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Wave equation and its
significance
Sandhya. S
29.04.2010
HISTORY OF THE ATOM
460 BC
Democritus develops the idea of atoms
he pounded up materials in his pestle and
mortar until he had reduced them to smaller
and smaller particles which he called
ATOMA
(greek for indivisible)
HISTORY OF THE ATOM
1808
John Dalton
suggested that all matter was made up of
tiny spheres that were able to bounce around
with perfect elasticity and called them
ATOMS
HISTORY OF THE ATOM
1898
Joseph John Thompson
found that atoms could sometimes eject a far
smaller negative particle which he called an
ELECTRON
HISTORY OF THE ATOM
1904
Thompson develops the idea that an atom was made up of
electrons scattered unevenly within an elastic sphere surrounded
by a soup of positive charge to balance the electron's charge
like plums surrounded by pudding.
PLUM PUDDING
MODEL
HISTORY OF THE ATOM
1910
Ernest Rutherford
oversaw Geiger and Marsden carrying out his
famous experiment.
they fired Helium nuclei at a piece of gold foil
which was only a few atoms thick.
they found that although most of them
passed through. About 1 in 10,000 hit
HISTORY OF THE ATOM
helium nuclei
gold foil
helium nuclei
They found that while most of the helium nuclei passed
through the foil, a small number were deflected and, to their
surprise, some helium nuclei bounced straight back.
HISTORY OF THE ATOM
Rutherford’s new evidence allowed him to propose a more
detailed model with a central nucleus.
He suggested that the positive charge was all in a central
nucleus. With this holding the electrons in place by electrical
attraction
However, this was not the end of the story.
HISTORY OF THE ATOM
1913
Niels Bohr
• Proposed that the electrons orbited the nucleus
• The further away the more energy was needed.
• Electrons only occupy orbits of certain energy.
Bohr’s Atom
electrons in orbits
nucleus
Bohr’s Assumptions for Hydrogen
• The electron moves in
circular orbits around the
proton under the influence
of the Coulomb force of
attraction
– The Coulomb force
produces the centripetal
acceleration
Bohr’s Quantum Conditions
• I. There are discrete stable
“tracks” for the electrons.
Along these tracks, the
electrons
move
without
energy loss.
• II. The electrons are able
to “jump” between the
tracks.
Ei-Ef=hf
In the Bohr model, a photon is
emitted when the electron drops
from a higher orbit (Ei) to a lower
energy orbit (Ef).
Bohr’s Model: Energy of the Atom
Orbit
• E =-kee2/(2r)
Elementary charge
Coulomb constant
The negative sign indicates that the electron is bound to
the proton!
Bohr Model: Orbit Radius
• Bohr assumed that the angular momentum of the electron was
quantized and could have only discrete values that were integral
multiples of h/2, where h is Plank’s constant
• mevr=nh/(2p); n=1, 2, 3,…Quantum number (or principal number)
• v=nh/(2p mer)
Bohr Model: Orbit Radius, cont.
• It follows:
2
2 2


me v
ke e
me  n h
k
e
e

 2 
2 2
2
2

r
r  4 me r  r
r
2

 2
h
rn   2 2 n Bohr orbit radius
 4 k e e me 
2
2
Orbital Radii and Energies (for the Hydrogen Atom)
2
kee
E
2r
2
 2

h
rn   2 2 n
 4 k e e me 
2 4
2
 2 k e e me  1

E n  
2
2

h
n

Specific Energy Levels
• The lowest energy state is called the ground state
– This corresponds to n = 1
– Energy is –13.6 eV
• The next energy level has an energy of –3.40 eV
– The energies can be compiled in an energy level diagram
• The ionization energy is the energy needed to completely
remove the electron from the atom
– The ionization energy for hydrogen is 13.6 eV
Energy Level Diagram
• The value of RH from Bohr’s
analysis is in excellent agreement
with the experimental value
• A more generalized equation can
be used to find the wavelengths
of any spectral lines
Generalized Equation
 1
1
 RH  2  2 

