AS Physics Unit 2
Waves and Quantum Behaviour
Superposition and Coherence
Waves are not physical objects; therefore they do not bounce off each other when they
collide.
Instead, they combine temporarily and then continue to travel as before. This
combination is called superposition.
The interference caused by superposition can be constructive, where two peaks meet
and make a bigger peak, or destructive, where a peak meets a trough and they cancel
out to some extent.
The phase of a wave, how far it is displaced from 0 at that point, can be represented
by a phasor.
The resultant amplitude when two waves meet can be found by joining their phasors
tip to tip.
If the waves are at the same point in their cycle, they are said to be in phase, with a
phase difference of a multiple of 2 π radians.
Waves in phase at a point in space interfere constructively. Their phasors point in the
same direction, giving the maximum possible resultant.
Waves at points in their cycle as far apart as possible are said to be in antiphase, with
a phase difference of an odd number times π radians.
Waves in antiphase at a point in space interfere destructively. Their phasors point in
opposite directions, giving a minimum resultant.
Waves at different points interfere with each other to a lesser degree, whether this is
destructive or constructive can be worked out by drawing phasors.
Interference patterns can be established when waves line up so that the phase
difference is always the same; this requires waves to be coherent. Sources are
coherent when they have the same wavelength and frequency and a fixed phase
difference.
The phase difference of two waves depends on the difference in the lengths of the
paths between the sources and the detector, known as the path difference. If the path
difference is a whole number of wavelengths the phase difference is 0, if the path
difference is a half number of wavelengths the phase difference is π/2.
Standing Waves
Standing waves are formed when waves are confined, such as on a length of string,
and a whole number of waves is produced in the time it takes one wave to go to the
end and back.
The effect of the moving pulses is to create a series of apparently stationary pulses
that move up and down instead of forwards and backwards.
The frequencies at which this occurs are called resonant frequencies, the relationships
between them is the basis of musical scales.
The stationary points are called nodes, the moving points are antinodes. Each
(anti)node is half a wavelength away from the next (anti)node.
On strings with fixed ends, the lowest resonant frequency, the fundamental frequency,
has a wavelength twice the length of the string.
Further resonant frequencies each add half a wavelength to the string (i.e. ½, 1, 3/2,
2). These higher resonant frequencies are referred to as harmonics.
Longitudinal standing waves can form in columns of air, as in wind instruments.
In a tube with a closed end the lowest resonant frequency is when the tube is one
quarter of the wavelength. Each subsequent resonant frequency adds half a
wavelength.
In open ended tubes the lowest resonant frequency is when the tube is half a
wavelength, with a node in the middle of the tube. Each subsequent resonant
frequency adds half a wavelength, with antinodes at both ends.
This phenomenon can also be demonstrated by reflecting electromagnetic waves off a
metal plate. In microwaves the wavelengths are convenient to be calculated by this
method.
Diffraction
When waves pass through narrow gaps they diffract, spreading out from the gap.
The effect is only noticeable when the width of the gap is similar to the wavelength.
This is also happens when waves pass around objects, immediately behind the object
there is a shadow, as the wave diffracts into the space the shadow becomes narrower
until the wavefront is complete again.
With light passing through a slit a diffraction pattern forms, the narrower the slit the
wider the pattern until the slit is narrower than the wavelength, when it cannot pass
through.
At bright points the path differences from all points in the slit are very similar so the
phasors give a large resultant.
At dark fringes the path differences are larger so the phasors give a small resultant.
Two Source Interference
If waves from two sources are coherent then an interference pattern can be formed.
With Young’s slits this interference pattern is a series of alternating bright and dark
fringes on a screen.
Points where the path difference from the two slits is a whole number of wavelengths
are maxima, points where it is some and a half wavelengths are minima.
𝐷𝜆
Young developed an equation to describe this situation: 𝑋 = 𝑑 , where X is the fringe
spacing, D is the distance from the slits to the screen, λ is the wavelength and d is the
slit separation.
Young’s experiments “settled” the wave vs. particle argument over the nature of light.
Diffraction Gratings
Diffraction gratings produce similar patterns to Young’s slits but are sharper.
Monochromatic light produces extremely sharp fringes as there are many beams
reinforcing them.
