Lecture 2 - CHEM F111 - 1sem 2019-2020 - Quantum Chem

General Chemistry (CHEM F111)
Lecture 2: 07.08.2019
From Classical to Quantum Description
" As for Planck himself, he strove hard

to keep his theory on the solid ground
of the classical physics that he loved so
much.
Planck became a revolutionary against
his will”!
-H. Kragh, Phys. World, Dec. 2000
It waited for other phenomena to clearly bring out the

inevitable quantized nature of the universe
RECAP
Quantum Mechanics
Black Body radiation
Wien’s Displacement Law
Stefan-Boltzmann Law
Rayleigh-Jeans formula
Planck Radiation Law
Success of Planck’s formula

Analysis of Planck’s formula
Quantization of energy
Emergence of Quantum theory: late 19th Century
Line Spectra
Hot gas emits photons with the characteristic wavelengths
corresponding to the transitions between different energy levels of the
atoms or molecules in the gas. This leads to bright lines in the spectrum.
•A hot source emits continuous radiation -- a cool gas in the way -- then
the cool gas absorbs photons with the characteristic wavelengths
corresponding to the transitions between different energy levels of the
atoms or molecules in the gas.
Most compelling evidence for QUANTIZATION

Atomic and Molecular Spectra
A region of spectrum of radiation Spectrum due to sulphur dioxide

emitted by exited iron atoms molecules
The energy of the atoms or molecules is confined to discrete

values, so energy can be discarded only in packets.
Atomic and Molecular Spectra
The energy of the atoms or molecules is
confined to discrete values.
Energy can be discarded or absorbed only
in packets as the atom or molecule jump
between its allowed states.
The frequency of the radiation is related to
the energy difference between the initial
and final states.
Theoretical background
for Line Spectra : Bohr
Theory
Bohr frequency
relation: E
Transitions between quantized energy levels of atom

or molecule, with absorption or emission of photon
accounts for line spectra.
BOHR FREQUENCY RELATION:
The frequency ν is directly proportional to the
difference in energy
ΔE = E2 – E1 = hν
Example: The bright yellow light emitted by sodium

atoms in some street lamps has wavelength 590 nm .
ΔE = hν = hc/λ
= [(6.626 x 10-34 Js)(2.998 x 108 ms-1)]/(590 x 10-9 m)
Yellow light emitted when a sodium atom loses an
energy ΔE = 3.4 x 10-19 J
Line Spectrum of Hydrogen atom
The frequencies (in wave numbers) at which the lines
occur in the spectrum of hydrogen : Rydberg’s
Empirical
= =1/ = RH(1/n1  1/n2
2 2)
formula
where RH = 109677 cm-1, is the Rydberg constant
n1 and n2 > n1 are positive integers
n1 n2 Region
Lyman 1 2,3,4,…. Ultraviolet

Balmer 2 3,4,5,…. Visible
Paschen 3 4,5,6,…. Near IR
Bracket 4 5,6,7,…. IR
Pfund 5 6,7,8,…. Far IR
Atomic Models
Rutherford’s Planetary Model
Bohr atom model
•Coulombic force and the centripetal force balance

Electron of mass m, in circular orbit of radius r, about
stationary nucleus of mass mN, charge Ze
mv2/r = Ze2/40r2
Bohr Model
1. Specific orbits, discrete quantized energies.
2. The electrons do not continuously lose energy –
gain or lose by jumping from one orbit to another
3. quantization of angular momentum
L = mvr = nh/2 = nħ, n = 1,2,3,….
 Energy occurs as ‘packets’ called quanta, of magnitude h/2π
Success
Could explain Rydberg’s formula (empirical)
Theoretical background for Line Spectra
Bohr model – Inadequacies
Primitive Model
Semi-classical Could not explain
•The spectra of larger atoms.
•The relative intensities of spectral lines
•The existence of fine and hyperfine structure in
spectral lines.
•The Zeeman effect - changes in spectral lines
due to external magnetic fields
Waves and Particles
Main experiment showing light as particles is
the Photoelectric effect and Black body radiation
Two properties of waves are:
1. Interference
2. Diffraction
The ability for something to behave as a wave and
a particle at the same time is known as wave-
particle duality.
 If electron is acting as a wave, We should see diffraction
and interference of matter waves
Photoelectric Effect
Emission of electrons from metals when exposed to (ultraviolet) radiation.

