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Aspects of Quantum Theory
www.curriculum-press.co.uk Number 155

FactsheetPhysics
It is often said of quantum physics that if you understand it you are missing the point. That is not quite true, but you do have to put aside
much of what you have learned about physics so far and start to see things from a new perspective.

Classical Physics
Most of A level physics studies the physics of the world around us. It is about how objects can be expected to behave, following rules like
Newton’s Laws of Motion. Everything interacts in a predictable and logical way. With a keen eye for angles, playing pool is simple
application of Newton’s laws. Provided that you can control the cue, you can predict exactly what will happen when you strike the cue ball.
If you were to play pool with quantum particles however you would find it far less straightforward and predictable.

How is Quantum Physics different?
It is not that the laws and rules of ‘classical’ physics do not apply on the small scale, it is more that things are more complicated when you
get down to looking at individual particles. There are more rules that have to be obeyed and often particles to not behave in a way which
could have been predicted by classical physics. Basically, we do not notice the effects of quantum physics in our everyday lives so it can
seem illogical to us.

The thing to remember is that it does make sense from the right perspective and more importantly, it is necessary for the universe to work
at all!

At a very simple level, the differences can be summed up as follows:

Measuring Quantities
Quantities are things like mass, energy, charge,
position -anything you can measure

Predicting Outcomes
By 'outcomes' we mean simply 'anything which
might happen'. Striking a pool ball is an action.
Where the ball goes and what it does on the way is
the outcome.

Classical Physics

Quantities can take any value. They
are continuous.

E.g. planets can orbit the sun with any
value of kinetic energy.

Outcomes are definite.

Events are totally predictable given
enough information.

E.g if you know the speed and angle of
a pool ball you can work out its exact
trajectory and rebound and the way it
will affect other balls on the table.

Quantum Physics

Quantities can only take specific values. We say
they are 'Quantised'. Quanta literally means
discrete bundles or chunks. A single chunk is a
Quantum.

E.g. electrons orbiting in an atom may only have
very specific energy values.

Outcomes depend on probability.

There are a number of possible outcomes from
any action and each has an associated probability.
The important thing is there is no way to know
for certain which outcome will occur.

Wave Particle Duality
One of the side effects of quantum physics is something called
wave particle duality. This is simply where waves have particle
properties and particles have wave properties. These are things
that can be proven by experiment:

••••• Photons
Although light acts like a wave (e.g. it diffracts), it travels in packets
(quanta) of energy called photons. The existence of these has
been proven by the photoelectric effect which is an example of
when light must be treated like a particle rather than in a wave in
order to match the experimental observations.

The energy carried by the photon depends on the
frequency:

E = hf E is the energy in Joules
h is Planck’s constant, 6.63x10-34 Js
f is the frequency in Hertz (Hz).

Example 1
An electron emits a photon as it loses energy. The energy has
frequency 6×××××1014 Hz which is in the optical part of the
electromagnetic spectrum.
(a) What is the energy carried by the photon?
(b) Another electron emits infra red, how does the energy emitted

change? [3 marks]

Answers
(a) E = hf = 6.63×10-34 × 6×1014 = 4.0×10-19 J
(b) Infra red is lower frequency than visible and therefore the

energy will be less.

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155. Aspects of Quantum Theory Physics Factsheet

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••••• De Broglie wavelength
All particles which are moving have an associated wavelength,
which depends upon their momentum, p (momentum = mass ×
velocity):

The De Broglie wavelength of a particle is given by:

λ = h/p = h / mv

λ is the associated de Broglie wavelength
h is Planck’s constant, 6.63×10-34 Js
p is the momentum ( kgms-1) or mv is mass × velocity

Electron diffraction is evidence for the wave nature of electrons.
This is where a beam of electrons create a diffraction pattern when
fired through a small hole, similar to that made by a laser when fired
through a slit.

Example 2:
(a) Estimate your own de Broglie wavelength when walking.

[2 marks]
(b) How will the wavelength of an electron change as it is

accelerated? [1 mark]
(c) A proton and electron are each travelling at speed v.

Without calculation explain what you know about the De
Broglie wavelengths of each particle. [1 mark]

(d) What speed must an electron be travelling at to have the
wavelength of 6×××××10-9 m (m

e
= 9.1 ××××× 10-31 kg)? [2 marks]

Answers
(a) Pick suitable, easy numbers. v = 1 ms—1, m = 70 kg

λ = h/mv = 6.63×10-34/(70×1) = 9.5×10-36 m
(Note that this is why on the scale of people and objects you
don’t notice the effects of your De Broglie wavelength)

(b) As v increases, λ decreases
(c) As the mass of the proton is greater it will have a smaller

wavelength
(d) Rearrange the equation

v = h/mλ = 6.63×10-34/(9.1×10-31 × 6×10-9) = 1.2×105 ms-1

Fig 1a Diffraction pattern created by electrons

This is simply a one dimensional version of the electron diffraction
pattern.

Fig 1b Diffraction pattern made by firing a laser through a single
slit

De Broglie Wavelengths of Electrons in the Atom
Further evidence for the De Broglie wavelength of the electron
comes in the form of the spectral lines produced when the electrons
jump between energy levels. You should be familiar with emission
spectra before continuing.

Classical physics says that an electron should orbit the nucleus of
the atom in much the same way as a planet orbiting the Sun. The
range of energies it could have should be continuous. In other
words it should be able to have any kinetic energy up to the point
where it escapes the atom altogether.

Observations of spectral lines emitted by electrons as they lose
energy (emitted at a photon) shows that they only exist in very
specific energy levels.

This means that only specific energy levels are allowed.

500 600 700
A(nm)

400

Only certain frequencies of light (and therefore energies of photon)
are emitted by hydrogen showing that the electrons can only move
between certain energy levels.

To explain this we must think of the electron as a wave. As it is
trapped inside the atom it sets up a standing wave (like a guitar
string vibrating as it is fixed at both ends).

Fig 2a Hydrogen emission spectra

In theory there are an infinite number as the wavelengths continue
to get smaller.

Fig 2b shows that only certain De Broglie wavelengths of the
electron can fit inside the atom. As the wavelength depends upon
the speed of the atom then this means that only certain speeds and
therefore certain kinetic energies are allowed.

Fig 2c. The allowed energy levels correspond to the allowed
wavelengths

Fig 2b Allowed wavelengths in the atom

nucleus nucleus nucleus
n =1 n =2 n =3

Not only does this allow for the discrete energy levels but it also
explains the pattern of the energy levels in an atom where the levels
get closer and closer together. This corresponds to the differences

ground state

n = 2

n = 3

n = 4
n = 5

n = ∞

n = 1

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Physics Factsheet

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155. Aspects of Quantum Theory

Pauli’s Exclusion Principle
Each energy level represents a position that the electron may occupy
in the atom, however most electrons will remain at the ground state
(n=1) unless excited (absorbing energy and moving to a higher
state). Even then they will immediately emit the energy as a photon
and drop back to the ground state. If the atom has several electrons
however, then the exclusion principle comes into effect.

There are several states on each energy level. The exact number is
given by: No. of states = 2n2 Where n is the energy level.

For example on energy level n=1 there are 2×12 = 2 states and on
n=2, there are 2×22 = 8 states.

Heisenberg’s Uncertainty Principle
Heisenberg stated that it is impossible to know everything about a
situation exactly.

There are two main ideas here:
• Firstly that you cannot know the exact position of a particle and

know its velocity. This is explained in more detail below.
• The second idea is to do with uncertainty in the energy of

particles. All we need to know is that there is an uncertainty in
the energy of a photon that is emitted by an atom. As the
frequency of the photon is determined by its energy this means
that there is an uncertainty in the frequency of the spectral line
produced. This explains why spectral lines are slightly thicker
than you might expect them to be if only one exact frequency
was being emitted. The tiny range of frequencies emitted for
each line depends upon the uncertainty.

Example 3:
(a) What is the maximum number of electrons that can exist on

the 5th energy level? [3 marks]
(b) What happens to any electrons which are added above this

number? [1 mark]

Answers
(a) Max no. of electrons depends on the energy states, one

electron is allowed per state ,
There are 2n2 = 2×52 = 50 states.
So 50 electrons on the 5th energy level.

(b) Electrons go onto 6th level until is full then go onto higher
levels.

The Pauli exclusion principle states that only one
electron can occupy a single state at any one time. Given that
an electron will tend to sit at the lowest possible energy level,
then an atom with many electrons will fill up the energy levels
from the bottom, not moving to the next level until all the states
have been filled up.

Fig.3. The Bohr atom, constructed using the energy levels
and placing one electron in each state. This shows possible
positions of electrons for the first 4 energy levels.

nucleus2

8
18

32

Uncertainty in Position and Speed

The first idea is that it is impossible to measure both the speed and
position of a particle. This may seem ridiculous, but it is a valid fact.
To understand it, you must think about the way that we ‘see’.

How do you see a ball?

Fig 4. Large scale, classical model of ‘seeing’ a ball

Light comes from a source, reflects off the ball and is detected by
your eye. As a result of this you can not only see exactly where the
ball is, but you can take measurements to determine its exact speed
too.

It makes no real difference whether we use the photon model or the
wave model to represent the light, the effect is the same in either
case.

How do you ‘see’ an electron?

Being tiny we could not see them with the naked eye of course, but
detecting them uses the same principle as seeing; to shoot
something (light or particles) at the object you wish to look at and
detect the particles as they bounce off it. It makes sense then that
we shine light at the electron.

At this scale we must use the photon model of light.

Fig 5. A photon hitting an electron

S

kinetic energy = hf

Electron

When a photon (carrying energy E = hf) strikes an electron, the
energy it carries is absorbed and becomes kinetic energy, causing
the electron to change velocity. This means that the act of trying to
observe the electron changes its velocity and position.

photon
E = hf

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155. Aspects of Quantum Theory

An alternative is to use a beam of electrons to detect the electron.
Of course even this cannot measure both the velocity and position
of an electron.

Fig 6a observing an electron using electrons

beam of electrons

electron being
'observe'

Fig 6 shows that using an electron microscope to measure position
and velocity of an electron is a bit like finding a tennis ball by
throwing many tennis balls at it.

The problem is not that it is a bit complicated to work it out; the
point of the uncertainty principle is that the very act of observing
the electron changes its velocity and position. Although it would
be possible to look at the scattering of the electrons and determine
where the observed electron once was, the only thing we can say
for certain is that it is no longer there and that we have no idea how
fast it is or was going.

Example 4:
Heisenberg is pulled up by the police. The officer steps out
and asks ‘Do you know how fast you were going?’ Heisenberg
replies ‘No, but I can tell you where I was.’
You might not find it funny, but what could Heisenberg’s other
response have been? [1 mark]

Answer
He could have replied ‘Yes, but I’ve no idea where I am.’
(sadly this is even less humorous)

Fig. 6b the observed electron is knocked out of position

electron being observed
is repelled

beam of electrons is scattered

Schrödinger’s Atom
Although we started with an electron looking like a small round ball,
we have discovered that it is actually more like a wave at times,
especially inside an atom. We have also discovered that it is very
hard to pinpoint where an electron is and what it is doing.

If we consider the De Broglie wavelength that represents an electron
and also apply the uncertainty principle, what we get is a probability
function in space which represents the probability of finding the
electron at a given position.

Fig 7. The exact position cannot be determined but the probability
of finding the electron can be calculated

Schrödinger took this one step further. Instead of imagining the
electron as buzzing around somewhere inside the probability function
he said that it actually exists at all points inside the wave at the same
time.

Schrödinger said that the electron actually occupies all
possible positions and states simultaneously and that only by
observing the electron did we force it into one state or another.

To help explain he used the famous Schrödinger’s cat analogy.

areas represent probability of finding the electron in that position

Fig 8 The cat in the box

Note that Schrödinger did not actually conduct this experiment.
Neither should you!

If you put a cat in a solid box with an open bottle of poison attached
to the lid, the cat will be in one of two states. It will either be alive or
it will have knocked over the poison and sadly died and there is no
way of knowing which without removing the lid and having a look.
The idea is that the cat is both alive and dead (occupying all possible
states). The only way to be sure is to open the box, but as the
poison is attached to the lid, the poison will spill and kill the cat,
thereby forcing the cat into one of the two states.

This new idea about electrons leads to a different view of the atom
from Bohr’s very clearly defined model.

Fig 9. Schrödinger’s atom

In the Schrödinger model of the atom, the electrons do not exist as
point particles orbiting the nucleus, but as a ‘cloud’ which represents
the electrons occupying their many possible states. When observed,
the electrons will be forced to occupy one of the states shown in the
Bohr model (fig 3).

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