Quantum Physics and
Nuclear Physics
13.1 Quantum Physics
Quantum Physics
Revision
• When an electron falls to a lower energy level, the
change in energy (ΔE) is emitted as a photon of e-m
radiation.
• ΔE = E2 – E1
• The change in energy is
proportional to the
frequency of the emitted
photon:
E = hf
( E = Photon energy
h = Plank’s constant
= 6.6 x 10-34 m2kgs-1 )
E.g. Energy levels in the hydrogen atom:
Thus the emission spectrum for Hydrogen (this
shows only the visible section)...
The Photoelectric Effect
Demo:
- u.v. light shone onto a zinc
plate will cause a negative
charged electroscope to
discharge .
- visible light will not cause it
to discharge (even if very
intense).
- if the charge is positive,
even the u.v. will not cause it
to discharge.
Conclusion
- u.v. light causes negative electrons to be emitted
from the surface of zinc (this is called photoelectric
emission) but visible light does not, hence the first
two observations.
- if u.v. causes negative electrons to be emitted, the
positive charge will not be discharged.
Applet link
Threshold Frequency
Electrons will only be emitted from zinc by photoelectric
emission if the electromagnetic radiation incident upon
its surface has a frequency of 1 x 1015 Hz or above. This
is called the threshold frequency of zinc.
Limitation of Wave Theory of Light
Wave theory would suggest that once enough visible
light energy had been absorbed by the zinc, the electron
would be able to escape. This is not the case. No
matter how intense the incident radiation, if its
frequency is below the threshold frequency for a
particular material, no photoelectric emission will occur.
Photons – the Quantum Model
In 1900 Max Planck came up with the idea of
energy being ‘quantised’ in some situations. i.e.
existing in small ‘packets’.
In 1905 Einstein suggested that all e-m radiation is
emitted in small quanta called photons rather than
in a steady wave.
- Intensity of radiation depends on the number of
photons being emitted per second (not amplitude as
suggested by the wave model).
- Energy per photon depends upon its frequency:
E = hf
Work Function and Photoelectric Emission
When u-v light is incident upon a zinc surface, each
photon gives its energy to a single electron on the
zinc surface. u-v photons have a high frequency. As
a result they give enough energy to the electron to
escape from the surface.
The minimum energy needed to just remove an
electron from a metal surface is called the work
function, .
Low intensity u-v light will still cause electrons to be
emitted. Because there are less photons per
second there will be less electrons emitted per
second.
If the incident light has a lower frequency, each
photon has less energy and so no electrons are
emitted, irrespective of the intensity.
Einstein’s Photoelectric Equation
If the photon energy (E=hf) is greater than the work
function (), any remaining energy becomes kinetic
energy of the electron (= ½mv2). Thus Einstein
stated...
hf =  + ½ mv2
m = mass of an electron
v = speed of fastest
electrons (ms-1)
This is one version of Einstein's photoelectric
equation.
Einstein’s theory was confirmed by Robert Millikan
in 1916. He realised that if the clean metal emitting
surface was given a positive potential, the electron
emission could be stopped.
Link - PhET simulation
Thus, if the p.d. applied was known, the KE
‘removed’ from the fastest electron can be found:
We know...
So...
V = W / q  W = eV
hf =  + eV
V = stopping voltage
e = charge on an electron
= 1.6 x 10-19C
 = Work function (Joules)
Testing Einstein's Photoelectric Equation
Einstein’s photoelectric equation can be rearranged to give...
V = h f - 
e
e
Thus plotting a graph of V against f enables us to determine
Plank’s constant.
• Incident radiation is shone onto a photoelectric cell
with a low work function
• This causes electrons to be emitted from the larger
emitting electrode. If they reach the small receiving
electrode a current is detected on the ammeter
• The stopping voltage V is increased until zero
current flows through the ammeter. The p.d. has
made it impossible for even the fastest electrons to
escape from the large electrode.
• The experiment is repeated with different
frequency incident radiation and a set of values for
frequency and stopping voltage are collected.
Results:
Q. Explain why this graph will always have the
same gradient, whatever metal is used for the
emitting electrode.
Q. Explain how you would determine a value for the
work function of the metal used to produce the
graph above.
Wave theory suggests that
Q.
Any frequency can emit electrons
from a certain metal surface
Current depends on intensity (i.e. the
number of photons per unit time)
Maximum energy of electrons is
independent of frequency
Maximum energy of electrons would
depend on intensity
Supported or Contradicted by
quantum theory?
Photocurrents
If the metal surface and frequency of incident
radiation are both kept constant, a graph can be
plotted showing how the photoelectric current
(photocurrent) in a photocell varies with applied p.d.
