Quantum Theory of the Atoms

1
Chapter 20

Quantum Theory of the Atoms
2
20-1 Rutherfords Experiment and the Nuclear
Atom ????????????
20-2 Atomic Spectra Bohr Model of Hydrogen
Atom ???? ????????
20-3 De Broglies Postulates and Matter Waves
??????????
20-4 The Uncertainty Principle ?????
20-5 Wave Function and Schrodinger Equation
Borns Interpretation ?????????
20-6 The infinite Potential Well ????????
20-7 Hydrogen Atom and Electron Spin ????????
20-8 Multielectron Atom and Periodic Table
???????
3
20-1 Rutherfords Experiment and the
Nuclear Atom
1. Thomsons model
Ze of an atom distributes uniformly in sphere.
electrons panel in the atom uniformly.
Plum pudding model
4

2. Rutherfords ?particles scattering
experiment
(1907)
?particles
atom
Some of the scattering angles are so large that
it cannot be interpreted with Plum pudding model
5
3. Rutherfords nuclear atom model (1911)
All the positive charge and almost all the mass
of the atom is concentrated in about 10-1510-14
m the volume of the atom.
6
4. Limitation Rutherfords model
?Cannot explain why the structure of atom is
stable .
? Cannot explain why the radiating spectrum is
discrete.
7

20-2 Atomic spectra and the Bohr model of
hydrogen atom

• 1. The experimental laws of hydrogen atom

6563Å
4861Å
4341Å
4102Å
3646Å
8

• J.Balmer found their regularities in 1885

-- Balmer formula
Wave number

• J.R.Rydberg proposed a general formula in 1890

-- Rydberg formula
-- Rydberg constant
R1.096776?107m-1
9
Same k, different n a series line (??)

• (1)k1,n2,3, Lyman series, ultraviolet
• (2)k2,n3,4, Balmer series, visble
• (3)k3,n4,5, Paschen series, infrared
• (4)k4,n5,6, Brackeff series, infrared
• (5)k5,n6,7, Pfund series, infrared

10
----T(k) , T(n) terms of spectrum
----Ritz combination principle
11
2.Bohrs postulates

• In order to support the nuclear model, Bohr
  proposed three postulates for explaining the
  experimental regularities of hydrogen atom.

(1) Stable states postulate
The motion of electron in a circular orbit is
stable
12

• (2) Postulate about the quantization of orbit
  angular momentum

The orbit angular momentum Lmvr of an electron
must satisfy ,
nquantum number
--simplified Planks constant
13
(3) radiation postulate
When an electron charges its orbit, a photon with
frequency is emitted or absorbed.
EngtEk--- a photon is emitted
EnltEk---a photon is absorbed

• Bohr got Noble Prize of physics in 1922.

14
3. The results of Bohrs model (1) The radius of
orbit
According to Newtons law
and
So
--quantization
15
For n1,
r10.53?10-10 m
----first Bohrs radius
(2) energy The energy of hydrogen
atompotentialkinetic
16
using
We get
---quantization

• The quantized energy ---- energy-leverl

• n----quantum number

17
discussion
(1)
For n1,
----ground state
Lowest energy, most stable state
For ngt1
(2)
----excited state
For
(3)
---The ionization energy of hydrogen atom in
ground state is 13.6eV
18
ground
Electronic orbits
Energy-level
19
(No Transcript)
20

• If an electron jumps En?Ek , the H-atom emits a
  photon.

Wave number is
21

• Here

---agree perfectly with the experiment.
Rexp1.096776?107m-1
Bohrs theory made great success in hydrogen
atoms and hydrogenlike ions (????)
But this theory cannot explain the experimental
results of more complex particles.
22
4. The limitation of Bohrs model
(1) using classical physics theory, but the
electron has acceleration and no radiation is
inexplicable.
(2) No any theory can explain the angular
momentum quantization.
(3) Cannot get the intensity of the spectra from
this theory.
23
20-3 De Broglies postulates and
matter waves

• 1. De Broglies postulates(1924)

• Just as radiation has particle-like properties,
  electron and other material particles possess
  wave-like properties.

---De Broglies formulae

• The waves related to the material
  particles----matter waves

• De Broglie won Noble prize in physics in 1929.

24
2. Testing De Broglie hypothesis

• The wave property of electrons was confirmed by
  Davisson and Germer in experiment in 1927.

E-gun
detector

• They observed that the diffraction patterns of
  the electrons are similar to with the ones of
  x-ray.

--electron have wave property.
Nickle crystal
25

• Meanwhile, Thomsons experiment gave another
  method to confirm the wave property of electrons.