This works because the path difference between the slit at the end and any other slit is
always a multiple of the path difference between the first slit ant the next one along,
because similar triangles are formed by the slit separation, the path difference and the
line between them.
A maximum therefore occurs whenever the path difference between two adjacent slits
is a whole number of wavelengths.
For a grating with slits distance d apart, the angle between the incident beam and the
nth order maximum is given by:𝑑 sin 𝜃 = 𝑛𝜆.
For larger values of λ sinθ and hence θ are larger. For larger values of d sinθ and
hence θ are smaller. If sinθ is greater than 1 a maximum of order n cannot exist.
If white light is passed through a diffraction grating a whit central maximum is
formed but at other maxima spectra form as the different wavelengths diffract
different amounts.
Gaps in these spectra (see below) are often searched for with diffraction gratings
rather than prisms as they are more accurate.
Light and Photons
Light travels at the universal speed limit, c.
In the late 19th century light was believed to travel as a wave, but this could not
explain the photoelectric effect:
When light is shone on a metal it may or not emit electrons. Experiment showed that:



For a given metal, no photoelectrons are emitted when exposed to radiation
below its own threshold frequency.
The kinetic energy of the electrons ranged from 0 to a maximum, this
maximum increased with the frequency of radiation but was not affected by
intensity.
The number of photoelectrons emitted per second was proportional to the
intensity of the radiation.
Classical attempts to explain the similar problem of blackbody radiation suggested
that everything above room temperature constantly emitted x-rays and everyone was
dead. Planck attempted to simplify the mathematics by pretending that
electromagnetic radiation could only be emitted in finite packets called quanta, he
later realised that this must actually be the case.
When Einstein looked at the photoelectric effect he realised that as well as only being
emitted in quanta, light could only exist in quanta, which he called photons.
Each photon had energy proportional to its frequency: 𝐸 = ℎ𝑓 =
ℎ𝑐
𝜆
.
He stated that photons give up all or none of their energy when they interact with
other particles.
In terms of the photoelectric effect, a photon interacting with an electron on a metal
surface will cause the electron to be ejected only if its energy is equal to or greater
than the work function φ, the energy necessary to remove the most loosely held
electron.
If the photon has more energy that φ it will either give the electron kinetic energy or
interact with and eject a more tightly held electron.
The threshold frequency is hence such that 𝜑 = ℎ𝑓.
Energy Levels and Photon Emission
Electrons in an atom can only exist at certain energy levels, each of which has a
number from n=1, the ground state, to n=infinity, being free.
Electrons can move up a level by absorbing a photon with the same amount of energy
as the difference or move down by emitting a photon with the same amount of energy
they gain.
When a gas is heated its electrons are excited and move up to higher levels. Over time
they fall back down and emit a photon, so the gas emits light.
When this light is split a spectrum of bright lines against a black background is seen,
the frequency of the light at each line corresponds to the energy of a gap between
levels in the atom.
Hot things emit continuous spectra of visible and infrared radiation. If this is shone
through a cool gas the electrons absorb photons of the right energy to move up a level
and so when the light is split dark lines appear matching the bright lines of the
emission spectrum.
Each element has a particular set of lines which can be used to identify it, this method
was used to discover helium in the Sun before it was found on Earth.
Sum Over Paths
Feynman realised that instead of taking one path between two points a photon would
take all paths, even the really wiggly ones.
Each path has a phasor, the different amount of time a path takes leads to phase
differences.
Adding up the phasors for all paths (or just a few important looking ones) gives a
resultant.
The probability that a photon will arrive at that point is proportional to the square of
the resultant.
As any light source chucks out loads and loads of photons, the probability a photon
will arrive there is essentially a measure of the relative intensity.
When the phasors for all possible paths are added up the probability photons will
arrive at that point by any path is given. The contribution of a small group of phasors
to the resultant (do they roughly line up with it or go around in a circle?) represents
the probability that a photon will take one of those specific paths.
The difference in probability is so great that it always appears that light takes this
path, even though a few photons go by other routes.
The paths that contribute most are always those close to the path that takes least time,
this gives rise to the laws of optics.
Quantum Behaviour of Electrons
Louis de Broglie made the bold suggestion in his PhD thesis that if light, which was
wave-like, could act like a particle, then electrons, which are particle-like, might act
like a wave. Which is pretty bold for a PhD thesis, but then again Marie Curie got her
first Nobel Prize for her PhD work.