(voltage).
Consider these situations and explain the
photocurrent that will flow in each case:
+
-
-
+
Photocurrent
Saturation current
-
Stopping potential, Vs
0
+
Applied p.d.
Photocurrent
High
intensity
Low
intensity
-
0
+
Applied p.d.
Photocurrent
Low frequency
(red)
Higher frequency
(blue)
-
0
+
Applied p.d.
The Wave Nature of Matter
We have seen that light behaves both like a wave (it
diffracts) and like a particle (in explaining
photoelectric emission). It has wave particle duality.
Wave - particle duality is the ability of something
to exhibit both wave and particle behaviour.
Demo 1: Shine a beam of laser light (a wave)
through a single diffraction grating (like a very fine
gauze) then through many gratings crossed at
angles to each other.
Demo 2: Fire a beam of electrons (particles of
matter) at a thin piece of graphite using a cathode
ray tube.
Control grid
Anode
Cathode
VA
Graphite
foil
Flourescent
screen
De Broglie’s Equation
Considering a photon of light, in 1924 Prince Louis
Victor de Broglie equated Einstein’s mass-energy
relation and Planck’s equation:
E = mc2 and E = hf
thus...
mc2 = hf
so...
mc2 = h c
λ
so...
λ = h
mc
or
But c = f λ
λ = h
p
This is the de Broglie equation.
( Where h = Planck’s constant = 6.6 x 10-34Js )
By analogy, this equation can be applied to any
other particle of matter. Thus de Broglie’s
Hypothesis:
All matter can behave like a wave, with the
wavelength given by Plank’s constant
divided by the matter’s momentum.
Q1 A year 13 student runs with joy to his physics
lesson. If he runs at 5 ms-1 and has mass 60kg,
determine the de Broglie wavelength of his motion.
Comment upon your answer.
Q2 An electron is accelerated in a cathode ray tube
through a potential difference of 2kv.
i. Determine the velocity of the electron (me = 9.11
x 10-31 kg)
2.65 x 107 ms-1
ii. Determine the de Broglie wavelength of the
electron.
2.7 x 10-11 m
Energy Levels
E = hf
h
= Planck constant
= 6.6 x 10-34 m2kgs-1
E.g.
hf = E2 - E1
Emission Line Spectra
Atoms of a gas emit e-m radiation if they become
excited. This means the electrons jump to a higher
energy level and then fall back, losing potential
energy and emitting it as a photon of e-m radiation.
This will have a frequency according to ΔE = hf
Experiment: Observing
emission spectra
- Place a slit in front of a
hydrogen lamp.
- View the light through a
diffraction grating or by
refracting it through a prism.
Results:
This is the image seen
when light from one
lamp is diffracted
through a grating:
Conclusion
Explain why...
a. only certain colours are seen for any particular
lamp.
b. coloured fringes are produced
Neon
Mercury
Argon
Xenon
Images from
this website
Hydrogen...
Helium...
Neon...
Absorption spectrum
If light with a continuous spectrum of frequencies (a
filament light bulb approximately emits this) is shone
into a gas its photons will interact with electrons in
the gas atoms, boosting them to a higher energy
level. Thus the gas will absorb only certain
frequencies of the light. These frequencies will be
missing from the light that passes through.
Absorption spectrum for Hydrogen.
Q. Why des the intensity not fall to zero for the
absorption lines? Energy is re-emitted as photons in all directions
Fraunhoffer Lines
This effect is visible in the Fraunhoffer lines seen in
spectra of light from the Sun. The Sun emits a
virtually continuous spectrum. The absorption lines
are due to gases in the Sun’s atmosphere
absorbing some frequencies of photon. Hence
astronomers can deduce what gases are in a star’s
atmosphere as well as their proportions.
The ‘Electron in a Box’ Model of an Atom
We have seen how the Bohr model of an atom
suggests that the atom has discreet energy levels.
Thus the energy of the electrons and of the atom is
not continuous.
Next we will look at how this idea is compatible with
the de Broglie hypothesis.
According to de Broglie, the electron must have
wave properties.
So how does this wave fit into an atom...?
We can create a quantum model of an electron
trapped in an atom by considering the electron as a
standing wave travelling back and forth across a
box of side L:
Clearly, if there are nodes at either
end, the electron can only have
certain fixed wavelengths and
frequencies.
i.e.
Java applet
λ = 2L
λ=L
λ = 2/3 L
(fundamental)
(2nd harmonic)
(3rd harmonic)
So in general λ = 2L
n
According to de Broglie...
