Planck’s Constant and Quantum Theory

The Constant

• Planck’s constant relates the energy of a photon to its frequency
• (It’s relevant to the photoelectric effect. See other post)
• The constant is 6.63 *10^(-34)
• It’s in units Js (not that it needs units), and it’s represented by the letter h
• The equation is E=hf, where E is the energy of the photon, and f is the frequency
• Photons are tiny, and don’t carry much energy
• You need loads of photons to carry decent amounts of energy, so it makes sense that Planck’s constant is very small
• Even rather high-frequency photons only carry small amounts of energy
• (It’s all relative, though. To the electrons, it must seem like higher-frequency electrons are carrying an awful lot of energy)
• Sometimes you’ll be told the power of a light source, and the frequency of photons it emits, and asked to work out how many photons it emits per second

Two other things

• The charge on a single electron is -1.6 *10^(-19)
• This is called the elementary charge, and represented by e
• An electron volt, eV is the energy of one electron in a potential difference of one volt

Anyway, Quantum Theory

• It seems light isn’t a particle, or a wave – it’s more complicated than that
• Light is clearly a particle (photons, anyone?), yet it clearly exhibits wave-like behaviours, some of which can’t be explained using the particle theory
• The current theory is that photons take literally all possible paths they could possibly take, at once
• They’ve got to end up somewhere, though, and the destination is all down to probability
• (The probability of something happening given that something else has happened is different from the probability that just the first thing happens, which is probably why photons do weird things when you’re checking up on them. That doesn’t seem to be part of the course, though. But I’m sure there are plenty of good books on it)

Working Out The Probability

English: Graph of a phasor as a rotating vector (Photo credit: Wikipedia)

 

• There are infinite paths to get from one particular place to another particular place, but, brilliantly, most of them will cancel out, because they’re symmetrical/mirror images of another path
• As a result, you can consider only a few very simple paths, and still get a good idea of the probability of a photon arriving somewhere
• You stick phasors on these paths and get them to rotate as they move from the start to the finish
• At the end, you remember what directions they were all pointing in, and add them together (the same way you’d add vectors) to get a resultant phasor
• The probability of a photon arriving in this place is proportional to the resultant phasor amplitude squared
• The amplitude and starting direction of the phasors don’t matter, as long as they are the same for each path#
• The places with the highest resultant phasor amplitude will be hit by the most electrons

Calculating Your Phasor Stuff

• The time a phasor should take to rotate once is the period of the photon (one divided by its frequency)
• The wavelength of a photon is the distance it will travel during one phasor rotation
• The number of times the phasor will rotate along the path is the time it takes to travel the path, divided by the time it takes the phasor to rotate once, which is also the time taken to travel the path multiplied by the frequency
• Notice how we’re still talking in terms of waves…

Photoelectric Effect, Photomultipliers and Resolution – AS Physics Revision

 

 

Photoelectric effect (Photo credit: Wikipedia)

Photoelectric Effect

• The photoelectric effect is one of the pieces of evidence which proves that light isn’t just a wave
• Photons (the quanta of light. I don’t think they need introduction) transfer their energy to an electron, when they hit one
• If an electron gains enough energy to break its bond with the atom, it will go flying off
• The electrons furthest from the nucleus are the ones that have the most energy to start with
• Because of relationship between energy and frequency of electrons (featuring the Planck constant) some frequencies/wavelengths/colours of light or electromagnetic waves work better for the photoelectric effect than others
• The wave theory could have predicted that the light intensity would dictate the number of electrons emitted, but it couldn’t explain that it actually depended on the frequency, not the intensity
• The photoelectric effect works best on a low work function (think of it as electron escape velocity), negatively charged (so there are plenty of electrons), metal plate, using high-energy (high-frequency) photons

Photomultipliers

• Photomultiplies (a type of vacuum phototube, according to Wikipedia) are devices that allow you to detect tiny amounts of light, by ‘amplifying’ a single photon
• They’re used in medical imaging, nuclear/particle physics and astronomy (you can tell I had homework to do on this)
• (Apparently, they’re also part of radar jamming and night-vision goggles. Oh, Wikipedia!)
• They rely on the photoelectric effect to work, and also something called secondary emission
• Secondary emission is when metal surfaces are hit by a beam of electrons, and emit more electrons than actually hit them
• Here’s what happens… (in an ordered list, needlessly…) <l>

Detecting Photons:

1. A photon arrives
2. It hits a negatively charged metal plate with a low work function (see photoelectric effect paragraph)
3. An electron is excited, and leaves the plate
4. It hits another metal plate, where secondary emission takes place
5. The electrons hit another metal plate, where secondary emission takes place
6. The electrons keep hitting metal plates, and secondary emission keeps taking place
7. Each time secondary emission happens, the electron beam has more electrons in it
8. Eventually, the beam is big enough, and it lands on a sensor, which informs a scientist that a photon has been detected