The de Broglie equation relates a wave property to a moving particle property, which
ℎ
was considered crazy: 𝜆 = 𝑚𝑣. However, he was later proved right by experiments,
which is why you have to try to pronounce his name.
If electrons are accelerated towards a graφte target they display a diffraction pattern
on a screen behind it .
E=hf still applies, but E is the kinetic energy of the electrons.
If the speed of the electrons is increased the rings of the diffraction patter become
smaller, which is predicted by the smaller wavelength given by the de Broglie
equation.
Electrons also behave like waves in other situations, such as in Young’s slits.
It has since been proven that all particles, including molecules and even large objects,
act like waves.
This only happens when the particle interacts with objects of a similar size to its de
Broglie wavelength, so for a tennis ball to diffract through a racquet the holes would
need to be 1019 times smaller than an atomic nucleus, which it wouldn’t fit through.
Unless it tunnelled, but in that case it still wouldn’t diffract. Which is why you don’t
need to study quantum physics to play tennis.
Space, Time and Motion
Scalars and Vectors
Scalars are amounts of stuff or things and stuff.
Vectors are amounts of things and stuff in a particular direction.
Vectors must always be given with a direction, usually stated at the beginning of the
question.
Vectors can be added by putting them tip to tail to find out what would happen if both
happened at once. The new arrow formed is the resultant.
If they are at awkward angles scale drawing is helpful, if not the resultant is easy to
calculate by Pythagoras’ Theorem.
A vector can also be split into two orthogonal components, which often makes
drawings and calculations easier. This is called resolving.
In this situation vx=vcosθ and vy=vsinθ.
Motion with Uniform Acceleration
When an object is has a constant acceleration (and hence by definition is going in a
straight line) four equations can be used to describe its motion.
𝑣 = 𝑢 + 𝑎𝑡
𝑣 2 = 𝑢2 + 2𝑎𝑠
1
𝑠 = 𝑢𝑡 + 𝑎𝑡 2
2
𝑠=
𝑢+𝑣
𝑡
2
Often, questions will give three values and tell you to find the fourth.
When an object is accelerating under gravity g is taken as 9.8ms-2, because accurate
measurement is for wimps.
If objects are projected horizontally you may have to work out how long they take to
fall with suvat and then how far they go in that time.
If objects are projected at an angle resolve the original vector into horizontal and
vertical vectors, use the vertical as the initial velocity upwards and then suvat with
a=g, then work out the horizontal distance in that time.
If an object is under thrust at an angle it sucks to be you.
Graphs
Displacement time graphs show displacement against time. A straight part represents
a constant velocity, a curved part indicates acceleration. Isn’t it nice when you find
something they didn’t lie to you about at GCSE?
The gradient at any point on a displacement time graph is the velocity of the particle.
The gradient on a velocity time graph is the acceleration at that point. A straight line
is hence a constant acceleration.
The area under a velocity time graph is the distance travelled in that time. The bit
below the x-axis must be worked out separately and taken away to give displacement.
Forces
Free-body force diagrams are used to show all the forces acting on an object and none
of the forces it is exerting.
Always remember the awkward forces like reaction and friction.
Like with other vectors, it is often handy to split a force into two orthogonal forces.
The resultant force on an object can be calculated by adding all the forces together.
When resolving forces on an object it’s a good idea to πck the direction you think it
will move in and 90o to this direction.
Objects will only accelerate when forces acting on them are not in equilibrium,
though this can mean they keep moving at equilibrium.
Acceleration is proportional to the resultant force by f=ma.
This means that under the same gravity all objects fall at the same rate disregarding
air resistance.
Work and Power
Work is simply the transfer of energy into another form, usually by application of a
force to overcome another force.
Work=force x displacement. The work is only the energy transferred, other types of
energy may be present in the system.
If the force is at an angle to the direction of motion only the component in the
direction of travel contributes to the work done.
Power = work done / time = force x velocity.
Conservation of energy means that a system can never lose energy; it is always
transferred to another part of the system in a different form.
This can be used in to calculate the amount of energy transferred in questions, usually
from gravitational potential to kinetic, but only works if air resistance is ignored.