 nf ni 
1
– For the Balmer series, nf = 2, ni=3, 4, 5,…
– For the Lyman series, nf = 1, ni=2, 3, 4,…
• Whenever an transition occurs between a state, ni to another
state, nf (where ni > nf), a photon is emitted
– The photon has a frequency f =(Ei – Ef)/h and wavelength λ
Modifications of the Bohr Theory – Elliptical Orbits
• Sommerfeld extended the results to include elliptical orbits
– Retained the principle quantum number, n
– Sommerfeld added the orbital quantum number, ℓ
• ℓ ranges from 0 to n -1 in integer steps
– All states with the same principle quantum number are said to
form a shell
– The states with given values of n and ℓ are said to form a
subshell
BOHR-SOMMERFIELD’S MODEL
According to the Bohr-Sommerfeld model, not only do
electrons travel in certain orbits but the orbits have
different shapes and the orbits could tilt in the presence of a
magnetic field. Orbits can appear circular or elliptical, and
they can even swing back and forth through the nucleus in a
straight line.
Heisenberg realised that
• In the world of very small particles, one cannot measure any
property of a particle without interacting with it in some way
• This introduces an unavoidable uncertainty into the result
• One can never measure all the
properties exactly
Werner Heisenberg (1901-1976)
Measuring the position and
momentum of an electron
• Shine light on electron and detect reflected
light using a microscope
• Minimum uncertainty in position
is given by the wavelength of the
light
• So to determine the position
accurately, it is necessary to use
light with a short wavelength
Measuring the position and momentum
of an electron (cont’d)
• By Planck’s law E = hc/λ, a photon with a short wavelength has a
large energy
• Thus, it would impart a large ‘kick’ to the electron
• But to determine its momentum accurately,
electron must only be given a small kick
• This means using light of long wavelength!
Fundamental Trade Off …
• Use light with short wavelength:
– accurate measurement of position but not momentum
• Use light with long wavelength
– accurate measurement of momentum but not position
Heisenberg’s Uncertainty Principle
The more accurately you know the position (i.e.,
the smaller ∆x is) , the less accurately you know the
momentum (i.e., the larger ∆p is); and vice versa
Classical Physics
•
Described by Newton’s Law of Motion (17th
century)
– Successful for explaining the motions of
objects and planets
H 
i
pi
U (r1 , r2 ,..., rN )
2mi
19th
– In the end of
century, experimental
evidences accumulated showing that classical
mechanics failed when applied to very small
particles.
Sir Isaac Newton
The failures of Classical Physics
•
Black-body radiation
– A hot object emits light (consider hot metals)
– At higher temperature, the radiation becomes shorter
wavelength (red  white  blue)
– Black body : an object capable of emitting and
absorbing all frequencies uniformly
The failures of classical physics
•
Experimental observation
– As the temperature raised, the peak in the
energy output shifts to shorter
wavelengths.
– Wien displacement law
1
Tmax  c2
5
c2  1.44 cm K
– Stefan-Boltzmann law
  E / V  aT 4
M  T 4
Wihelm Wien
Rayleigh – Jeans law
•
First attempted to describe energy
distribution
• Used classical mechanics and equipartition principle
dE  d

8kT
4
• Although successful at high wavelength, it
fails badly at low wavelength.
• Ultraviolet Catastrophe
– Even cool object emits visible and UV
region
– We all should have been fried !
Lord Rayleigh
Planck’s Distribution
•
Energies are limited to discrete value
– Quantization of energy
E  nh
•
, n  0,1,2,...
Max Planck
Planck’s distribution
dE  d

8hc
5 (e hc / kT  1)
• At high frequencies approaches the Rayleigh-Jeans
law
(e hc / kT  1)  (1 
hc
hc
 ....)  1 
kT
kT
• The Planck’s distribution also follows StefanBoltzmann’s Las
Wave-Particle Duality
-The particle character of wave
•
Particle character of electromagnetic radiation
– Observation :
• Energies of electromagnetic radiation of frequency v
can only have E = 0, h, v 2hv, …
(corresponds to particles n= 0, 1, 2, … with energy = hv)
– Particles of electromagnetic radiation : Photon
– Discrete spectra from atoms and molecules can be explained
as generating a photon of energy hn .
– ∆E = hv
Wave-Particle Duality
-The particle character of wave
•
UV
Photoelectric effect
– Ejection of electrons from metals
when they are exposed to UV radiation
– Experimental characteristic
• No electrons are ejected,
regardless of the intensity of
radiation, unless its frequency
exceeds a threshold value
characteristic of the metal.
• The kinetic energy of ejected
electrons increases linearly with
the frequency of the incident
radiation but is independent of the
intensity of the radiation .
• Even at low light intensities,
electrons are ejected immediately
if the frequency is above threshold.
electrons
Metal
Wave-Particle Duality
-The particle character of wave
•
Photoelectric effect
– Observations suggests ;
• Collision of particle – like projectile that carries energy
• Kinetic energy of electron = hν - Φ
Φ : work function (characteristic of the meltal)
energy required to remove a electron from the metal
to infinity
• For the electron ejection , hν > Φ required.
• In case hν < Φ , no ejection of electrons
Wave-Particle Duality
-The particle character of wave
• Photoelectric effect
Wave-Particle Duality
-The wave character of particles
• Diffraction of electron beam from metal
surface
– Davison and Germer (1925)
– Diffraction is characteristic property of
wave
– Particles (electrons) have wave like
properties !
– From interference pattern, we can get
structural information of a surface
LEED (Low Energy Electron Diffraction)
Wave Particle Duality
•
De Brogile Relation (1924)
– Any particle traveling with a linear
momentum p has wave length l
Matter wave: p = mv = h/
– Macroscopic bodies have high
momenta (large p)
 small wave length
 wave like properties are not observed
Schrödinger equation
• 1926, Erwin Schrödinger (Austria)
– Describe a particle with wave function
– Wave function has full information about the
particle
Time independent Schrödinger equation
for a particle in one dimension
Schrodinger Equation
General form
H = E 
H= T + V
: Hamiltonian
operator
The Schrodinger equation:
Kinetic
energy
+
Potential
energy
For a given U(x),
• what are the possible (x)?
• What are the corresponding E?
=
Total
energy
For a free particle, U(x) = 0, so
 (x)  Ae
ikx
Where k = 2
= anything real

