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Chapter 2
Intro to quantum mechanics
Why do we need QM?

Classical physics is great for large objects,
but fails when it comes to atomic particles
like electrons and EM waves.

QM picks up where classical mechanics
fails.

Semiconductor materials -> properties
governed by behavior of electrons in crystal
lattice.

QM and more specifically wave mechanics
can explain the behavior of electrons in
semiconductor crystals.

Schrodinger’s wave equation
Ultraviolet catastrophe

Till 1800 all light problems could be solved
by treating light as waves.

The UV catastrophe is an experiment that
wave theory couldn’t explain.

To understand the UV catastrophe we need
to understand thermal radiation.

You could say...the UV catastrophe was the
hottest problem a century ago. Bad pun!
Why do hot objects glow?
Radiation from hot objects
What did classical physics say
about thermal radiation?
I

1
4

So as you go to shorter wavelengths the
intensity really takes off. (rayleigh and jeans)

According to the Rayleigh and Jeans law at
UV the intensity approaches ∞ ! And thus
this was called the UV catastrophe.
Max Planck
Max planck (1858-1947)
Nobel prize in 1918 for work
on quantum theory

Came up with an equation that mimicked
thermal radiation but could not explain how it
worked.

Planck introduced the concept of quantization
to explain thermal radiation.

E = hν(nu). (h = 6.625 x 10-34 J-s known as the
plack’s constant).

If atomic oscillations are the source of the
spectra then the energy of each of these
oscillators is given by the above equation.

E = hν is also called a packet of energy or a
quanta.
Planck’s radiation law as a
function of frequency and
wavelength
Photoelectron energy
Photoelectric effect

Fig 2.1

Monochromatic light incident on a material,
under certain conditions leads to the ejection
of electrons called photoelectrons.

Classical physics says light is a wave –


ITHRESHOLD
Incident intensity
Below threshold intensity – no emission
Above threshold intensity – emission that
increases with intensity – this effect has
nothing to do with the frequency of the
light.
PHOTOELECTRIC EFFECT
(cont.)

Blue = 480 nm
BKE-Max
What is observed is that “At a constant
incident intensity, max KE of the
photoelectron varies linearly with frequency,
with a limiting threshold frequency of ν= νo”
Green = 510 nm
Red = 800 nm
GKE-Max
RKE-Max
IR = 1000 nm
IRKE-Max = 0
BKE-Max>GKE-Max>RKE-Max>IKE-MaxR for same incident intensity!
Max KE for a photo e-
PHOTOELECTRIC EFFECT
(cont.)
I constant

If intensity is constant and the frequency is
changed then the MAX KE of the
photoelectron changes linearly with the
frequency.

If the frequency is constant and the Intensity
is the changed, the rate at which the
photoelectrons emit is changed but the MAX
KE is not affected.
νTHRESHOLD
Frequency
Einstein’s interpretation of the
photoelectric effect in 1905

Light can be treated as a particle.

From planck’s theory we can say that light is
a particle or a packet of energy with E = hν.

This packet is called a photon.

Higher the frequency greater the energy. So
the energy of a blue photon is greater than
than of a red photon.
Explaining the photoelectric
effect with photons.

A minimum energy is required to dislodge
the photoelectron from the material. This is
Emin=hνMin=work function.

Energy in excess of the wf goes into the KE
of the electron.

As the frequency gets higher there is more
energy directed to the KE.

KEMAX = (1/2)mv2= hν-hνMin
Compton effect
Photons behave like billiard/pool balls –
energy and momentum are conserved.
More proof of particle behavior.
De Broglie's theory

If waves can be treated as particles can
particles be treated as waves?

Principle of wave-particle duality
p
h

h

p


What is the De-broglie wavelength of an
abrams tank? Hint: it weighs 65 tons and
travels at a speed of 42 mph.
Davisson and germer – wave
nature of electrons.

Read text.
Uncertainty principle
1st statement

It is impossible to describe with absolute
accuracy the position and momentum of a
particle.
2nd statement

It is impossible to describe with absolute
accuracy the energy and the instant at which
the particle had that energy.
p.x  h

E.t  h
h
h
 1.054 1034 J  s
2
This is only reasonable for sub-atomic particles.
Proof: speeding ticket.
Probability density function

If you cant specify where an electron is, then
the alternative is to state the likelihood of
finding an electron at a particular place i.e
probability.

We can also develop probability density
functions to determine that an electron has a
particular energy.
Schrodinger’s wave equation
Schrodinger' s equation :
h2 2


 (x, y,z,t)  V (x, y,z)(x, y,z,t)  ih
2m
t
where,
 is the wave function and V is the potential (energy)

Wave eq. (cont)
This is a partial differential equation
_
_
   .
2
 _  _  _
  ax  ay  az
x
y
z
_
2 _
2 _
2 _



 2  2 ax  2 ay  2 az
x
y
z
Equation 2.1 can be re - written as :
h2  2   2   2 


( 2  2  2 )  V  ih
2m x
y
z
t
Writing equation 2 in 1 - D :
h2  2 (x,t)
(x,t)

(
)  V(x,t)  ih
2
2m
x
t
(2.2)
(2.3)
What does the Ψ mean?

The wave function Ψ(x,t) “describes” the
behaviour of a particle of mass m “acted
upon” by the potential V.

Ψmay be complex

This is new physics. It cannot be derived
from other more basic laws. It can be
rationalized for simple cases, such as the
free particle (V=0), which is represented by
a plane wave.

The “truth” of QM can be judged only by its
ability to predict results which agree with
observation.