Observations
1. No emission of electrons below a threshold value
characteristic of the metal – Work function
2. Kinetic energy varies linearly with the frequency
3. Above the threshold value, emission of electrons is
instantaneous.
Explanation (EINSTEIN 1905)
1. Light : collection of particles, called photons, each
of energy h.
2. If h < , no emission of electrons occurs.
3. Threshold frequency 0 ,  = h0
4. For  > 0, the kinetic energy of the emitted
electron Ek = h   = h(  0).
Ek = h   h = 6.626 x 10-34 J s, Planck’s constant

h can be calculated from the slope
Example:
The work function of rubidium is 2.09 eV (1 eV = 1.602 x
10-19 J). Can blue (470 nm) light eject electrons from the
metal?
Ans: Need to find out energy of radiation, convert
470 nm to eV.
hν = hc/λ = (6.626 x 10-34 J s) x (3.00 x 108 m/s) = 4.23 X 10-19 J = 2.63 eV
(470 x 10-9 m)
Energy of blue light > work function 2.09 eV thus Photoelectrons will be ejected
Electron Diffraction
Firing electron at an object and observing the scattering
(analogous: X-ray and neutron diffraction): Davisson and
Germer 1925
Investigation of the angular distribution of electrons

scattered from nickel :
electron beam was scattered by the surface atoms at the exact angles
predicted for the diffraction of waves, with a typical wavelength
Constructive interference Destructive interference
Bragg’s law A pattern of sharp

nλ=2d sin(θ) reflected beams
from the crystal
λ-wavelength of the
electron wave
Duality
Girl or Granny? From duality comes singularity

Matter-Wave Duality de Broglie matter waves
• If light (radiation) can be viewed as a collection of particles,
then can entities considered as particles also be seen as waves?
• de Broglie postulated the existence of matter waves associated
with particles, with a wave length given by λ = h/p
• Confirmed by the scattering experiments of Davisson and
Germer, and G P Thomson.
 = h/p
1.  should decrease with particle’s speed
2. Low momentum - long 
3. High momentum - small 
Definite wavelength →
definite momentum,
but since wave extends
over all space, no
information on
position
Two Slit Experiment with Electrons etc.
Experiment with electrons/photons etc. :

Arrive in identical lumps – particles, but distribution
shows interference – wave behavior.
Consequence of Duality
No physical phenomena can be described by

only classic point ‘particle’ or ‘wave’ .
Neither the wave or particle description is fully

and exclusively accurate
Uncertainty Principle
Heisenberg Uncertainty Principle
• It is impossible to specify simultaneously, with arbitrary
precision, (a given Cartesian component of) the
momentum and position of a particle
x px  ħ/2
• Complementary variables, increase in the precision of
one possible only at the cost of a loss of precision in the
other
• Trajectories not defined precisely, unlike classical
mechanics
• Other such examples of complementary variables too
Heisenberg Uncertainty Principle
For macroscopic objects, position accuracy is high, so

there should be a lot of uncertainty in momentum ---->
more curvature ---> insignificant smaller wavelength
Wavefunction for particle with precisely
defined position
Superposition
A sharply localized wavefunction can be generated by

adding/superposition of large/infinite number of wavefunctions
We can get localization of particle only at the expense of/loss of precise information
about the momenta
Calculate the wavelength of an electron in a
10-MeV particle accelerator. (1 MeV = 106 eV)
• Solution: We need to find the momentum, p,

from the energy, E.
• The relationship between them is p = (2mE)½ =

(2 x 9.11 x 10-31 kg x10 x106 x 1.602 x 10-19 kg
m2 s-2)½ = 1.7 x 10-21 kg m s-1
• λ= h/p = (6.626 x 10-34 J s)/(1.7 x 10-21 kg m s-1)

• λ = 3.9 x 10-13 m = 0.39 pm
Some Typical de Broglie
wavelengths
An electron accelerated through a potential

difference of 50 V ~ 1.73 x 10-10 meters.
A ball weighing 100 g, and moving with a speed of

25 m/s ~ 2.65 x 10-34 meters.
A human being weighing 70 kg, and moving with a

speed of 25 m/s ~ 3.79 x 10-37 meters.

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General Chemistry (CHEM F111)

" As for Planck himself, he strove hard

It waited for other phenomena to clearly bring out the

Success of Planck’s formula

Most compelling evidence for QUANTIZATION

A region of spectrum of radiation Spectrum due to sulphur dioxide

The energy of the atoms or molecules is confined to discrete

Transitions between quantized energy levels of atom

Example: The bright yellow light emitted by sodium

Lyman 1 2,3,4,…. Ultraviolet

•Coulombic force and the centripetal force balance

Emission of electrons from metals when exposed to (ultraviolet) radiation.

Ek = h   h = 6.626 x 10-34 J s, Planck’s constant

Investigation of the angular distribution of electrons

Bragg’s law A pattern of sharp

Girl or Granny? From duality comes singularity

Experiment with electrons/photons etc. :

No physical phenomena can be described by

Neither the wave or particle description is fully

For macroscopic objects, position accuracy is high, so

A sharply localized wavefunction can be generated by

• Solution: We need to find the momentum, p,

• The relationship between them is p = (2mE)½ =

• λ= h/p = (6.626 x 10-34 J s)/(1.7 x 10-21 kg m s-1)

An electron accelerated through a potential

A ball weighing 100 g, and moving with a speed of

A human being weighing 70 kg, and moving with a

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