The atom
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Stationary states for central potentials
Solutions for the hydrogen atom
Magnetic dipole moment of hydrogen
The independent electron model
Ground state of atoms
The Zeeman effect
Spin-orbit coupling
(X-rays and Auger electrons)
(Fluorescence)
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Spherically symmetric potentials
We will look for stationary states describing
the 3d motion of an object in a central potential V(r)
r We will not consider the time dependence
of the wavefunctions (measurements)
Our stationary states will be used to describe
the electron states of the hydrogen atom.
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They are also eigenstates of L and L z .
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The Schrödinger equation
Stationary states: ψ ( r, φ, θ )
1  h 2
– ------- ------
∂
2
2
2
∂
1
1
+ --ψ + ------- ----- L ψ + V ( r ) ψ = Eψ
2 r ∂r
2m  2π
2m 2
∂r
r
but we already have the spherical harmonics
so we can write
ψ ( r, φ, θ ) = R ( r )Y lm ( θ, φ )
and solve the radial equation:
 1  h 2
- ----- – ----- 2π
2m

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 d2 2 d  1

+ ---  – ----- l ( l + 1 ) + V ( r ) R ( r ) = ER ( r )

 d r 2 r d r r 2

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Application: the hydrogen atom
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e 1
V H ( r ) = – ------------ --4πε 0 r
r = distance between the electron and the proton
The radial equation can be use for V H ( r ) provided
that
mp me
m → µ = -------------------mp + me
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The solution
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Hydrogen atom: the results
1. The energy levels:
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E n = – -----E
, E R = 13.6 eV
R
2
n
where n = 1, 2, … is the principal quantum number
2. The radial wavefunction: R nl ( r ) depends on n, l
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probability density for electron P ( r ) = r R nl ( r )
2
2
average position varies roughly as n (application: shells)
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Hydrogen atom : the results (cont’d)
3. The spatial dependence of the wavefunction:
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ψ nlm ( r, φ, θ ) = R nl ( r )Y ( θ, φ ) but ψ nl ( r, θ )
lm
l
4. The parity of hydrogen: Pψ nlm = ( – 1 ) ψ nlm
5. ‘Orbitals’ instead of ‘orbits’
6. Spin: the complete wavefunction is ψ nlm m
l s
7. Degeneracy of energy levels: 2n
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Magnetic dipole moment of hydrogen
In general, µ is due to angular momentum
The hydrogen atom has:
(a) orbital angular momentum, l
(b) spin orbital momentum, s
We thus have ‘two’ magnetic dipole moments:
eh
eh
µ L, z = – -------------- m l = – m l µ B and µ S, z = – -------------- m s = – 2m s µ B
4πm
2πm
e
e
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The independent electron model
We describe many-electron atoms by using an
‘effective potential’ in the Schrödinger equation
- wavefunction: ψ nlm m as in hydrogen atom but
l s
- energy levels: depend on n, l
l = 0, 1, 2, 3, ...
name of the level: s, p, d, f, ...
- notation for electronic energy levels: nl, e.g. 1s
- energy ordering: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, ...
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The independent electron model
Energy levels
E
5p
4d
5s
4p
4s 3d
3p
3s
2p
2s
1s
s and p orbitals ‘go’ closer to the nucleus
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Application of the independent electron model
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The Pauli principle
‘For each set of values for the quantum numbers
n, l, m l, m s , we can have only one electron’
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Applications of the model (i.e.m)
- electronic structure of atoms
- determine the chemical and electromagnetic
properties of the atom
- calculate angular momentum of atom (L-S coupling)
- explain optical transitions
- ground state for light atoms
r cannot be used for the study of X-rays (heavy atoms)
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(Optical) transitions
frequency of emitted/absorbed photon:
1
ν = --- ( E i – E f )
h
selection rule:
∆l = ± 1
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The ground state of atoms
Hund’s rules
1. add spins of valence electrons so that the total spin
is maximized
2. add the orbital angular momenta of valence
h- =  ----h- m is max.
electrons so that Lz = ∑ m l  ----
  L
2π
2π
3. S and J are added to calculate J
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The Zeeman effect
Assume ‘spinless’ valence electron with n=2, l=1 in
a magnetic field (along z-axis)
Interaction energy between the magnetic moment and
the magnetic field: U = – µ L, z B = m l µ B B
field off
E2
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field on
E2 + µB B
E2 - µ B B
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Spin-orbit coupling
The nucleus produces a magnetic field that
interacts with the spin magnetic dipole
moment of the electron
U = – µ S, z B = 2m s µ B B = ± µ B B
2
2
2
Electron states: eigenstates of J , L , S , J z (i.e.m.)
notation:
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2s + 1
Aj
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