• Contact Us
• Programming Blog
• Tutoring Services
• Bio

  Search by keyword:  

Last modified: July 20, 2022 ✓

Schrodinger Equation for Hydrogen Atom
--------------------------------------




The electron in the hydrogen atom sees a spherically 
symmetric potential, so it is logical to use spherical 
polar coordinates (r, θ, φ) to develop the 
Schrodinger equation. 

Solving the SE involves separating the variables into 
the form,

ψ(r,θ,φ) = R(r)(P(θ)F(φ)

This leads to the following:

R(r) -> n = 1, 2, 3 
 
This is the PRINCIPAL QUANTUM NUMBER and corresponds 
to the electron shells.

P(θ) -> l = 0, 1, 2, ..., n - 1 
 
This is the ORBITAL (AZIMUTHAL) QUANTUM NUMBER and 
corresponds to the magnitude of the orbital angular 
momentum, L, given by:

   |L|  = √l(l + 1)h 

The orbital quantum number is used as a part of 
the designation of atomic electron states in the 
spectroscopic notation. 

s (sharp)          l = 0
p (principal)      l = 1
d (diffuse)        l = 2
f (fundamental)    l = 3

Thus n = 2, l = 1 is designated as 2p.

F(φ) -> ml = -l, -l + 1, ..., l 

This is the MAGNETIC QUANTUM NUMBER and corresponds 
to the quantization of the z component of angular 
momentum.  It is given by:

  |Lz| = mlh

The application of an external magnetic field 
causes a splitting of spectral lines called the 
Zeeman effect.  The orbital quantum number plays 
a role in the Zeeman interaction since the orbital 
motion contributes a magnetic moment, and is 
important as an indicator of subshell differences 
in electron energies. 

Electron Shells:

n = principal quantum number
l = angular quantum momentum number = 0 -> n-1  
    (0 = s, 1 = p, 2 = d, 3 = f, 4 = g, ...)
m = magnetic quantum number = -l -> l
s = +/-1/2

n      l            m           s    Total   Name
-      -            -           -    -----   ----
1      0            0          +/-    2 } 2   1s

2      0            0          +/-    2 }     2s
2      1        -1, 0, -1      +/-    6 } 8   2p

3      0            0          +/-    2 }     3s
3      1        -1, 0, -1      +/-    6 }     3p
3      2    -2, -1, 0, -1, -2  +/-   10 }18   3d   

In general:

n = 1 => 1s   
n = 2 => 2s, 2p   
n = 3 => 3s, 3p, 3d 
n = 4 => 4s, 4p, 4d, 4f
n = 5 => 5s, 5p, 5d, 5f, 5g

Wavefunctions:

The wavefunction of the electron is the product 
of 3 functions:

ψ(r,θ,φ) = R(r)P(θ)F(φ)

Generating the wavefunctions for each state is 
extremely math intensive and will not be attempted 
here.  Instead, the results of the these computations 
for different values of n, l and ml are 
shown below:

Newsletter Sign-up:

Copyright © 2022 Redshift Academy

website built by Redshift Academy using the Codeigniter MVC framework