Resolution

• The resolution of a sensor is the smallest change in input it is able to detect (and produce a measurable change in output for)
• When you’re talking about interference patterns, it’s more to do with how close two patterns can be before they’re indistinguishable
• i.e. are they far enough apart that you can tell one from the other?
• The closest two interference patterns can get before they’re indistinguishable is when the first minimum of one lines up with the peak of the other
• To improve the resolution, you could increase the slit width, or decrease the wavelength of the wave (like they did when they switched from red ray to blu-ray, to fit more information on a disk)
• I seem to have written an equation in my book. I’m not sure quite what it’s for, but it’s next to the stuff on resolution. Here it is:
• θ = width/L
• I might work out what that means and get back to you…

Standing Waves – AS Physics Revision

So… Standing Waves

Standing Wave (Photo credit: Wikipedia)

 

• Progressive / travelling waves are waves that go somewhere (it looks like the peaks are moving)
• Standing waves are waves that just go up and down on the spot
• They have parts that don’t move, called nodes…
• …and parts that go up and down the whole amplitude, called antinodes
• At nodes, the displacement from equilibrium is always zero
• At antinodes, the average displacement is zero, but that’s only because it goes right  to the top and right to the bottom, cancelling out
• Standing waves are formed when two travelling waves going in opposite directions superpose

In pipes and on strings

• Standing waves can be formed on a string (such as a guitar string)
• The fixed ends of the string become nodes
• There’s no non-fixed end on a guitar, but if there was, it would be an antinode
• Similarly, sound waves can be generated inside pipes (such as in musical instruments)
• Sound waves are longitudinal, but for ease of drawing diagrams, you can get away with pretending they’re transverse, in this case
• Open ends of the pipe will become antinodes, and closed ends will become nodes
• By drawing a diagram of the string/pipe, labelling where the nodes and antinodes would form, and drawing a wave over it, you can work out the number of wavelengths of standing wave that fit
• If you know the length, and the number of wavelengths in that length, you can work out the wavelength of that wave 

Fundamentals and Harmonics

• The longest (wavelength) standing wave you can fit on a string or in a pipe (bearing in mind the rules above) is called the fundamental
• There will be further harmonics that can fit – double, triple, quadruple the fundamental, and so on…
• The fundamental on a guitar string is half a wavelength, because it has a node at each end and an antinode in the middle. λ=2L, where L is the length of the string.
• The first harmonic on a guitar string is a whole wavelength, because it has a node at each end, a node in the middle and antinodes between the nodes, just like one full cycle of a sine wave. λ=L
• The second harmonic has nodes at each end, two nodes between those, and antinodes between the nodes. λ=2L/3
• A pipe open at both ends would have a fundamental with two antinodes, and a node in the middle, making it half a wavelength, so λ=2L
• Its first harmonic would have antinodes at either end, and in the middle, with nodes between the antinodes… you get the picture
• A pipe open at one end and closed at the other (or a string with one fixed end, and a loose end, I guess) could have a fundamental with only one node and one antinode, making it a quarter of a wavelength, so λ=4L

Superposition, Interference and Phase – AS Physics Revision

Wave Collisions

• When multiple waves occupy the same space, at the same time, they add together
• This is called superposition
• The resultant wave’s amplitude could be more or less than their individual amplitudes
• It doesn’t permanently affect the waves – once they’ve gone through the other waves, they just carry on exactly as they were before the encounter (because we’re in an ideal world where they don’t seem to lose any energy)
• If two identical waves going the same way, at the same speed, time, etc, superpose, the result will be a wave of double their individual amplitude, but the same frequency
• However, if one of those waves is half a wavelength ahead of the other, its negative bits are exactly where the other’s positive bits are, so you get a flat line as the result
• This is called antiphase, and happens when the waves are 180 degrees, or Pi radians, out of phase, as will be covered in the next unordered list

Interference, Coherence and Phase

• When the resultant is bigger than the individual waves superposing, you have constructive interference
• When the resultant is smaller, it’s destructive interference
• Absolute destructive interference, where there’s no wave left (until the superposing waves separate) happens when the superposing waves are in antiphase
• Complete constructive interference, where the wave doubles, happens when the superposing waves are in phase
• The wavelength of two coherent waves, and their phase difference, determines how they will interfere (constructively or destructively)
• Phase difference is the amount by which one wave lags behind another
• Only waves that are coherent will interfere with each other
• Coherent waves have a constant phase difference between them
• Waves have to come from the same source to be coherent (for all intents and purposes)
• The coherence length is the distance over which the waves are still coherent