λ = h = h
p mv
v= h
mλ
The KE of the electron is given by KE = ½ mv2
Substituting the equations for v and λ gives…
KE = n2 h2
8mL2
n = integer
h, m and L are all
constants.
The energy of the electron ‘wave’ is clearly
quantised (it has discreet values depending upon
n), thus fitting scientific observations e.g. work
function in the photoelectric effect.
Limitations of the Bohr Model of the Atom
The Bohr model of the atom worked well with basic
ideas of quantum theory (as we have seen).
However it has limitations.
E.g. It only works well with the Hydrogen atom
It does not predict intensities of different
spectral lines.
Schrödinger's Model of the Atom
Schrödinger developed a new model that accounted
for all these limitations. He suggested that...
- electrons exist in the atom with a position determined by
the wavefunction (ψ – psi), a function of position and time.
- the position of the electron at any time is undefined
- the probability of the finding the electron at any particular
position in the atom is determined by the square of the
amplitude of the wavefunction i.e. ψ2
Thus for a given energy there are some places
where the electron is more likely to exist. This can
be represented by ‘probability clouds’.
Additional note:
This illustrates an important feature of quantum
mechanics. Its outcomes are probabilities and not
certainties; it is a probabilistic model.
Classical mechanics (e.g. Newtons laws) are
deterministic: the future is determined by the past
and is therefore predictable with some certainty!
Heisenberg’s Uncertainty Principle
Consider one electron amongst a beam of electrons
moving with identical momentum towards a narrow
slit:
If the slit is wide, the electron will
pass straight through without
diffracting. Hence we can know its
direction and thus its momentum
beyond the slit. However we cannot
be sure where exactly it passed
through the slit so the uncertainty in
position is large.
If the slit width is similar to the
diameter of the electron, it will diffract
as it passes through.
So we are not sure about its direction
and thus momentum although we can
be quite sure of its position.
This is an example of the Heisenberg uncertainty
principle which states that...
It is not possible to measure the exact position and
the exact momentum of a particle at the same time.
Note: This is a fundamental property of the universe and
is nothing to do with our ability to measure accurately.
This imposes a minimum uncertainty on the product
of the uncertainties of position and momentum:
Δp Δx ≥ h
4π
h = Plank’s constant
This can also be written in terms of energy and
time:
ΔE Δt ≥ h
4π
Clearly the value of h is so small that the effects of
this uncertainty are not seen in everyday events.
E.g. Quantum tunnelling
You tube link
As a result of the uncertainty principle energy can
be ‘borrowed’ to overcome potential barriers:
- Alpha particles should not be able to escape from
the nucleus due to the ‘strong force’ between
quarks. However they escape by borrowing energy
and paying it back within a time limit defined by the
equation.
- Two protons (hydrogen nuclei) should not be able
to fuse together in the Sun at its current density and
temperature. They do fuse together by borrowing
energy and then paying it back later on.
Q1
Consider a football moving through the air. If the
uncertainty in its momentum is 2.25 x 10-2 kgms-1,
Determine the uncertainty in its position (Δx).
Δx Δp ≥ h
4π
so… Δx ≥ h
4π Δp
Δx ≥
6.6 x 10-34
4π x (2.25 x 10-2)
Δx ≥ +/- 2.33x10-33m
Q2
A proton in the LHC takes 1.0 x 10-10s to collide.
Assuming a 1% uncertainty in measurement of time,
determine the uncertainly in measurements of
proton energy (ΔE).
1% uncertainty in time = 1/100 x (1.0 x 10-10)
= +/- 1.0 x 10-12 s
ΔE Δt ≥ h
4π
so… ΔE ≥
ΔE ≥
h
4π Δt
6.6 x 10-34
4π x (1.0 x 10-12)
ΔE ≥ +/- 5.3 x 10-23 J
Q3
Assuming 1% uncertainty in velocity, determine the
uncertainty in the position of a proton moving at
i.
2.5 x 107ms-1
ii. Close to the speed of light (3.0 x 108 ms-1)
( mp = 1.673 x 10-27 kg )
Q4
Outline by reference to position and momentum
how the Schrödinger model of the hydrogen atom is
consistent with the Heisenberg uncertainty principle.
- the Schrödinger model assigns a wave function to
the electron that is a measure of the probability of
finding it somewhere;
- therefore the position of the electron is uncertain;
- resulting in an uncertainty in its momentum;
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Probability Waves
From the de Broglie hypothesis we can see that our
picture of the electron as a small ‘ball’ is too
simplistic. It
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http://www.physics.uq.edu.
au/people/mcintyre/applets/
cathoderaytube/crt.html