Complex Atoms
4 Quantum numbers
Several quantum numbers: n, l, ml, s, ms
Lifting the l -degeneracy
Magnetic effects
Spin
Otto Stern
Pieter Zeeman
Hendrik Lorentz
The Helium atom; semi-classical approximation
a
Without repulsion:
a/2
a/2
E  2E1  2Z 2 E0  8  13.6eV  108.8eV
With repulsion; electrons at fixed distance ~a0,
Repulsive energy:
e2
4 0a0
Calculated:
Experimental:
 c
Etot  81.6eV
Etot  79eV
mec

 2 E0  27.2eV
Intermezzo
Complex Atoms and the Schrödinger Equation

+2e

2
2e 2
2e 2
e2
2
2
H 
1   2 


2me
4 0 r1 4 0r2 4 0r12
Ekin
Two-particle Schrödinger equation:
attraction
NB:
A non-central
force
repulsion
 
 
Hr1, r2   Er1, r2 
First approximation: unperturbed system (no repulsion)
H 0  H1  H 2
2
2 2
2e 2
Z
4
H




Each:
has solution
i
i
E


R


R
i
2me
4 0 ri
n12
n12
 


r1, r2    nlm r1  nlm r2   1 2
Wave function:
 
H 0  r1, r2   H1  H 2  1 2  H1 1  2  1H 2 2 
 E1 1 2  E2 1 2  E 1 2
Hence:
 1
1 
E  4 R 2  2 
 n1 n2 
General rule: complex atoms  Product wave functions of hydrogen orbitals
Magnetic Dipole Moments; Space quantization
Stern-Gerlach experiment
-nonuniform magnetic field
-classically a continuum of deflection angles is expected
Instead, the angles were quantized, corresponding to the
quantized values of the magnetic moment.
dU
dBz
Fz  
 z
dz
dz
General for a magnetic dipole:
-Force on a dipole in a gradient
-Orientation of a dipole in a
Homogeneous B-field
Otto Stern
The Nobel Prize in Physics 1943
"for his contribution to the development
of the molecular ray method and his
discovery of the magnetic moment of the proton"
Magnetic Dipoles
Potential Energy of a magnetic dipole
 
U    B
Choose the magnetic field along z;
U   z Bz
Force = gradient of the potential:
dU d z Bz
dBz
Fz  

 z
dz
dz
dz
1. Only a force if there is a magnetic field gradient
2. In a homogeneous magnetic field
Stern-Gerlach
space quantisation
Zeeman effect
Both effects related to quantisation of angular momentum
Intermezzo
Angular momentum and the unit of the Bohr magneton


-e
+Ze
v

L
Current (charge passing per time)
e ev

T 2r
Magnetic moment
evr 2
  IA 
2r
Angular momentum
I
L  me vr
Hence
e 

L
2me

Define
B 
e
2me
Bohr magneton
We find a relation between
the angular momentum and the
magnetic moment of a classical
point charge

L

   B

Hydrogen Atom:
Schrödinger Equation and Quantum Numbers
If no magnetic field: energy does not depend on m
ml degeneracy
In a magnetic field
 
U    B   z Bz
 z   B
Lz

For atomic wave functions;
Lz   m
Hence splitting
U   B ml B
Hydrogen Atom: The Zeeman effect
In a magnetic field, the energy levels
split depending on m .
For an 1P  1D-transition
m2
m 1
m0
m  1
m  2
2
1902
U  g L  B B  ml
m 1
m0
m  1
 1
Dm=1
Dm=-1
Dm=0
9 transitions
3 spectral lines (in hydrogen)
Pieter Zeeman
Hendrik Lorentz
"in recognition of the extraordinary service
they rendered by their researches
into the influence of magnetism
upon radiation phenomena"
Normal Zeeman effect
For an 1S  1P-transition
Energy splitting for an nl state
m
m   1
n
nm
m  
Energy splitting:
Em  g B M
s sz s
Recall selection rules for m
and polarization of light
Electron spin
No classical analogue for this phenomenon
s
Origin of the spin-concept
-Stern-Gerlach experiment;
space quantization
1

2
There is an additional “two-valuedness”
in the spectra of atoms, behaving like
an angular momentum
This may be interpreted/represented
as an angular momentum
For a (relativistic) spin

S

S   g S B

-Theory: the periodic system requires
an additional two-valuedness
gS 2
This is peculiar !
Hydrogen Atom:
Schrödinger Equation and Quantum Numbers
The spin quantum number, ms, for
an electron can take on the values
+½ and -½. The need for this
quantum number was found by
experiment; spin is an intrinsically
quantum mechanical quantity,
although it mathematically
behaves as a form of angular
momentum.
An additional quantum number !
Hydrogen Atom:
Schrödinger Equation and Quantum Numbers