Stationary phase condition: searching for an understanding

Having noted that Isaac Newton made a lot of progress during self-isolation, I've been back to the large Quantum Mechanics text book I was working through last year. Luckily, I kept a diary so one year later I have been able to pick up the threads. I got stuck on the stationary phase condition for deriving the wave packet needed to represent a particle in QM.

 It's the integral above that was bothering me. p stands for momentum and by the De Broglie equation, p=h/lambda. But since lambda is inversely proportional to frequency, then p is another way of choosing different frequencies. Instead of sine or cos, the wave is represented in its more obscure complex exponential form. Integration means adding up. The clue is superposition. So all we are doing is putting waves of different frequencies one on top of the other.

I drew what we'd do for superposition of different frequencies with a L6 class. Three waves are shown, all with different wavelengths and thus different momenta. Notice that they all coincide constructively at x=0 but don't elsewhere. The green curve is only just out with the red curve and just those two added would still give a reasonable tall wave. But the blue waves interferes destructively in a lot of places. So this is my take on the stationary phase condition - in one place and across a narrow range of frequencies, you get constructive interference. Since we are adding an infinite number of frequencies, elsewhere we get total; destructive interference. Hence overall we get a wave packet - a narrowly defined space with waves. It does bother me that it would seem that constructive interference at x = 0 would give infinite amplitude. I have drawn all 3 with the same amplitude but the equation implies that amplitude depends on momentum. So if most momenta have a very small amplitude, perhaps in the limit you get a defined amplitude.

Does concentration affect pH?

When I made my red cabbage indicator, I thought that the pH for bicarbonate of soda was too high so I decided to see if it was because I had made the bicarbonate to strong.

 1 Mole is an amount of chemical that was invented about 200 years ago when atoms were too small to weigh. They reasoned that if you put a large number of atoms together, it would be heavy enough to weigh. Now it turns out that if you get the formula for the chemical and add up all the mass numbers (top numbers on the Periodic Table), then that mass in grams is worth 1 Mole. Next, if you dissolve that in 1000 cubic centimetres of water, the concentration comes out to be 1 Molar. I calculated that 0.84 grams in 100 cubic centimetres (same as millilitres) would be 0.1M.

So I zeroed the scales with the measuring jug on. I noticed that the RESOLUTION of these scales is + or - 1 gram - it measures to the nearest gram so I decided 1g would be close to 0.84g. I kept adding more and more powder and it still said zero.

 Turns out that it won't measure anything less than 10 grams, so that's what I did. I tried to dissolve it in 100 cubic cm of water, but it wouldn't all dissolve so I went for 200 cubic cm. That works out as a concentration of 0.25M. I made 0.125M by doubling the amount of water to 400 cubic cm and stirring well.

 I made 0.0625M by doubling the amount of water to 800 cubic cm. Concentration is INVERSELY PROPORTIONAL to volume of water: double the volume of water, halve the concentration.

 Then I ran out of measuring jug so I poured half into another jug and doubled both to get 0.0313M.

 Look! It doesn't matter how dilute you make it, the pH stays the same.

The reason is that pH depends on how much the molecule can dissociate into H+ and OH- ions. The more dissociation, the stronger the acid or alklai. The less dissociation, the closer to neutral. That is a property of the chemical and is not affected by diluting it.

Making a red cabbage indicator

I took some red cabbage

 I chopped it quite finely and boiled it in a little water.

 Then I crushed it with the potato masher for a minute or two.

 Then I strained it.

When it cooled I used sherry glasses because I don't have any test tubes at home.

I tried it with lemon juice, vinegar, bicarbonate of soda and toothpaste. The little pot on the end is to show the colour of untested red cabbage juice. So acids tend to be pink and alkalis like bicarbonate of soda go green. What I hadn't realised is that the red cabbage is like a universal indicator in that it not only tells you whether it is acid or alkali but also how strong an acid it is. Strength of acid is given as a pH number. The full colour scale is shown here (scroll down when you get there) https://www.compoundchem.com/2017/05/18/red-cabbage/ pH 8 for toothpaste is probably not far off the mark (I'd expect 7.5) , lemon juice as pH 2 is about right, as is vinegar. I used bicarbonate of soda powder so I might have got a different result if I'd watered it down more.

L6 Estimation question A

Here are the electricity cables behind Wigton cemetery. If you enlarge this photograph, you'll be able to see the next pylon in the distance. Here are two questions to answer:
1. Why does the top wire have no insulator when the lower 6 wires hang from large insulators?
2. Estimate some quantities, look other quantities up and use the resistivity equation. Then calculate the TOTAL resistance between 2 pylons.

