Transcript 6.5

6. Atomic and
Nuclear Physics
Chapter 6.5 Quantum theory
and the uncertainty principle
Atomic spectra

When a atomic gas, like hydrogen, is either heated at a high
temperature or exposed to a high electric field it will glow and
emit light.

The emitted light can be analysed using a spectrometer that
will split the light into its component wavelengths

The lines
produced make
the emission
spectrum of
hydrogen.
Atomic spectra


Other gases will have similar spectra.
But different gases will have emission lines at different wavelengths
Atomic spectra

A similar phenomenon takes place when white light is
allowed to pass through hydrogen gas.

Most light will pass through but a series of black lines are
also seen.

This constitutes an absorption spectrum, where the black
lines correspond to the radiation absorbed by the gas
Sun’s absorption spectrum
Atomic spectra
The
striking feature of
emission and absorption
spectra is the fact that the
emission and absorption
lines are at specific
wavelengths for a particular
gas.
Explanation of spectra

In 1885 Johann Balmer, a Swiss physicist,
discovered, by trial and error, that the
wavelengths in the emission spectrum of
hydrogen were given by the formula:
Johann Balmer
1 1 
 R  2 

4 n 
1
(1825-1898)
where n may take integer values 3, 4, 5, … and R is a
constant number
Explanation of spectra


Since the emitted light from a gas carries energy, it is reasonable
to assume that the emitted energy is equal to the difference
between the total energy of the atom before and after the
emission.
Since the emitted light consists of photons of a specific
wavelength, it follows that the emitted energy is also of a specific
amount since the energy of a photon is given by:
E  hf 

hc

This means that the energy of the atom is discrete, that is, not
continuous.
The “electron in a box” model

If the energy of the atom were continuous the emission of light
wouldn't always be a set of specific amounts.

The first attempt to explain these observations came with the
“electron in a box” model.

Imagine that an electron is confined in a box of linear size L.

If the electron is treated as a wave, it will have a wavelength
given by:
h

p
the electron can only be found somewhere
along this line
x=0
x=L
The “electron in a box” model

If the electron behaves as a wave, then:
 The wave is zero at the edges of the box
 The wave is a standing wave as the electron does not
lose energy

This means that the wave will have nodes at x=0 and x=L.
This implies that the wavelength must be related to the size
of the box through:

2L

n
Where n is an integer
The “electron in a box” model

Therefore, the momentum of the electron is:
h
h
mh
p 

 2L 2L
n

The kinetic energy is then:
2
 mh 


2
2 2
p
n h
2L 

Ek 


2m
2m
8mL2
The “electron in a box” model

This result shows that, because the electron was treated as
a standing wave in a “box”, it was deduced that the
electron’s energy is quantized or discrete:
Ek 

h2
1
8mL2
h2
4
8mL2
h2
9
8mL2
n 1
n2
n 3
However, this model is not correct but because it shows that
energy can be discrete it points the way to the correct
answer.
The Schrödinger theory



In 1926, the Austrian physicist Erwin Schrödinger
provided a realistic quantum model for the
behaviour of electrons in atoms.
The Schrödinger theory assumes that there is a
wave associated to the electron (just like de
Bröglie had assumed)
This wave is called wavefunction and
represented by:
Erwin Schrödinger
(1887-1961)
 ( x, t )

This wave is a function of position x and time t. Through
differentiation, it can be solved to find the Schrödinger function:


i  (r, t )  
 2 ψ(r, t )  V (r ) ψ(r, t )
t
2m
2
The Schrödinger theory

This wave is a function of position x and time t. Through
differentiation, it can be solved to find the Schrödinger function:

2
i  (r, t )  
 2 ψ(r, t )  V (r ) ψ(r, t )
t
2m
where

r (x, y z) is the particle's position in three-dimensional space,
 ψ(r, t ) is the wavefunction, which is the amplitude for the
particle to have a given position r at any given time t.
 m is the mass of the particle.

V (r) is the potential energy of the particle at each position r.
The Schrödinger theory

The German physicist Max Born interpreted Schrödinger's
equation and suggested that:
 ( x, t )
2
can be used to find the probability of finding an electron
near position x at time t.