• They won Noble prize in physics in 1937

26

• The results of slits diffraction experiments in
  1960

• It was confirmed by great amount of experiments
  that all micro-particles such as electrons,
  neutrons, proton, as well as atoms and molecules
  have wave properties. And their wave properties
  agree with De Broglies formulae.

27
Example an electron is accelerated by electric
field. If the Acc. potential is U, the electron
is at rest before it is accelerated and its speed
VltltC after it is accelerated. Find its wavelength
?
Solution
28
Scanning electron microscope---SEM
---observe the microscopic morphology.
Mosquito Head with X 1,000
29
A shell of a radiolarian a single-celled animal
with X 2,000
30
Transmission electron microscope --TEM
---study the microstructure.
a leaf of a green plant.
31
20-4 The Uncertainty Principle

• For microscopic particles their position ,
  velocity and others cannot be determined by
  actual experiment at same instant because of wave
  properties.

32

• The single slit diffraction of electrons

A beams of electrons is incident on a slit
normally,
considering the central bright fringe only,
The first dark fringe satisfies,
33
Considering all diffraction fringes,
--estimated result

• Hesbreg deduced the precise results in 1927.

• Hesbreg won Noble prize in physics in 1932.

34

• Any experiment cannot determine simultaneously
  the exact values of a component of momentum and
  its corresponding coordinate.

---Coordinate-momentum uncertainty relation

• Similarly, measuring the energy of a particle in
  a time interval ?t, the uncertainty ?E of the
  energy is

35
Example according to classical electromagnetism
and mechanics, the speed of the electron moving
around its nuclear is about 106 m/s. the size of
an atom is 10-10m. Estimate the uncertainty of
the speed.
Solution the uncertainty of the position of
the electron is
?v ? v. the uncertainty of the speed of the
electron in atoms is very apparent.
36
20-5 The wave function and Schrodinger
equation Borns interpretation of wave
function

• 1.wave function
• A classical plane wave function travels in x
  axis,

• Write it in complex form,

• Similarly, for a free particle with energy E and
  momentum p, its wave function,

37
2. Schrodinger equation

• Free particlemoves in x axis

---S-Equation of a free particle
38

• The particle has potential energy V(x,t), its
  total energy is

i.e.
39

• If a particle moves in three dimension space,

• Introducing energy operator,

--Hamiltonian operator
then
40

• Stationary state potential is independent on time

Let
41
Left side of the equation is the function of x,
and the right side is the function of t,
If it is useful for any t or any x, it must
satisfies,
left side right side a constant
42
i.e.
?(1)
?(2)
the solution of Eq. (1) is
E has the demission of energy.
43
Rewriting (2),
or
----Schordingers equation in stationary state

• The wave function of the particle is

44

• 3.The physical meaning of wave function

---Borns interpretation of wave function
? --- complex function
no physical meaning

• M.Born postulated

represents probability density,
45
? continuous, single-valued and finite.
---Standard conditions
And ?
---Normalized condition
46
? The superposition principle
i.e
then
is also the solutions of the S- equation.
Here, c1 and c2 are normalized factor.
47
? The conservation of probability
---probability density
---probability current density
From Sch. Eq., we can get
---probability is conservative.
48
4. Operators and physical observables
The value of a physical observable
Such as
Energy operator is
So
momentum operator is
So
49
If the particle is in stationary state,
-- probability density has nothing to do with t.

• Schrodinger won Noble prize in 1933.

50
20-6 The infinite potential well

• A particle with mass m moves along x axis. It
  potential is

Out well
?
In well
51
Let
Its general solution is
C?? ---integral constants
Using the continuous condition of wave function,
52
Using normalized condition of wave function,
We get
--quantized energy
nquantum number
53

• Notes

? n1
-- zero point energy
-- the lowest energy of a particle cannot be zero
in quantum mechanics.
54

• ?For an eigen state ?n with quantum number n,
  there are n1 nodes and n antinodes in the well.

55
Example A particle with mass m locates in a
infinite potential well with length a. Calculate
? the probability of finding the particle in the
range of 0?x?a/4 for the two different states
with n1 and n?. ? the positions of maximum
probability for the state n2.
Solution ? As
? the probability of finding the particle in the
range of 0?x?a/4 is
56
For
For
57
?
Let
We get
58
Tunneling effect

• The potential distribution is

A particle move along x axis,
For classical particle,
59
For macroscopic particle,
Using Schrodinger equation,
Area 1
Area 2
Area 3
60
The solutions of S.Eq. are
Area 1incident waves reflected waves
Area 2transmitted waves reflected waves
Area 3 transmitted waves
No matter EgtU0 or EltU0
---Tunneling effect
61
Application
Scanning Tunnel Microscope (STM)
62
??????
??????? ??????????? -- ???
63

• 1993????STM???????????????