Rainbows

I have been given these pictures of rainbows spotted from exercise walks in Wigton.

https://www.msn.com/en-gb/lifestyle/lifestyleshoppinghomegarden/childrens-handmade-rainbows-are-popping-up-in-windows-all-over-the-uk-%e2%80%93-heres-why/ar-BB11CWsd?li=AAnZ9Ug&ocid=mailsignout
One of the things that Isaac Newton did whilst he was self-isolating from the plague in 1665 was to experiment with prisms. He shone a narrow beam of sunlight through one prism and got the rainbow spectrum. He was convinced that the colours were present in the light, but he was aware that the prism itself could have been behaving like stained glass. To rule this out, he took the spectrum from one prism and shone it into a second upside down prism. The colours joined together again to make white light. If the prism was like stained glass, that would have been impossible. So in self-isolation, Isaac Newton proved that white light really is made up of the colours of the spectrum. https://micro.magnet.fsu.edu/primer/java/scienceopticsu/newton/ As I understand it, he chose 7 colours because 7 is seen as a propitious number in religion. That's why I struggle to pick out blue from indigo from violet.

GPE with Joe

Here's one of this morning's newspapers! Joe Wicks has been helping me get fit for running for a couple of years now so I was very pleased to see that he has stepped up to be the nation's PE teacher. https://www.youtube.com/channel/UCAxW1XT0iEJo0TYlRfn6rYQ Go on! You know you want to.
Today he was doing jumps that he likened to Pikachu. We think he got at least 30cm off the ground so he emptied his store of chemical energy from his food and filled his store of gravitational potential energy (GPE). GPE = mgh so for a 65kg man, gain in GPE for such a jump = 65 x 9.8 x 0.3 = 191J. The pathway through which one store empties and the other fills is mechanical work. Work done = force x distance so 191 = force x 0.3. The average leg force then becomes 637N. I can do 10 in 18 seconds so 1.8 seconds per jump. Power = work done/ time taken = 191/1.8 = 106 Watts, so basically 100W. Now he had 800000 computers connected so feasibly 2 million people involved. Suppose they were all adult men - that would be 0.2GW. He made some stab at a number like that at the end on Wednesday - the real number is perhaps 0.1GW - so he's not that far off.

How to set up the pendulum experiment

 Tie a small object onto the bottom of the string. It has be heavy enough to make the string go tight. I used a rubber but you could try a small stone or a lump of Blu-Tack.

 Use a ruler and a pen. Measure from the MIDDLE of the object that you tied on. Put a mark at 20cm from the middle of the object, then 25cm, then 30cm, then 35cm etc.

Next hang your pendulum at the 20cm mark. Try to get it so that it doesn't catch too much as it swings. Pull it a couple of cm to one side (10 or 20 degrees - as shown in the photo) and let go.

One whole swing is across AND back, called the TIME PERIOD T. But you are going to time 10 whole swings. We'll call that 10T. The reason we do that is because T is very small and too close to your reaction time. Doing 10T means that there is less RANDOM ERROR in the experiment.

Year 10 Cooling by evaporation - why does hand sanitiser feel cold?

Look at this photograph of the ball bearing model of Kinetic Theory. Kinetic Theory says that all atoms are always moving. But look carefully at the ball bearings in the photograph. Most of them are at the bottom. They have low energy. A few are at the top because they have high energy. In fact, one or two had so much energy that they jumped out of the tube.
Now, the kinetic energy of atoms tells you the temperature. High kinetic energy = high temperature. Low kinetic energy = low temperature. If the high energy atoms jump out, then what is left are the low energy atoms. High energy atoms jumping out is called EVAPORATION. If you put hand sanitiser onto your hands, it evaporates. the high energy atoms jump out into the air leaving the low energy atoms behind. Low energy = low temperature. Hand sanitiser feels cold.

Year 10 Air pressure and kinetic theory experiment

 We started with a can full of air and with a small amount of water in the bottom. We heated the water until it boiled. Steam was coming out of the top of the can and it pushed the air out of the can. So now we had a can full of steam.

Next we very quickly tipped the can upside down into cold water. The cold water made the steam condense and turn back into water. Water doesn't take up as much space as liquid water so now most of the inside of the can was an empty space - a VACUUM. The air particles outside hitting the can and causing pressure (kinetic theory) crushed the can because there were no air particles inside the can pushing outwards to stop it crushing.

Year 10 Kinetic Theory and Pressure

We saw this model of atoms in the air. The tiny metal ball bearings are like the atoms and they are constantly moving at high speed. The fact that atoms are always moving is called KINETIC THEORY. In the picture above, the atoms crash at high speed into the polystyrene block. This pushes the block up. Pressure in the air exists because atoms in the air are crashing into objects at high speed all the time.

What happened when Isaac Newton was self-isolating

Here's a picture of Isaac Newton's famous apple tree from his home at Woolsthorpe, in the countryside near Grantham. When the plague hit Cambridge, the university students were sent home. He did his best work there. I'm working out how we can do some home experiments during this time. https://www.washingtonpost.com/history/2020/03/12/during-pandemic-isaac-newton-had-work-home-too-he-used-time-wisely/

Marie Curie book

This month's Wigton Library Book Club has been "Marie Curie and her Daughters" by Shelley Emling. It deals with the second half of Marie Curie's life, starting on 1911 with her second Nobel Prize for Physics. It was written by an American author and deals therefore with the American angle on the Curie family story. I did once see a documentary about this - about how a friendship with an American journalist led to American charity donations funding Marie Curie's research institute. One thing that I really admire about Marie Curie is that she believed obstinately in research being made freely available so she never patented her work. The other thing I took from the book is that I have long neglected the work of her daughter and son-in-law who were awarded the Nobel Prize as well for discovering artificial radioactivity - when bombarding an unradioactive element makes it then become radioactive.

Paul Dirac on In Our Time

Well, not in person. He was the subject https://www.bbc.co.uk/programmes/m000fw0p

Range and resolution in the Victorian kitchen at Penryn Castle

This lovely spring balance has a range up to 200 lb. I'll call that a Full Scale Deflection like I would for an analogue ammeter because the needle is actually deflected. The resolution is usually taught to be the smallest scale division: in this case +/- 1 lb. But this is a single point measurement so +/- 0.5 lb would be better. For a ruler where it is being lined up at either end, the +/- 0.5 divisions error at each end then gives you the +/- 1 scale division resolution.

Anticline or unconformity in Thornthwaite Forest

At first sight, this looked like an unconformity. Vertical layers in the shale with a horizontal layer on top. The lowest horizontal layer seemed to be quartz.

But looking from a different angle, it seems that the vertical curves round to become the horizontal. I actually think this is a very tight anticline that has been broken through its peak.

Steam locomotive pistons

The steam goes through the pipe from the boiler into the piston where it pushes. The piston is connected to an incompressible rod that is attached the wheel, which means that the horizontal motion of the piston pushed on the wheel. The heavy engine pushes down on the wheel so much that there is enough friction for the wheel not to slip. The push of the piston can thus turn the wheel. The drive wheel is coupled to the front wheel. I guess that this means the friction at the front can be utilised as well.

Steam locomotive boiler

I've seen models of this set up before but never really understood what they were trying to show. The picture is of the front of a steam locomotive with its front open. This space opens into the chimney above. Smoke goes up the chimney,not steam, but I'd been left with the impression that steam was coming through the pipes shown. In fact, very hot air and smoke particles from the firebox are coming through. There is water behind the white wall in the top picture. The arrangement means that thermal energy is applied to the tank of water over a much larger surface area and so the process is more efficient. Water boils and steam builds up at the top. Then the two substantial pipes that come from the middle top and go down each side actually carry the steam down to the pistons. https://www.explainthatstuff.com/steamengines.html

A brake on the railway pedal car

The pedal carriage on the Penryn railway had a brake that was a semi-circular metal band that was forced onto the wheel by a lever. The ratio on the lever looks to be about 10:1. Suppose a burly quarry worker could lift 30kg with one hand. That would be a 300N force. The brake would be applied by 3000N. Eight 100kg workers in a heavy car would be at least 1 1/2 tonne. That would mean acceleration of a = F/m = 3000/1500 = 2 m/s/s. That's a fast decrease in speed so is probably unrealistic. The brake is semi-circular to give a large surface area to increase the friction since friction is a contact force.

Reflective clothing at the Watchtree Seasonal Series

We went running in the dark last night. I understand that the reflective tape on my shoes must work by Total Internal Reflection, but how do they make it? I found this lovely explanation https://www.chinareflective.com/faqs/How_does_reflective_fabric_work.html The one described uses tiny glass beads part-coated in aluminium to make sure that the light bounces around inside the sphere and then bounces back the way it came.

Pedal power on the railway

This amazing pedal car for a quarry railway was in the museum at Penryn Castle. One detail is hard to see in the photographs of the back of the carriage. Each of the 4 foot cranks from the front is connected to the drive shaft at 90 degrees to the others. This means that there will always be drive from one of the shafts and enough momentum to push the opposite shaft back into position. It's a bit like a 4-cylinder engine.

A leap year can of worms

The more I read about leap years the more I found out.
1. I knew that the calendar had been reformed in the Middle Ages and that the British had been late to adopt this but I didn't know the reason for the reform. It has to do with the correct calculation of the date for Easter, which was calculated from the Spring Equinox.
2. The reason that this was out of sync was that a year lasts 365.2422 days not 356.25 days. The calendar was slipping out of sync with the sky by about a day every 200 years. I knew that the solution was to drop the leap year day at the turn of the century but that didn't happen on the Millennium. You drop 3 leap years every 400 years.
3. The year is called a "tropical year" https://en.wikipedia.org/wiki/Tropical_year. The definition involves precession so that will take a bit of thinking about.
4. There is an astronomical year numbering system that involves giving a Year 0 to the usual western calendar dates. https://en.wikipedia.org/wiki/Astronomical_year_numbering