table, the main shells of electrons are labled K(n=1),L(n=2),M(n=3), etc. based on the principal quantum number.

The Orbital Quantum Number From constraints on the behavior of the hydrogen wavefunction in the colatitudeequation arises a constant of the form

where n is the principal quantum number. This defines the orbital quantum number, which determines the magnitude of the orbital angular momentum in the relationship

This relationship between the magnitude of the angular momentum and the quantum number is commonly visualized in terms of a vector model. The orbital quantum number determines the bounds on the magnetic quantum number. The orbital quantum number is used as a part of the designation of atomic electron states in the spectroscopic notation.

The Magnetic Quantum Number From the azimuthal equation of the hydrogen Schrodinger equation comes a quantum number with the constraint

While the azimuthal dependence of the wavefunction only requires the quantum number to be an integer, the coupling to the colatitude equationfurther constrains that integer to be less than or equal to the orbital quantum number. The direct implication of this quantum number is that the z-component of angular momentum is quantized according to

It is called the magnetic quantum number because the application of an external magnetic field causes a splitting of spectral lines called the Zeemaneffect. The different orientations of orbital angular momentum represented by the magnetic quantum number can be visualized in terms of a vector model.

Spectroscopic Notation

Before the nature of atomic electron states was clarified by theapplication of quantum mechanics, spectroscopists saw evidence of distinctive series in the spectra of atoms and assigned letters to the characteristic spectra. In terms of the quantum number designations of electron states, the notation is as follows: