Quantum Physics – with a Coin

Find a coin.

Go on, rummage in a pocket, purse or wallet and find something a reasonable size like a UK 2p or 10p coin, about 1 inch or 2.45cm in diameter. This is all you need to explain quantum physics to someone.

Yes I did just say quantum physics. Surely that’s only for geniuses you cry! Quantum physics is a subject that intrigues and is hopelessly misunderstood in equal measure. This little demonstration will really impress your friends and explain all the basics of quantum physics in simple language. Let me show you how.

I’m assuming you can toss a coin. Throw it up and catch it then slap it down on the back of one hand. Did you get heads or tails?

Quantum Physical States and Probability

The only outcomes possible in a coin toss are heads or tails. You can balance a coin on its rim if you are careful and have a flat table, but that outcome never occurs in a coin toss, due to the slapping-it-flat-on-the-back-of-your-hand action.  There is no way to get heads and tails at the same time as they are on opposite sides of the coin. So I’m going to go ahead and say that the result of a coin toss is one of two possible states, either heads or tails.

Toss the coin a few times and you’ll find out that the number of heads and number of tails usually balance out. You should have an unbiased coin. This would mean the chance of getting a head each time you toss is the same as the chance of a tail, 50:50 chance of either state.

What if we suspected one side had been weighted and so the coin was biased? Tossing the coin hundreds of times would give us a very reliable way of finding if it had been tampered with. The numbers of heads and tails should balance out. If 40% of the time the coin’s state was found to be heads and 60% tails we know it is biased towards tails.

Tossing the coin allows us to find out information about what states are possible and what is the chance of ending up with any one of those states.

Now we’ve established that, what state is the coin in before you catch it?

You might say “no state” but that would mean there was no head-ness or tail-ness there at all, nothing. That’s not quite right as the coin is there. You can see it spinning in the air. How do you describe that?

 

Superposition in Quantum Physics

Quantum physics has a neat word for describing the mixture of head-ness and tail-ness that a spinning coin has before you catch it. The word is superposition. Superimposing is a technique used in photography or film where one image is put on top of another. The final image is a composite of the two. Superposition is a bit like that.

The spinning coin is not devoid of states, we can’t say it has no heads or tails to it. Instead it is a constantly changing superposition of the two states. This superposition tells us nothing about what state this unique toss will end in. That depends on all sorts of things like how fast it is spinning, how high we throw it, when we catch it etc. It won’t help us work out the outcome of an individual throw.

The superposition does contain information about any bias in the coin. As it spins, a heavier side to the coin will spend more time at the bottom of the spin and the lighter side would spend more time at the top. There is some information there we could extract about how the pattern of heads and tails would look after 100s of throws, if only we had a way of measuring it.

 

Taking A Quantum Measurement

You already know how to measure the state of the coin. You catch it and slap it down on your hand. Look at the coin and there is your measurement.

But this involves permanently altering the condition of the coin. You are forcing it into one of two possible states. When I measure my height I don’t force a mixture of all possible heights into 5 foot 10 inches. There is no probability of getting anything other than 5’10”. This measurement of the coin is different to normal measurements. The act of measuring the system we are interested in (the spinning coin), fundamentally alters the system we are interested in. Bummer.

There is no way around this I’m afraid. To get information about the possible probabilities of either heads or tails we toss the coin 100s of times. Then we interfere with the spinning superposition each and every time, forcing it into a single state. Only by repeating this measurement over and over do we get enough data to work out a pattern.

This is true of quantum objects like electrons. It is called The Measurement Problem. Only in quantum physics does the act of measuring the system permanently change the system. Or to put it another way, extracting information from the superposition destroys it. This is called collapsing the wave function.

What is a Wave Function?

Waves bob you up and down in the sea or make use of your hand to signal your departure or arrival. In both cases something is oscillating between two positions over and over. Up and down on the sea and side to side with your hand.

Lots of natural phenomena can be described using the language of waves. The variation in light intensity due to day and night, seasonal mean temperature changes, hormone levels in menstruating women, anything which has a repeated variation across a range of values is wave-like.

