GENERAL PHYSICS III
GENERAL PHYSICS III
Optics
&
Quantum Physics
Chapter XXII
Chapter XXII
Atom
Atom
ic
ic
Structure
Structure
§1. Hydrogen atom
§2. Angular momentum of electron
§3. Electron spin
§4. Many-electron atoms
At present it is well known for you that
The Atom - electrons confined in Coulomb field of a nucleus

Au

v
Rutherford (also Geiger-Marsden)
Experiment (1911):
Measured angular dependence of
particles (He ions) scattered
from gold foil.
The results:
• Mostly scattering at small angles. But…
• Occasional scatterings at large angles

1.1
Potential energy of the electron
Potential energy of the electron
in the hydrogen atom
in the hydrogen atom:
r
e
)r(U
2


r
U(r)
0
229
0
C/Nm109
4
1



2
2
)(
r
e
r
rU
f

m






EzyxU
z
x
y
x
x
x
m











































 )(
sin
1

but must accept a discrete set of values: “the energy levels”.
The energy levels of the electron in the hydrogen atom is found to be
22
o
2
n
n
eV6.13
n
1
a2
e
E





,...3,2,1n
2
2
0
em
a



= “Bohr radius” = 0.053 nm
-15
-10


2 2
1 1
13.6 eV
i f
n n
i f
E
n n

 
  
 
 
 
we have
3 2
1 1
13.6 eV 1.9eV
9 4
photon
E E

 
   
 
 
1240eV nm
656nm
1.9eV

E
1
E
3

From
eVEEEE
n
6.130
11


Note: The ionization energy of atomic hydrogen at its ground state is
1.4 The eigenfunctions and physical interpretation:
Due to the spherical symmetry, the solution to
the SEQ in spherical coordinates is found
by the method of separation of variables,
and has the form:
x
y
z
r


),()(),,(

lmnlnlm
YrRr 
with quantum numbers: n l and m
principal orbital magnetic

0
0
r
R
20
10a
0
0
0
0.5
R
10
0
0
4a
0
0
3/
2
00
0,3
3
2
2
3)(
ar
e
a
r
a

1)(










)r(RE)r(R
r
e
r
r
r
1
m2
0nn0n
2
2
22








“2s state”
“1s state”
Probability density of electrons:
P(r) is plotted below for the two s-states with n=1,2:
The horizontal axis is in (r/a
0
) units. At
r = a
0
(the Bohr radius) the probabilty
of finding the electron is highest for 1s-state.
The angle dependent part of the eigenfunctions:
The investigation of the
angle dependent part
of the eigenfunctions of
the electron in the hydrogen atom gives the following results:
This part can be represented in the following form:
,)(


im
m
e
)(cos.sin)(

ml
m
lm
F
where



im
eY 
→ the “p-states” (l=1)
§2.Angular momentum of electron:
 Angular momentum L
Z
around the z axis depends on how fast
the phase changes as you rotate around the z-axis.
The phase change is expessed by the factor exp(im) in the angle
dependent part of the eifenfunction
 A state with an exact value of L
Z
is of the form:
Re()

r
,
where ( ) ( , )
im
Z l m
L m r Y e

 
  


0, , 2 etc
Z

LLml
.2,,0,,2,6
2,1,0,1,2,2
 

Z
LL
ml
(the “p-states”)
(the “d-states”)