Download Lecture 24: Quantum mechanics

Lecture 24: Quantum mechanics
Overview of Enzyme Kinetics
The issue of vision
Three experiments that changed physics
o Black body radiation
o Spectrum of Hydrogen
o Photoelectric Effect
Notions of quantum mechanics
o
o
o
o
o
o
o
Energy is quantized
Light can be thought of as a particle
Electron can be thought of as a Wave
Wave-particle duality
Location of a particle, how certain can we be?
Heisenberg’s Uncertainty principle
Probabilistic view of the microscopic world
Vision revisited
Black body radiation
As we know, when we heat coal, initially it becomes red
hot, and as it’s temperature increases further its color
becomes almost white. The “spectrum” of light that is
emitted by such body is shown below (Think of prism).
More quantitatively, if we were
to plot the intensity of emitted
light as a function of
wavelength it would look like
the series of spectra shown to
the left. Note it has a distinct
maximum characteristic of its
temperature. The position of
the maximum shifts towards a
lower wavelength (Blue shift)
as the body temperature is
increased. In the late nineteenth
century, Raleigh showed that the energy density per unit
wavelength should follow:
I ( ) 
8k BT
4
The corresponding simulated line is shown as dashed line
in the above figure. Note the equation predicts intensity to
increase continuously as the wavelength decreases. It does
not predict a maximum at finite wavelength. This is known
as ultraviolet catastrophe!
Non-classical Explanation: Planck hypothesis.
Planck assumed that the radiating substance was composed of electric
dipoles that acted as simple harmonic oscillators. His suggestion was
as follows:
The energy of an oscillator must be discrete.
E = n h
where n is an integer, h is a constant of proportionality called Planck’s
constant having value 6.6256 10-34 Js, and  is the frequency of
oscillation. The integer value of n cause the energy to be in multiples of
h and results in discrete energy spectrum, as opposed to continuous
energy spectrum of classical physics.Energy of the S.H oscillator
following Planck’s hypothesis is said to be quantized, the allowed
energy states are called quantum states, and the integer n is called the
quantum number.
His result yielded:
Note the excellent fit with
experimental results!
The spectrum of hydrogen
What is typically observed is a series of lines. If we were to
calculate associated energies one finds:
1 
 1
E  R 2  2 
m 
n
This is inconsistent with the classical picture. Niels Bohr
explained this result by considering the angular momentum
of electron-proton pair. He suggested, following Planck,
that the angular momentum cannot be varied continuously,
but it is quantized:
 
nh
i  p  r  mvr 
2
This is a very profound statement that can be experimental
verified.
Bohr’s Quantum Numbers and Spectroscopy
To see how we can test above hypothesis, consider an
electron and proton pair. Since the mass of the electron is
much smaller than electron, let us assume that the electron
revolves around proton with certain velocity v, tracing an
orbit of radius r. However, to maintain an orbit we must
balance centrifugal force, which can be due to Columbic
interaction. Equating the centripetal force with coulombs
law:
mv 2 e 2
e2
2
F
 2 v 
r
mr
r
2
but mvr 
nh
2
2
 nh 
1  nh 
2
  r  

v  
m  2 e 
 2 mr 
2
4
2
2
2

me
e
e
E  T  V  1 mv 2 
1

2
2
r
r
n2h2
Using this equation it’s possible to explain the entire
absorption spectrum of Hydrogen! But it also asserts
o Electron orbit radius is also quantized (r—n2)
o Electron energy is also quantized consistent with
Planck theory
Photoelectric effect
It was observed that the kinetic energy of the electrons
ejected from clean metal surface by illuminating it with
light follows somewhat remarkable behavior:
In classical theory, the kinetic energy should be
proportional to the intensity of light. Thus Einstein
suggested following simple equation:
EKE    hv
In this way he proposed that light can act as a particle;
back to Newton again!
Wave-particle duality
If the light photon can act like particle then it would
appear that particles such as electron should exhibit wave
characteristic. These ideas were succinctly unified by de
Broglie, who suggested that the electron wavelength be:

h
h

p mv
Question then arose how do we prove the wave nature of
electrons. In a brilliant experiment, Stern and Gerlach
proved this. They exploited Huygen’s principle that was
used to prove light is wave. The primary effect of wave
nature is the observation of diffraction or interference
phenomena. Using single crystals of Nickel, and beam of
electrons they observed the diffraction caused by single
crystals just as in case of X-rays!
A corollary of dual nature of light is that if it is particle
then it must have mass. We can use Einstein relation to
estimate a mass of photon:
E  mc  hv 
2
m
hc

h
c
These developments generated tremendous excitement in
physics. Because this meant we have to discard classical
mechanics, developed by Newton, to describe strange
new properties at microscopic scale.