Thin Film and Path Difference

• When light encounters a thin film some of it will reflect, but some will travel through the film to the other end, and reflect there
• Because this light has travelled a bit further than the other light, it lags behind the other light, so there is a phase difference
• This means that it will interfere with the other light as it reflects
• Since white light is made up of many different colours, with different wavelengths, they all interfere differently, creating a nice light show
• (Phase changes when during reflection, which adds another layer of complexity to this phenomenon)
• The thickness of the film, the wavelength of the light, and the angle of incidence hitting the film all have an effect on the pattern
• If the film is too thick, though, thin-film interference doesn’t work

Path Difference and Wavelength

• A reliable way to get coherent waves (from the same source) interfere is to them travel different distances
• The distance one has travelled further than the other is called the path difference
• If the path difference is equal to the wavelength , two wavelengths, three wavelengths, or so on, the waves are in phase and will interfere constructively
• If the path difference is equal to half a wavelength, 1.5 wavelengths, 2.5 wavelengths, or so on, the waves are in antiphase and will interfere destructively
• In other words, if the path difference = nλ, where n is an integer (and lambda is the wavelength, obviously), the waves are in phase
• And, if the path difference = nλ + 0.5λ, or λ(n+0.5), then the waves are in antiphase
• They can, of course, be somewhere between antiphase and completely-in-phase (and usually will be) – leading to patterns of increasing and decreasing amplitude in the interference pattern

Phasors

• Phasors are a way of working out superposition, and representing phase difference
• They’re more important in quantum theory, but are used for waves, too
• They’re arrows, which represent the phase at a point on the wave as a direction
• They have amplitude, which is the length of the arrow (in a similar way to how vectors can be represented as arrows)
• As a wave goes along, the phasors rotate ANTICLOCKWISE
• They start from the right, pointing to three o’clock
• Each phasor is slightly behind the one in front of it, in its rotation
• When you’re working out superposition, instead of adding the amplitudes at each point, you can add the phasor arrows at each point, to get a resultant

Related articles

• Waves – AS Physics Revision (mattg99.wordpress.com)

Waves – AS Physics Revision

Terms

Sine waves of different frequency. (Photo credit: Wikipedia)

• Transverse waves are the ones that go up and down as they go along, and look like a sine wave when you draw them (oscillation perpendicular to direction of travel)
• [Longitudinal waves oscillate in the same direction they travel, so they're sort of pressure waves, like sound waves (and difficult to draw)]
• The amplitude is the maximum displacement from equilibrium, of a wave. How far it is from the middle, at its highest/lowest point, basically
• The period is the time taken for one complete oscillation (to the top, to the bottom and back to where it started, not necessarily in that order)
• The frequency is the number of oscillations per second

Measuring and Equation-ing

• The symbol for wavelength is lambda, λ
• Wavelength is measured in metres, m (often nanometres, nm)
• The period is measured in seconds, s
• The frequency is measured in hertz, Hz (often kilohertz, kHz)
• Frequency and period are reciprocals: fT=1, f=1/T, and T=1/f, where f is the frequency and T is the period
• That’s why Hz^-1 = s, and Hz = s^-1. Hertz are kind of ‘per-seconds’
• The speed / velocity, v, of a wave is measured in metres per second, ms^-1 (when it’s velocity, it’ll have a direction attached, remember)
• v=fλ is the wave equation. It’s useful. Remember it.
• (I’m going to assume you know how to use it. Hint: substitute values in)

On Graphs and in Radians

• Ever seen a graph of y=sin(x)?
• It’s a transverse wave of amplitude 1
• The period of the graph depends on whether you’re measuring the angles along the bottom in degrees or radians
• See my AS Maths Revision post on radians if you don’t know what they are
• The period of a sine wave in degrees is 360
• The period of a sine wave in radians is 2 * Pi
• If the scale along the bottom is in metres, it’s not a graph of y=sin(x) – it’s just a drawing of a perfect sine wave

Related articles

• Radians – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)

Forces, Falling and Projectiles – AS Physics Revision

Parabolic Projectile Paths

• The theoretical path for a fired object is a parabola
  

  English: Trajectories of three objects thrown at the same angle (70°). The black object doesn’t experience any form of drag and moves along a parabola. The blue object experiences Stokes’ drag, and the green object Newton drag. (Photo credit: Wikipedia)

• You may remember parabolas from such courses as GCSE Mathematics – they’re various transformations of the y = x^2 curve
• Something fired at a steep angle will follow a tall, narrow parabola
• Something fired horizontally will follow a path that’s only half a parabola
• This assumes that the horizontal and vertical components of the vector are independent, and that there’s no air resistance or anything like that!
• You can use the SUVAT equations to calculate the distance, velocity, acceleration or time of the x and y components separately
• (See the SUVAT equations post)
• The initial velocity will be zero, because this thing was fired from rest
• (I guess projectiles can be fired from moving vehicles, but relative to the vehicle, the initial velocity is zero)
• Full parabolic paths are symmetrical, so if you know something about one half of the path, it’s probably true for the other (we’re ignoring air resistance, remember?)