• ??48????????????? 71.3Å?????????????????,????????,
  ??????????????

64
20-7 The Hydrogen atom and Electron spin

• 1.The Schrodinger equation of hydrogen atom

In the hydrogen atom system, the potential energy
is

• Schrodinger equation is

65

• introducing the transitions,

• S.Eq. changes into

66

• Let

We get
67

• 2.the results of the solution of Schrodinger
  equation for H-atom

?Quantized energy and principle quantum number n.
The energy of H-atom system can be gotten from
Eq. (3)
---- principle quantum number n.

• It is agrees with Bohrs result.

68

• ? Quantized angular momentum and azimuthal
  quantum number l.

The magnitude of orbital angular momentum about
the electron moving around the nuclear can be
gotten from Eq.(2)
--azimuthal quantum number(????)
l has n possible values for a given value of n.
69

• ? The direction of orbital angular momentum about
  the electron moving around the nuclear can be
  gotten from Eq.(3)

---magnetic quantum number(????)
The component Lz of L along z axis is
ml has (2 l1) possible values for a given value
of l .
70
3. The electron cloud
The distribution of the probability density of
finding the electron.
71
4. Electron spin

• ?Unlenbeck and Goudsmit arranged a experiment to
  check whether the orientation (??) of is
  quantized (1921)

72

• The results of the experiment
• No magnetthere is one track of the atoms on the
  film.
• With magnet there are two tracks of the atoms on
  the film.

So the number of the tracks of the atoms should
be a odd number.
73

• ?. In order to explain the experimental results,

Unlenbeck and Goudsmit proposed a postulation in
1925
An electron not only revolves around a nucleus
but has a spin. This is analogous that the earth
revolves around the sun and meanwhile rotating
about its own axis.
74
here
----spin quantum number
75

• 5. Four quantum numbers (n, l, ml, ms)
• ---determine the state of an electron.
• (1) Principle quantum number n(n1,2,?)---determin
  e the energy of the atom system.
• (2) azimuthal ?? l(l0,1,2,?,n-1)---determine
  the magnitude of orbital angular momentum of an
  electron.

(3) magnetic ?? ml(ml0,?1,?2,?,?l)---determine
the orientation of orbital angular momentum of an
electron.
(4) Spin ?? ms(ms?1/2)---determine the
orientation of spin angular momentum of an
electron.
76
20-8 Multi-electron atoms and
The Periodic table

• Multi-electron

Average field theory
Ze protons (Z-1) electrons forms a uniform
field.
Hydrogenlike (??) motion
77
Conclusions
?The state of each electron can be specified by
n, l , ml, ms .
?Pauli exclusion principle(???????)no two
electrons in any one atom can have the same four
quantum numbers n, l , ml , ms .
Corresponding same n, l , ml ,
ms has two values
i.e. only 2 electrons with same n, l , ml .
78
Corresponding same n, l ,
ml has the values ml0,?1,?2,?,?l
i.e. 2(2l1) electrons with same n, l .
The electrons which have same l are said to
belong to a subshell (???).
Corresponding l 1, 2, 3, 4, 5, 6 ? With
code letters s, p, d, e, f, g, ?to specify.
79
So the maximum number of the electrons with same
n is
The electrons which have same n are said to
belong to a shell(??).
Corresponding n 1, 2, 3, 4, 5, 6 ? With
code letters K, L, M, N, O, P ?to specify.
80
? minimum energy principleIf an atom is in
ground state, the electrons must fill in such way
as to minimize the total energy of the atom.
Minimum energy corresponds the minimum values of
n, l .
the configuration of an atom is represented by
symbol
81
For example,
H-atom has one electron,
its configuration is represented by H 1s1
Helium-atom has 2 electrons,
its configuration is represented by He 1s2
Lithium-atom has 3 electrons,
its configuration is represented by Li 1s2 2s1
Beryllium-atom has 4 electrons,
its configuration is represented by Be 1s2 2s2
82
Boron-atom has 5 electrons,
its configuration is represented by B 1s2 2s2
2p1
?
Neon-atom has 10 electrons,
its configuration is represented by Ne 1s2
2s2 2p6
Sodium-atom has 11 electrons,
its configuration is represented by Na 1s2 2s2
2p6
?
83
It is a great triumph for atomic theory to
understand The periodic Table of the Elements in
terms of atomic levels (or quantum states).