Lecture 10
1
The Hydrogen Atom
Rutherford’s model:
A single electron moves around a single proton (or nucleus)
in a circular orbit.
Assuming that the proton does not move, due to its large mass
compared to the electron, the energy of the system is given by
We also have
1
e2
2
E  K  P  m0 v 
2
4 0 r
e2
2
v2
1
1
e
2

m

m
v

0
0
2
4 0 r
r
2
2 4 0 r
e2
v
4 0 m0 r
e2
r
4 0 m0 v 2
Eq. 1
Eq. 2a
Eq. 2b
2
Energy of the Atom
Therefore, we have
1 e2
P
E
 K 
2 4 0 r
2
Since
v   r  2 fr
we have
and v from Eq. 2b
2
e
r
4 0 m0 v 2
from Eq. 2b
Eq. 3
1
e2
f 
2r 4 0 m0 r

4

2 1/ 3
1 m0e f
E
2
 02 / 3
3
Reading Assignments
• Chapter 4, sections 4-1 and 4-3.
4
Radiation from the Electron
According to the classical electromagnetic theory, the electron
in a hydrogen atom produces electromagnetic radiation, just
like an oscillating electric dipole, with the frequency of the
radiation corresponding to that of the orbital motion f,
1
e2
f 
2r 4 0 m0 r

4

2 1/ 3
1 m0e f
E
2/3
2
0
5
“Classical Dilemma”
As the electron loses energy, its orbit shrinks and the frequency of
radiation increases, until the electron crashes onto the proton.
• The spectrum of the radiation should be continuous, with
the frequency goes as
f  r 3/2
• The atom is unstable, with the electron eventually crashes
onto the proton.
Rutherford’s model implies an unstable
atomic structure and is, therefore, wrong!
6
Spectrometer
Spectrometer is an instrument that is capable of measuring the
wavelength (or frequency or energy) of each incident photon.
7
Spectrum
A spectrum is simply a histogram of photons binned by
their wavelengths (or frequencies or energies).
8
Spectral Components
•
Continuum: smooth, featureless part of a spectrum
•
Emission line: a discrete feature above the continuum that
is localized at a certain wavelength
•
Absorption line: a discrete feature below the continuum
that is localized at a certain wavelength
Observations show that each element has its own set of
characteristic spectral lines and that the formation of the
lines depends strongly on the physical conditions of the
emission region such as temperature and density.
9
Spectral Lines
10
Hydrogen Lines
Balmer series:
m2
m  364.6 2
nm, where m  3, 4,5,
m 4
OR
1 
 1
 RH  2  2 
m
2 m 
1
where
RH  109677.5810 cm 1 is the Rydberg constant
11
Hydrogen Lines Series
Rydberg-Ritz formula:
1 
 1
 RH  2  2  , where n  m

n m 
1
Balmer series is only a special case where n = 2
Other common series:
Ritz Combination Principle:
•
•
•
•
•
The difference of the frequencies of
two lines in a line series is equal to the
frequency of a spectral line which
actually occur in another series from
the same atomic spectrum.
Lyman series: n=1
Balmer series: n=2
Paschen series: n=3
Brackett series: n=4
Pfund series: n=5
12
Bohr’s Postulates
• The classical equations of motion are valid for electrons in
atoms. However, only certain discrete orbits with the
energies En are allowed. These are referred to as the energy
levels of the atom.
• The motion of the electrons in these quantized orbits is
radiation less. An electron can be transferred from an orbit
with lower (negative) binding energy Em (i.e., larger r) to
an orbit with higher (negative) binding energy En (smaller
r), emitting photons in the process. The frequency (or
wavelength) of the photons is given by
Em  En  hf 
hc

13
Comparison with Experiments
To obtain the Rydberg-Ritz formula for hydrogen atoms, Bohr
identified the energy terms as
Rhc
Rhc
En   2 , Em   2
n
m
where n and m are referred to as principal quantum numbers.
It is shown, using the first postulate, that the orbital radius
is proportional to the principle quantum number squared.
• With increasing orbital radius r, the laws of the quantum
atomic physics approaches those of classical physics, i.e.,
Correspondence Principle.
14
Classical Limit
Considering the emission of photons according to the first
two postulates for a transition between neighboring orbits,
i.e., for m – n = 1, and for large n (or m). We get
1
 1
hf  Rhc  2  2
m
n
2n  1
 Rhc 2
n ( n  1) 2
For large n, we have
1 
 1

  Rhc  n2  ( n  1) 2 



2Rc
f 3
n
15
Comparison with Classical Physics
Applying the correspondence principle, for very large n (or
orbital radius), we know that the frequency of the emitted
photons is equal to that of the orbital motion.
Also, according to the classical theory, the total energy of
an electron is given by
1  m0 e f
E
2
 02/3
4

2 1/3
Plugging in the expression for f and comparing it to the
formula in quantum theory (i.e., Bohr’s second postulate),
16
Deriving Rydberg Constant
We have
Rhc 1  m e 

4
0
n2
finally,
2
 02 / 3
1/ 3
 2 Rc 
 n3 


2/3
m0 e4
R  2 3  109737.318 cm1
8 0 h c
which is slightly different from
RH  109677.5810 cm1
17
Homework set #5
Due date Feb.21. 2014.
Tipler 6th Edition, Modern Physics, page 148,149,150, 186
Chapter 3. Problems: 3-33
3-38
3-39
3-58
Chapter 4. Problems 4-8
4-9
4-10
18