This means that the equation cannot tell exactly where to
find the electron.
This notion represented a radical change from classical
physics, where objects had well-defined positions.
The Schrödinger theory


When Schrödinger's equation is applied to the electron in a
hydrogen atom, it gives results similar to those found using the
“electron in a box” model.
It predicts that the total energy of the electron (Ek + Ep) is given by:
C
E 2
n
where n is an integer and C a constant equal to:
2 2 me 4 k 2
C
h2
where:
- k is the constant in Coulomb’s law
- m is the mass of the electron
- e is the charge of the electron and
- h is Planck’s constant
The Schrödinger theory

Substituting the constants by their values, it is found that:
18
C  2.17910 J  13.6eV
So,
13.6
E   2 eV
n

In other words, this theory predicts that the electron in the
hydrogen atom has quantized energy.

The model also predicts that if the electron is at a high
energy level, it can make a transition to a lower level.

In that process it emits a photon of energy equal to the
difference in energy between the levels of the transition.
The Schrödinger theory
Because
the energy of the photon is given by E = hf, knowing the
energy level difference, we can calculate the frequency and
wavelength of the emitted photon.
Furthermore, the theory also predicts the probability that a
particular transition will occur.
This
is essential to
understand why
some spectral lines
are brighter than
others.
Thus, the
Schrödinger theory
explains atomic
spectra.
high n
energy
levels very
close to
each other
energy
0 eV
n=5
n=4
n=3
n=2
-13.6 eV
n=1
The Schrödinger theory
The
graph below shows the variation of the probability distribution
function with distance r from the nucleus for the energy level n=1 of
the hydrogen atom.
The
shaded area is the probability for finding the electron at a
distance from the nucleus between r = a and r = b.
Probability density
ψ2
a
b
r (x10-10m)
0.50
1.00
The Heisenberg Uncertainty Principle
Werner
Heisenberg, a German physicist and one of
the founders of Quantum Mechanics, discovered the
principle in 1927.
Heisenberg
said that if the electron behaves
simultaneously as a wave and as a particle, we
cannot divide physical objects as either particles or
waves.
Applied
to position and momentum, Heisenberg
uncertainty principle states that:
It is NOT possible to measure simultaneously
the position and momentum of something with
indefinite precision.
Werner Heisenberg
(1901-1976)
The Heisenberg Uncertainty Principle
This
has nothing to do with imperfect measuring devices or
experimental errors.
It
represents a fundamental property of nature.
The
uncertainty in position Δx and the uncertainty in momentum Δp
are related by:
h
x p 
4
This
means that making momentum as accurate as possible makes
position inaccurate and vice-versa.
If
one is made zero the other has to be infinite.
The Heisenberg Uncertainty Principle
Imagine
that we create an electron beam and try to make it move in
a horizontal straight line by inserting a metal with a small opening of
size a.
If
the opening is very small, this means that we know very
accurately the vertical position of the electron and the uncertainty in
its vertical position will be no bigger than a, so Δx  a.
BUT
if the opening has the same size of the de Bröglie wavelength
of the electron, the electron will diffract like waves of wavelength λ
diffract when passing an aperture of size similar to λ.
There
The
is an uncertainty in the electron’s momentum Δp.
electrons will spread out with an angular size of 2θ.
The Heisenberg Uncertainty Principle
The
angle by which the electron is
diffracted is given by:


p

p
But
a
Therefore:
Δp
θ
a
electrons
observed
within
this area
p

p

a
p
we take the opening a as the uncertainty in the electron’s position
in the vertical direction, we have
If
Δp Δx  p = h
This
is a very simple explanation of where the uncertainty formula
comes from.
The Heisenberg Uncertainty Principle

We can write Heisenberg's principle in terms of Energy and time
as it also applies to these measurements.

If a state is measured to have energy E with uncertainty ΔE, there
must be an uncertainty Δt in the time during which the
measurement is made, such that:
h
 E t 
4
The Heisenberg Uncertainty Principle
Heisenberg is pulled over by a policeman whilst
driving down a motorway.
The policeman gets out of his car, walks
towards Heisenberg's window and motions with his
hand for Heisenberg to wind the window down, which
he does.
The policeman then says ‘Do you know what
speed you were driving at sir?',
Heisenberg responds ‘No, but I knew exactly
where I was.