2
2
k
E
2m
= any value from
0 to infinity
The free particle can be found anywhere, with
equal probability
The Born interpretation
of the Wave Function
•
The Wave function
– Contains all the dynamic information about
the system
– Born made analogy with the wave theory
of light (square of the amplitude is
interpreted as intensity – finding
probability of photons)
– Probability to find a particle is
2
proportional to    *
Probability Density
– It is OK to have negative values for wave
function
Max Born
Born interpretation of
the Wave Function
Born interpretation of
the Wave Function
Normalization
• When ψ is a solution, so is Nψ
• We can always find a normalization const. such that the
proportionality of Born becomes equality
N 2  * dx  1
*

  dx  1
*
*


dxdydz



  d  1
Normalization const. are
already contained in wave
function
Quantization
• Energy of a particle is
quantized
 Acceptable energy can be found
by solving Schrödinger equation
 There are certain limitation in
energies of particles
The information in a wavefunction
• Simple case
– One dimensional motion, V=0
 2 d 2

 E
2
2m dx
Solution
  Aeikx  Be ikx
k 2 2
E
2m
Probability Density
B=0
  Ae
ikx
  A
2
2
A=0
  Be
 ikx
 B
2
2
A=B
  2 Acos kx
  4 A cos 2 kx
2
nodes
Eigenvalues and eigenfucntions
• Eigenvalue equation
(Operator)(function) = (constant factor)*(same function)
̂  
Operator
Eigenfunction
Solution : Wave function
Eigenvalue
Allowed energy (quantization)
(operator correspond ing to observable )  (value of observable ) 
Operators
̂  
• Position
x
x̂  x 
px
pˆ x 
• Momentum
• Potential energy
V
• Kinetic energy
• Total energy
1 2
kx
2
p x2
EK 
2m
 d
i dx
1
Vˆ  kx2 
2
2
2

d
Eˆ K  
2m dx 2
2
2

d
Hˆ  Eˆ K  Vˆ  
 Vˆ
2
2m dx
Quantum Numbers
• Definition: specify the properties of atomic orbitals and the
properties of electrons in orbitals
• There are four quantum numbers
• The first three are results from SchrÖdinger’s Wave Equation
Orbital Quantum numbers
An atomic orbital is defined by 3 quantum numbers:
–
n
l
ml
Electrons are arranged in shells and subshells of ORBITALS .
n  shell
l
ml
 subshell
 designates an orbital within a subshell
Quantum Numbers
Symbol
Values
Description
n (major)
1, 2, 3, ..
Orbital size and
energy = -R(1/n2)
l (angular)
0, 1, 2, .. n-1
Orbital shape or
type (subshell)
ml (magnetic)
-l..0..+l
Orbital orientation
in space
Total # of orbitals in lth subshell = 2 l + 1
Quantum Theory Model
Orbitals
One “s” orbital
Three “p” orbitals
Five “d” orbital
Problems
(1) Calculate the number of photons emitted by a 100 W yellow lamp in 1.0s.
Take the wavelength of yellow light as 560 nm and assume 100 percent
efficiency.
Each photon has an energy hυ, so the total number of
photons needed to produce an energy E is E/hν.
The number of photons is
Pt
=
h(c/)
=
N = E/hν
t
hc
Substitution of the data gives
N
=
(5.60 x 10-7 m) x (100 J s-1) x (1.0 s)
(6.626 x 10-34 Js) x (2.998 x 108m s-1)
=
2.8 x 1020
(2) Show that eαx is an eigenfunction of the operator d2/dx2.
What is the eigenvalue?
f(x) = eαx
d2/dx2(eαx) = α d/dx (eαx)
= α2 eαx
Thus eαx is the eigenfunction of the given operator d2/dx2.
α2 is the eigenvalue
 
(3) Show that eikx is an eigenfunction of a operator ^Px = -ih
x
F(x) = eikx
= -i h  eikx
x
= -i2 hk2eikx
= h k2eikx
Thus eikx is an eigenfunction
(4) For the wavefunction ψ(φ) = Aeimφ, where m is an integer.
Calculate A so that the wavefunction is normalized
∫ ψn*(x) ψn(x) dx = 1
∫ Aeimφ Ae-imφ dφ =1
= A2 ∫ eimφ e-imφ dφ = 1
= A2φ =1
Thus A = 1/√φ
(5) For the wavefunction of ψ(x) = B sinnπx/a, Evaluate B so that the
wavefunction is normalized. The permitted values of x are 0 ≤ x ≤ a.
a
 xx* = 1
0
a
 (Bsin nx/a) (B sin nx/a) dx =1
0
a
2
B  sin (nx/a) dx = 1
0
2
B2a n
 1 - cos2z dx = 1
2n 0
B2a  x - sin2z/2n = 1
2n
0
Hence B = √2 / a