Physicists use the mathematical language of waves to describe quantum states. The spinning coin has a repeated variation in a range of values, it flips from heads to tails and back again as it spins. We can use wavy maths to describe this variation. The mathematics of waves is even more helpful as it allows us to add waves on top of each other and work out the resulting wavy shape. Like different tides or ripples interfering with each other at a point on the sea.

The spinning coin has a changing superposition of the heads and tails states. The chance of measuring either state depends on which way up the coin is pointing when you catch it. As the coin rotates it is found in a 100% heads state only once each rotation and only for a very short time, likewise for the tails state. The the rotation moves one side of the coin slightly higher and the amount of “heads” reduces slightly and the amount of “tails” increases. Or more correctly the probability of getting heads reduces slightly and the probability of getting tails increases slightly. So the chance of ending up with heads or tails is continuously changing as the coin rotates. If you plotted the probability shape on a graph it would rise and fall and rise and fall just like a wave.

The wave function is the name given to a mathematical wave shape which describes this varying chance of getting a particular state in a measurement.

Collapsing the wave function forces a single state on the coin, in this case either of the extreme positions, giving a head or tail.

 

Now the Quantum Physics Bit!

All this talk of coins is preparing us for the big moment to tie all this in with teeny tiny quantum particles.

There is an experiment which shows exactly what we have been talking about with coins, but with electrons. It’s called the double slit experiment. It is the classic way of introducing people to the weirdness of the quantum world. Electrons behave like a wave superposition of states until you force a measurement on them and then they behave like a lump with a fixed state.

Instead of throwing electrons up and down and catching them with some clever laser trap equivalent of a coin toss (or something!), the electrons are thrown at a screen. The screen can detect the hits from the electrons and emit a tiny light burst. So a pattern builds up showing where the hits occur.

When you shine a wavy source, like a torch, on a screen you get a bright spot in the middle and it gradually dims as the light spreads out at the edges. If you fire a lot of blobs at a screen there is a big build up of bobs in the middle and fewer out to the sides. It is hard to tell one result from the other.

So the physicists put something in the way of the screen. A big sheet with two narrow slits in it.

quantum double slit experiment

Diagram of the double slit experiment from Wiki-Commons

Now you can tell the difference between the pattern produced by wavy things and the pattern produced by blobs. Waves overlap and produce a new wavy pattern as shown in the picture above. The blobby particles are simply split into two streams of blobs by the slits and produce two splats on the screen. No superimposing happens for them.

Electrons are supposed to be blobs not waves. They are particles. When a beam of electrons is fired at the slits they start to build up a pattern on the screen. The pattern is a wavy interference pattern!

quantum double slit interference pattern

Each dot represents the impact of an electron on the screen. Many hits build up to produce an overall wave interference pattern, even though electrons are particles. This is evidence of the wave-particle duality of electrons. Experiment performed by Dr Akira Tonomura in 1989 at Hitachi. Creative Commons ShareAlike licence.

Here we are looking at information contained in the wave function of the electrons. Only when hitting the screen are they being forced into a state. Then their position state is fixed to a definite value. Just like the coins spinning in the air contained information about the probability of heads or tails, so the electrons shooting towards the screen contain information about the probability of where they will show up in the interference pattern.

We can collapse the wave function earlier on and force the electrons into a position state much sooner if we measure them. We can put a sensor on the double slits and record which slit the electrons go through. This destroys the supposition of location states and forces the electron to be in the left or right slit. Like catching a coin destroys the supposition of heads or tails and forces the coin to be one or the other. Then the electrons behave like blobs and build up in two patches on the screen. The waviness of the electrons has been lost.

This experiment was astounding. To actually see the theoretical wave nature of the electrons supposition and then be able to collapse the wave function and bring their blob nature back again has boggled many a mind over the years. Tiny particles can be waves or can be blobs depending on how they are interacting with their surroundings. Some interactions collapse the wavy behaviour and force the electrons into a fixed state. What we understand as measurements always collapse the wavefunction because our measurements require a value, a fixed state to give a result.

Conclusion

Tossing a coin will never look the same again! Each time you catch a coin you can think about how every photon of light hitting your eyeball has just collapsed its wavefunction, every click of the TV remote, every interaction of microwaves with food in your kitchen is a quantum measurement happening in front of you.

Now we need to talk about Schrodinger’s Cat…

schrodinger's cat experiment

A thought experiment proposed by Erwin Schrodinger. Picture from https://www.nobelprize.org/educational/physics/quantised_world/

Make the Most of Your Specification – Top Revision Tips

It is good practice for teachers to share the content of the examination course with their students. Each year the exam boards print the content of the exams you are taking. This content specifies precisely what you are expected to learn, hence they are called the specification. Here in the UK the main exam boards are AQA, OCR and Edexcel.

If you aren’t sure which exam board your exams are with the first thing to do is ask your teacher. The exam boards also publish textbooks and revision guides to help you with learning and revising the course content.

Getting Hold of Your Specification

If by some oversight your teacher hasn’t given you a photocopy of your specification, don’t worry! They are all freely available on the internet for anyone to look at. Past papers, markschemes and examiners’ reports are also available.

Here are links to the major exam boards’ different GCSE and A Level physics specifications.

AQA GCSE

AQA AS/A Level Physics

Edexcel

GCSE Combined Science and Physics  – use the tabs to find the single GCSE physics specification

AS/ALevel Physics

OCR GCSE

OCR AS/ALevel

The specifications are large PDF documents over 100 pages long. Most of it is irrelevant to you. Tedious details of how each exam paper is arranged and assessed fill many, many pages. Go to the index and find “subject content”, this is the bit you want. In a combined science GCSE there will be separate biology, chemistry and physics sections. Most specifications have useful internal links which allow you to jump straight to the physics content from the index.

Be careful, the page number in the index rarely (if ever) coincides with the page numbers on the PDF reader you use to look at it. The document may say physics is between pages 59-74, but if you type those pages into the printer dialogue box and hit print the wrong pages will print out. Select the correct pages from the sidebar thumbnails and print the selected pages. I print the pages back to back and two per sheet to save paper. It’ll look something like this:

 

Understanding Your Specification

Let’s assume you have printed your own copy or been given a copy of your course content. You will find it is conveniently divided into sections by topic. AQA Trilogy which is the example given above, has all the physics content in chapter 6 of the specification and individual topics are numbered 6.1, 6.2, and so on.

Generally speaking, these topics aren’t taught in the exact order that they appear in the specification. This is not a teaching plan. Your teacher can divide up this course content however they like. So don’t be surprised if you were taught topic 6.4 before 6.1.

Your exam questions can be on anything in the first column. The second column lists key experiments, science skills or maths and ICT skills. It gives details about how to use equations and the sorts of data/graphs and typical experiments. Ignore the codes like WS 1.2, 4.3 etc.

At the end of the physics chapter is a short summary of the key ideas that cover the whole subject. This is surprisingly helpful and condenses the content down to overarching principles which you should always have in mind when tackling questions.

Using Your Specification

There are so many ways you can use a specification! I could spend a whole morning showing what you can do with this document. Here are my 5 top revision tips for making the most of a specification.

1 – Definitions and Equations

Go through section by section and write out the definitions the examiner provides you with. For example “A system is an object or group of objects”. There will be definitions for all major physical quantities like current, pressure, momentum etc and for qualities such as insulator, conductor, solid, liquid, gas etc.

Examiners often set one mark questions which ask “What is the definition of X?” and you need to be able to correctly write a short sentence which answers that. Writing out definitions is a good way to get used to concisely answering these questions.

Quantities are often stated as relationships between other quantities, e.g. current is the rate at which charge flows past a point in a circuit and is measured in Amps. These relationships lead naturally to equations; current = charge/time or I = Q/t. The specification includes all of these.

At GCSE all the formula are given in a data sheet which you can refer to in the exam. However they DO NOT give you the units that these quantities are measured in. You have to remember that current is Amps (A), charge is Coulombs (C) and time is seconds (s). The specification lists all the quantities, their units and symbols. Grab a highlighter pen and highlight every equation in your chosen colour. Then you can make a quick and easy list to test yourself make sure you have learnt all the units correctly.

2 – Examples and Scenarios

The specification gives the examples that are asked about on the exam paper. In the pages shown above the section on energy starts with a bullet point list. These are the common situations where a student should be able to describe energy changes.

When revising the topic you can use these examples to check that you can indeed describe the energy changes in an object moving upwards or a vehicle slowing down.

Here it is useful to use the specification to test yourself. Get some paper or a notebook and write out the answer to each example as if it were a question.

3 – Key Facts to Learn

Each section includes the key facts that students need to use to solve problems and answer questions.

There are three parts to this subsection, the last two clearly relate to possible exam questions. Show the directions of the force, current and magnetic field using Fleming’s left-hand rule. List that factors that affect the size of the force. Can you do this? Anything that starts “students should be able to…” is obviously something you can be asked.

Note how they don’t actually give you those factors that affect the force or describe Fleming’s left-hand rule. This isn’t a textbook, it’s a what you should know guide. You’ll need to look these things up in your notes, a textbook or on a reliable science website.

So what do we do with the first paragraph? These parts of the specification are key key pieces of understanding which you should be able to reproduce in answers asking for an explanation. For example “Explain what happens to a current carrying wire in a magnetic field  – 3 marks” could be answered by the first paragraph. You should be able to write these ideas in your own words.

4 – Ticking Off Your Revision as You Go

I encourage students to systematically read through their specification and tick off every sentence, equation and definition that they are sure of. Then go back and put a star or double line next to the bits they need to go over because they don’t remember it or aren’t sure. Using a pencil is helpful because you can rub out stars and tick off bits as you work through each section.

Ticking off the content as you go allows you to be systematic and cover every section completely. The specification is better than your notes from class as notes often have missing bits. You can’t always rely upon class notes to have covered every single point. Many times teachers demonstrate or talk through something but don’t have time to write up formal notes on all of it. It would take far too long and writing notes isn’t the point of a lesson anyway. That’s not taking into account the times you were absent, day dreaming or mucking about! The specification is the bible of your course and is the go-to document for checking your have covered everything.

 

5 – Understanding is the Key to Remembering

There is no point remembering incorrect facts and your brain won’t retain information that you can’t make sense of. Use your specification to find the gaps in your understanding then spend your time targeting those gaps. Everything else will fall into place around them. There is no point revising things you understand and remember already. You must find and work on the gaps.

Then use your class notes, reliable webpages and textbooks to study and understand the sections you have identified that need work. Create your own notes, labelled diagrams or mind maps of topics you have identified as weaker. Just reading a page of text will not miraculously make you understand and recall it. You have to do something with the material to make it stick.

Take a list of bits you just don’t understand yet to your teacher or tutor and see if they can break down the topic and explain it. Sometimes all you need is to hear the explanation put in a slightly different way for it to click.

Go and Do This Now!

How incredibly helpful is this document? It lists everything you need to know in concise and manageable chunks which let you learn key facts and check off every single item as you go. You can find which bits you don’t understand and there are all kinds of ways you can colour, highlight and annotate the sections to flag up bits you need to work on or remember.

Coloured pens, stickers, highlighters and post-its are your friends when it comes to revision.  That’s why I chose the picture at the top of the page. Make good use of your specification and you will feel so much more confident and prepared. I remember getting hold of my A Level chemistry specification a month before the exam (the teacher was reluctant to hand it over) and it made such a difference. Go and use it now.

Tackling Long Answer Questions – Top Revision Tips

In the UK the new A Level and GCSE exams include 4-6 mark questions which require an in-depth written answer several paragraphs long.

These long answer questions come in certain types and test the higher level thinking skills of interpretation and explanation (creating a logical explanation for a scenario), synthesis (pulling ideas from different areas together) and evaluation (weighing up options or outcomes). They also award a mark for clear written English and a well structured argument.

Long answer question can put the fear of God into many students who never seem able to break the 3/6 mark barrier and achieve the higher marks.

Thinking Skills in Long Answer Questions

Some questions on exam papers are easy, 1 mark for recalling the charge of an electron for example. These questions usually come with a certain prompt or command word which indicates you have to remember a fact but not actually do anything with it. “State the definition of electric current” is a typical, “name”, “give”, “what is…” etc are all command words for rote recall questions.

Long answer questions are different. Usually they give quite a lot of information before stating what you must do with it. They are setting the scene and every word used to set that scene is important. Examiners spend hours getting the phrasing of exam questions just so. If they have told you about the efficiency of a power station in the introduction to the question, you HAVE TO mention that in your answer.

The skills you need to answer these questions go beyond just including the larger amount of information these questions give you. Here is a famous pyramid called Bloom’s Taxonomy. Benjamin Bloom was an American educational psychologist who chaired a group which first devised this taxonomy (or classification) of command words into groups according to the level of brain power needed to complete each task.

Long answer questions are targeting the top three layers; analysing, evaluating and creating. These skills involve selecting the most important information, working out which order to put it into, identifying the key bit of physics the question is about, linking in related principles that may not be explicitly stated in the question and offering a solution to the problem solved. What makes this hard is holding all these aspects in mind at the same time.

Let’s look at an example:

6 mark A Level physics question – OCR exam board

  • Identifying the most important information: They tell you up front it is about the Big Bang. They mention the CMB radiation, they give you wavelengths.
  • Key bits of physics: the origin of the CMB, probably need to relate this to wavelengths somehow, the idea of the expanding universe, OK that’s Hubble’s Law, then wavelengths from galaxies which must mean cosmological redshift tying in with Hubble’s Law. OK sorted.
  • What order to put them in: “Explain” command word, 2-3 marks for explaining the role of CMB in Big Bang theory. Go on to discuss how present day Universe must still be expanding and how we could show this, 1-2 marks, relate data from the table to this. Have to use an equation relating wavelength at rest and in galaxy to redshift or recessional velocity. Need to use data they have given me, 2 marks for working out the redshifts. “Comment” command word, finish off with remark about how the data is relevant.
How to Tackle Long Answer Questions

The main strategy is not to read the question and immediately start writing, DON’T DO IT!

Here is a printable infographic which breaks down how to tackle these questions.

Step 2: Annotate the question is identifying the most important information

Step 3: Jot down ideas is recognising the key bits of physics to use

Step 4: Sequence your ideas is working out what order to put them in

 

Types of Question

Now we know how to go about answering these questions let’s consider certain types of question that crop up.

1) Experimental Design

Describe how you would compare … in a laboratory. Billy Bob wants to determine Planks Constant using LEDs, outline an experimental method which would….

The main things to include are apparatus, preferably a labelled diagram, brief method stating what you change, what you measure and what needs to be controlled. Give the range over which you take readings using the equipment you chose. Mention how many repeat readings you take. State two (or more) measurements will be plotted and what data processing you may need to do to find the desired result. This could be a simple as saying we divide voltage by current to find resistance, or you plot an I-V graph and work out the gradient. But you must say what you intend to do. Experimental method does not mean JUST apparatus and method. You must talk about range of data, reliability and data processing as well.

6 mark experimental question, A Level Physics – OCR exam board

The marks for this question were awarded as follows:

Equipment used safely (E)

  1. Wire fixed at one end with load added to wire
  2. Suitable scale with suitable marker on wire
  3. Micrometer screw-gauge or digital/vernier callipers for measuring diameter of wire
  4. Referencing to safety concerning wire snapping

Measurements Taken (M)

  1. Original length from fixed end to marker on wire
  2. Diameter of wire
  3. Measure of load
  4. New length of wire when load increased

Calculation of Young modulus (C)

  1. Find extension (for each load) or strain (for each load)
  2. Determine cross-sectional areas or stress
  3. Plot graph of load-extension or graph of stress-strain
  4. Young modulus = gradient x original length/area or Young modulus = gradient
  5. Calculate Young modulus from single set of measurements of load, extension, area and length.

Reliability of results (R)

  1. Measure diameter in 3 or more places and take average
  2. Put on initial load to tension wire and take up ‘slack’ before measuring original length
  3. Take measurements of extension while unloading to check elastic limit has not been exceeded
  4. Use log wire (to give measurable extension). Scale or ruler parallel to wire

To get 5-6 marks you had to include all points E1, 2, 3 and 4 for equipment, all points M1, 2, 3 and 4 for measurements and for the calculations you were expected to show C1, C2, C3 and C4.

Each specification says which experimental methods students should be familiar with and you can practise these questions by ticking off each experiment in turn and writing your own diagram, method, data collection and calculation notes.

 

2) Big Physics

We have seen an example in the previous section. Big Physics is particle accelerators, the Big Bang, life cycle of stars, fission and fusion, MRI scanners etc. These questions ask for descriptive explanations of the phenomena. Quite a lot of which is recall from lesson notes and reading. However exam questions increasingly throw in a data or a graph to make the answer more specific to a situation.

These are best revised for by practise writing the “story” of a particle through a mass spectrometer, the “story” of the interstellar dust cloud which becomes a blue giant or whatever.

 

3) Analysis of a given scenario

These questions are the trickiest in my opinion as they throw a piece of data or a novel set up at you and you have to think on your feet a bit more. These questions are often designed to identify the A* students as they are the ones who score 5-6 marks most easily on these.

Here are two example, one GCSE and one A Level

6 Mark GCSE Physics – AQA exam board

6 Mark A Level Physics – OCR Exam Board

These question include apparatus and/or graphs and tables. You must refer to the data in the graph or table. Describing the overall trend in words, identifying relevant data points on the graphs and using the numbers given on the axes i.e. quantifying the data will all score points.

In the first example the axes are not numbered so you would have to point out key features or trends. Most particles have an average speed much lower than the fastest particles forming a large hump or maxima on the graph, and that small number of particles have very high speeds forming a tail. You should fully describe the graph in other words. Do not say “the graph goes up at first and then it goes down”. The fact that a small number of particles have a slow speed is irrelevant to the question about evaporation (why?) and so you wouldn’t need to include a comment on that as it is not important information.

The second question outlines a device which student won’t be familiar with but they will have seen similar circuits before. The important information includes the stated range of temperatures and this should be linked back to values on the top graph because the graph has gridlines and numbered axes meaning you will be expected to read values from it.

The circuit is a potential divider (a fixed and a varying resistor in series) with two resistances and two voltages necessary to use the potential divider equation. Explaining how it works now becomes a more general question on explaining how a potential divider works.

Using the data to work out the voltage across the thermistor will be the route to take to score the second half of the marks.

Tackling an analysis long answer question involves more of the identifying the key physics step than some of the others. You have to be able to look at the question from a distance and spot the topic area they are asking about, recall the main points and then go back in to the detail and apply that to the scenario. You always get marks for spotting the topic area and identifying the key features/equations to use.

 

4) Pros and Cons

These style questions are often a straightforward comparison of two outcomes, for example should nuclear power replace fossil fuel power stations? Or they ask you to consider the wider impact of a certain choice given data and context in the question, for example is converting to an electric car a worthwhile choice?

6 mark GCSE Physics Question

This particular question about Europa came with some additional data showing where it is (a moon around Jupiter) and how far away it is compared to the Earth’s Moon.

These questions are best tackled as a list of pros and cons, 3 reasons with an explanation for each side and an overall judgement on the best outcome. The specific reasons you choose for or against will of course be led by the important information you have identified from the question and any key physics ideas (distance, speed and time, energy costs etc) that you know relate to the topic.

 

Conclusion

Following these tips and thinking about what type of question you have been handed will enable you to get the most out of your answer. Every question style can be tackled using the 1-6 Step method shown in the infographic, this is the basic cake, tailoring your important information and key physics topic facts to the question style is the icing on the top.