Hydrogen atom
quantum numbers
l  0,1,..; m  0,.., l ; n  l  1,..
energy spectrum
E1
En  2
n
l 0
m0
l 1
m  1,0,1
E1  13.6 eV
l 2
m  2, 1,0,1,2
1
Li atom
in other atoms energy levels En depend on angular momentum
Li: 3 electrons: (1s)2 (nl )1
(Li core)+ + 1 electron in En state
(Li core)+ = Li3+ nucleus + 2 electrons in 1s state
En
 high energy levels (n large) are hydrogen-like (i.e. degenerate)
2
 electrons in low energy levels penetrate the core: lowers their energy
other than H atoms (He,Li,etc.)
same quantum numbers l  0,1,..; m  0,.., l ; n  l  1,..
energy spectrum
l 0
Enl
l 1
function of n and l
l 2
4p
3d
4s
energy
3p
3s
2p
2s
m  l ,..., l
levels are
degenerate
result of spherical symmetry
3
1s
(normal) Zeeman effect (1902)
place atom in magnetic field
no magnetic field
breaks the symmetry
lifts the degeneracy
magnetic field
m 1
energy
m0
p-level
m  1
transitions
m0
s-level
optical spectrum
4
the “Dutch” angular momentum connection
Lorentz
1853-1928
Zeeman
1865-1943
5
classical model (Lorentz, 1902)
charged particle in orbit
L  r  mv
magnetic dipole
q

L
2m
r
charge
mass
energy in a magnetic field
q
m
Emag
v
q
 B    
B L
2m
6
(normal) Zeeman effect (1902)
no magnetic field
magnetic field
m 1
En
energy
m0
p-level
En  BBm
m  1
m0
s-level
7
angular momentum horror
in spherical coordinates

Lˆz 
i 
Lˆ  Lˆ  Lˆ  Lˆ  
2
2
x
2
y
2
z
2
 1  
 
1 2 
 sin   sin    sin2   2 




we had before
 
1 2 
 1 1  
  r 2  sin   sin    sin2   2 





1   2   Lˆ2
 2
r
 2 2


r r  r 
r
1   2 
2  2
r

r r  r
Hamiltonian in spherical coordinates

Hˆ  
  V (r )  
2m
2mr 2 r
2
2
2
 2 
 r r

radial
kinetic energy
Lˆ2 
 
  V (r )  2mr 2 
 

centrifugal
kinetic energy
8
the angular momentum ladder
eigenfunctions | lm
ml
m  l 1
operations:
L̂
L̂
Lˆ2 | lm 
2
l (l  1) | lm
Lˆz | lm  m | lm
L̂
L̂
Lˆ | lm  Alm | lm  1
Lˆ | lm  Blm | lm  1
L̂ L̂
m  l  1
enable to calculate everything !!
no. steps bottom to top = 2l ,
L̂ L̂
must be integer !!
m  l
9
(normal) Zeeman effect (1902)
orbital angular momentum
no magnetic field
l = 0,1,... integer!
magnetic field
m 1
energy
m0
p-level
m  1
transitions
m0
s-level
then no. lines is odd !!
10
(anomalous) Zeeman effect
optical emission of Na atoms (street lights)
no magnetic field
magnetic field
no. lines even
l is half-integer ??
11
the “Dutch” angular momentum connection
Lorentz
1853-1928
Zeeman
1865-1943
orbital angular momentum
ℓ is integer
Goudsmit
1902-1978
Uhlenbeck
1900-1988
but there exists another
angular momentum
s
1
2
electron spin
12