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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)
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 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) -> m_{l} = -l, -l + 1, ..., l
This is the MAGNETIC (AZIMUTHAL) QUANTUM NUMBER and corresponds to the
quantization of the z-component of angular momentum. It is given by,
|L_{z}| = m_{l}h
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 m_{l} are shown below: