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June 2013 Archives

How long does it take you to load the dishwasher? Placing all those tricky-shaped objects in position to maintain the perfect balance between getting the maximum numbers of objects into the machine and placing them so that they clean optimally. 

Most likely, less time than it takes to load ours. This isn't because it's a particularly difficult machine to load (and we don't have to worry about it cleaning optimally - because it doesn't - if we want something really clean then it goes in the washing up bowl). No, it's because the baby is getting rather adept at unloading it. That's his latest bit of amusement. Pull out all the items of cutlery, one by one, and scatter them over the kitchen floor. 

One can construct a simple theoretical model of the process, to see what happens. Suppose I can place objects into the dishwasher at a rate m (for me, Marcus). Suppose baby can unload them at a rate b (for baby, Benjamin). Then the rate of increase of objects is given by m - b. Simple enough. If m is greater than b, I eventually win and the dishwasher ends up fully loaded and I can shut the door.  If b is greater than m we end up with all our dirty dishes on the kitchen floor. If m equals b, we will go on forever. 

But it's slightly more complicated. The rate at which I can load them diminishes as more objects are in the dishwasher. This is because 1. there are fewer dirty objects in the kitchen to grab, so I have to take longer hunting down the next bit of washing-up, and 2. the dishwasher is more crowded so it takes longer to negotiate the tangle of dishes to place the plate into position. So the rate m depends on the number of objects in the dishwasher, N. We might suggest that m reduces linearly with N, so that m = c - kN, where c and k are constants. The maximum number of objects that one can load is then c/k, because when N=c/k, m will be zero. 

But it's worse than that. The more objects in the dishwasher the more quickly baby can grab the next and remove it. So b grows with N. Let's assume it grows approximately linearly, so b = a N, where a is another constant. 

What will happen now? Let's look at the difference between m and b? m minus b is now (c-kN) - a N, which equals c - (k+a)N. At the start, N is low, and I have the upper hand. More objects go in, so N increases. But as N gets larger, the game swings in favour of Benji. We can see that when c = (k+a)N, that is N = c/(k+a), m - b is exactly zero. Then we have reached a dynamic equilibrium - the rate at which I can load the dishwasher is exactly balanced by the rate Benjamin unloads it. It's then time to give up, shut the door, and start the machine going, with c/(k+a) objects inside.  

So, if we want to maximize our use of the dishwasher, we simply need to increase 'c' (practise grabbing objects quickly), reduce 'k' (practise manoeuvering objects into the last remaining slot at the back) and reduce 'a' (distracting the bubble with something more interesting, like emptying the entire contents of our cereal packets onto the floor). 

Or we could just wait until the baby's in bed.

 

 

 

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People who think well, write well. Woolly minded people write woolly memos, woolly letters and woolly speeches. David Ogilvy.

There's nothing like reading through and marking students' exam scripts. Mostly it is terribly boring, but sometimes it is enlightening. 

One of the questions I asked on an exam this semester involved getting the students to describe and explain what happens in a particular situation. The exact question is immaterial - but what the students had to do was to write sentences. It was clear that this task is very difficult for a good many of our students. Their responses are a reminder to me that we don't specifically teach writing in our science degrees. 

Well, we do, to some extent, in some papers. Students have to write things. But we don't have a specific course on how to write scientifically. Student answers were plagued by bad grammar and spelling, but, more worryingly, were vague and woolly*. There are a lot of physics words with very specific meanings, that can be used to describe the movement of something unambiguously. Force, centre-of-mass, momentum, angular acceleration, etc, all have well-defined meanings. Instead of containing such words, used correctly, many answers were couched in vague, ill-defined language, or (maybe worse still) used good-sounding physics words but incorrectly. 

There are two issues I see here:

1. Is it time we  taught students explicitly how to write? (In particular, how to write technically). 

2. Woolly writing is a sign of woolly thinking. A badly phrased response is indicative that the student hasn't really got their head around what is going on. And that's suggesting that there is learning still to do. It is easy to hide behind mathematical calculations if you don't know what's going on. But having to abandon the calculator and resort to descriptions may really show up how a student is really thinking.

I've come across the  ten tips for good writing from David Ogilvy (of Ogilvy and Mather advertising agency) on the BrainPickings website. Have a read. They seem obvious, and they're not difficult steps to follow. But it's clear that following them doesn't come naturally.

*I'm not sure where the term 'woolly' comes from, but it evokes the image of something ill-defined like the surface of a sheep: where does the fleece stop and the air start? It's hard to pin down anything definite. 

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With an exam imminent, I've had a queue of students outside my door wanting help with their quantum mechanics. This semester, they've come across the Schrodinger equation and the wavefunction for the first time and, unsurprisingly, some are struggling to grasp it. "But what IS the wavefunction?", they say. "How do you derive the Schrodinger equation?"

Very good questions. While we use Schrodinger's equation quite happily to make predictions (and extremely accurate ones at that) about the behaviour of things at small length scales, a physicist won't be able to give you a decent answer as to why it works. There are hand-wavy explanations about wave-particle duality and energy balance, but the honest answer to the questions "What is the wavefunction?" and "Why does the Schrodinger equation work?" is I don't know. 

They are questions that philosophers have seized upon, and have resulted in many 'thought' experiments, such as the famous Schrodinger's cat, who is neither alive or dead, and the Einstein Podolsky Rosen 'paradox'. How does the wavefunction correspond to reality?

In the May Physics World magazine (unfortunately not accessible free online), a great article from Jon Cartwright explores some of the thinking and possibilities for 'untangling' the 'uncertainty' in all this (puns intended). What are the possibilities for interpreting the wavefunction? Four are listed, broadly along the lines below.

1. Quantum mechanics just happens to be descriptive of the real world, but it's not a real thing in itself. Reality happens only when we look at something. It's a recipe for making predictions. (This is part of the 'Copenhagen Interpretation' of quantum mechanics - from Niels Bohr and others). For a physicist, there's something deeply disturbing about this interpretation since it is not a realist theory - reality only exists when we are observing it. The problem is that it has worked really well.

2. Quantum mechanics does describe reality, but the wavefunction is just describes probabilities of finding answers. There's something deeper underlying it.  (Einstein's view). 

3. Quantum mechanics and the wavefunction are both real things, but that's not the entirety of reality. 

4. Quantum mechanics and the wavefunction are both real, and that's all there is folks. 

The first two situations are considered 'epistimic' interpretations, since they are concerned with knowledge of things. The second two are considered 'ontic' interpretations, since they are concerned with reality. (See my rant about social science for more about reality). The key question is "Are wavefunctions real?" Are they things in their own right, or are they simply descriptive?

The article explains that there's mounting evidence for the ontic case. The Pusey, Barrett and Rudolph theorem (something I won't try to elucidate here) says that the only way for quantum mechanics to be epistemic if for it to be wrong. And given that it appears to be right in all its experimental predictions, it looks as if we may have to abandon the epistemic view. So the wavefunction may well be real (hooray!), and both Einstein and Niels Bohr are wrong.  In one way, that will make most physicists happy (reality is back), but one has to ask now what sort of reality is it? Given that option 4 includes Hugh Everett's 'many worlds' interpretation of quantum mechanics, in which every possible outcome is played out in an array of parallel universes (great for science fiction writers but freaky for everyone else - I've always thought this viewpoint as laughable) there might be some more discomfort to come.  

Though it might be reassuring for my students to know that out there somewhere is a universe in which they have straight 'A+' grades in their exams.

 

 

 

 

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 Yesterday I had the pleasure of visiting the Measurement Standards Lab at Lower Hutt. We were talking about making measurements of electrical impedance. In practice, if you want a decent measurement, it's rather less straightforward than whacking on a multimeter. It was interesting to have a look at their labs, including several New Zealand 'standard' things (such as resistors) - and apparatus for measuring them really, really accurately. 

But that's another blog entry. On the flight from Hamilton to Wellington we had a fantastic view of the snow-capped Tongariro volcanoes, in some really great visibility conditions.  There was a fair bit of photography  going on. The guy in front of me got out his cellphone, but before snapping his picture noted a rather interesting phenomenon. He was viewing the image on his phone screen partly through a rotating propellor. For the eye, since the propellor was rotating so fast, all this meant was that the background viewed through it was simply slightly darker than the rest. However, the camera screen told a different story. On this one could see almost stationary propellor blades. They were rotating very slowly and a little distorted in shape.  

I assume this would be an example of aliasing. The rotation rate of the propellor, and the refresh rate of the camera screen, would be close to integer multiples of each other. So every time the screen refreshed, the propellor had rotated such that there was a blade in approximately the same position. Consequently, a blade shape appeared on the screen. However, on the final photograph, with a long exposure time, no such blade was apparent. 

With a given sampling rate (in this case that of the camera screen) there is a maximum frequency of oscillation which one can detect. This is half the sampling rate, called the Nyqvist frequency. Anything that is oscillating more quickly than that won't appear at its true frequency - instead it will be aliased to a frequency that lies between the negative of the Nyquist frequency or the positive of it. So the oscillating thing will appear to move either forward or backwards at a slower frequency. One sees it on movies with accelerating cars - the wheels can appear to be rotating slowly backwards (or forwards) or even be stationary.  Here's a nice example from YouTube. Very disturbing visually. Fortunately the eye doesn't have a regular refresh rate and so isn't susceptible to aliasing - it just fails to see the rotation at all once it gets to fast.

 

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 Pseudoscience - the packaging of absolute gibberish in clever sounding scientific terms - is nothing new. Here's an example my father-in-law has sent me, from the West Coast Times of 27 October 1869: Tidal Waves and their Causes. It's a newspaper report of a lecture given in Melbourne on the causes of tsunami. I get the impression that the writer of the article wasn't taken by the explanation. I love the phrase "Having satisfied himself on that point...", rather suggesting that he hadn't satisfied anyone else in the room. Enjoy. 

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