Weight, Mass and Gravity

• Forces are vectors (see the vectors post), because they act in a direction
• Mass is a scalar, the amount of matter in an object. It’s not a force.
• Weight is the force that pulls things down. Not to be confused with mass.
• Gravity is an acceleration, not a force. It has magnitude g, 9.8 or 9.81 m s^-2 (on Earth, anyway) and acts in the direction towards the centre of the Earth (i.e. down)
• Newton’s second law, F=ma, states that force is the product of mass and acceleration
• Thus, weight = mass multiplied by gravity

Terminal Velocity

• Whilst we’re talking about stuff going down, I might as well mention it
• Terminal velocity is the velocity (in the downwards direction) at which the weight (force) is equal to the drag (air resistance, or whatever force is acting in the upwards direction), so the falling object stops getting faster and just goes at a constant speed
• That’s because the weight and drag are the same magnitude, in opposite directions, so there’s no resultant force and no acceleration (no vertical component, anyway)
• Newton’s first law states that an object will continue at a constant speed, until it’s acted upon by a (resultant) force
• Terminal velocity depends on weight and air resistance (so mass, gravity and cross-sectional area)

Miscellaneous

• You can add forces up to work out the resultant as described in my post on vectors
• When there’s a resultant force, there’ll be an acceleration, and vice versa (unless the mass is zero, I guess)
• The reaction force of a surface is the force with which a surface pushes back against something pushing against it
• I suppose that force is what explains why I’m not falling through the chair I’m currently sitting on
• The retarding force is the term for the combined effects of friction, air resistance, etc
• You can get forces generated by vehicles, etc, propelling them forward. I don’t know what they’re called. Propulsion or thrust, perhaps. Not “push” and “pull”, though.
• Is it me, or is “parabolic projectile paths” a tongue twister that gets easier the more times you say it quickly in a row?

Related articles

• SUVAT Equations – AS Physics Revision (mattg99.wordpress.com)

Geometric Series – AS Maths Revision – Core Mathematics (C2)

Geometric Series(es?)

An illustration of geometric series. Each of the purple squares is 1/4 of the area of the previous square, with the total area being : \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots \;=\; \frac{1}{3}. SVG redraw of original image. (Photo credit: Wikipedia)

 

• Hi, I’m ‘a’, the first term in a geometric series
• You may remember me from such series as the arithmetic series from Core 1
• The difference here is that instead of adding something to get from one term to the next, you multiply by something
• That something is the common ratio, ‘r’
• Thus, the series goes a, ar, ar^2, ar^3…
• So the nth term, if you will, is ar^(n-1)
• Sometimes you’ll be given a and r and asked to work out terms
• Other times you’ll be given some terms and asked to work out a and r
• Other times you’ll be given a or r and asked to work out the other, given a term or two
• Simultaneous equations are sometimes involved (solving by substitution, or even elimination)
• Hopefully the Core 2 examiners won’t get any more inventive than that, though

To Infinity (no further)

• If the common ratio is greater than 1, no sum to infinity for you. Go away.
• Wait, come back. Is it less than -1? If so, go away.
• If you’re still here, your common ratio is between (not including) -1 and 1
• That means it’s a convergent function, so a sum to infinity is possible
• Otherwise, it’s a divergent function, in which case the sum to infinity would literally be infinity, so it’s not worth calculating
• Convergent series get closer and closer and closer and closer (you get the picture) to a number
• You can find out what number your series is aiming at using the formula:
• Sum to infinity = a / (1 – r)
• It’s the first term divided by… 1 minus the common ratio
• And it only works for convergent series! (Ones which get smaller each time)

Proving the Sum to n

• Actually, I can’t be bothered to prove it here (mainly because to do so in ASCII would be rather difficult indeed)
• You do seem to need to be able to prove the formula for the sum of a geometric series to n, though, so it might be worth memorizing
• Anyway, what you get is this… (and it is in the Core 2 formula booklet)
• Sum to n = (a * (1 – r^n)) / (1 – r)… or something like that
• It’s pretty easy to use. You may have to use it in an inequality, too, just as a heads up.

Related articles

• Logarithms and Exponential Stuff – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)
• The Binomial Expansion – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)
• Factor, Remainder Theorems and Algebraic Long Division – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)