Brillouin-Wigner Perturbation Theory Part 1

I've been working on trying to understand part of Quantum Mechanics. There are only certain simple systems for which it is possible to solve the maths of Schrodinger's Equation. Ordinary maths won't solve more complex problems so the idea of Perturbation Theory is to change the energy by a small amount - give it a nudge - do some maths on that and come up with a solution written in terms of the original eigenstates of the system, because it won't have changed much. To work out the Brillouin-Wigner version of it (as opposed to the Rayleigh-Schrodinger version that we have looked at in the past) I've been using this document http://www.phys.ufl.edu/~kevin/teaching/6646/04spring/bw.pdf  What I decided  to do was to write the ket-vectors as if they were 3-dimensional real vectors and do matrix work based on them to get a feel for what the symbols actually mean. The ket vectors |n> are orthonormal eigenstates - in other words, if you do the inner product of one with another, the answer is zero. Each has its dual, the bra

 The inner product of a ket with a bra turns out to be a matrix. The document calls this one P1 and calls it a PROJECTION OPERATOR. (Remember that operators are matrices)

 The second projection operator is Q1. It is the identity matrix - P1. It says that P1 and Q1 are complementary.

 I have also proved that Q1=P2+P3 so for any number of n, Pn=sum of Qm, provided you miss m=n.
Then I tackle the idea that the Hamiltonian is also the sum of these projection operators multiplied by a factor. He calls this the SPECTRAL REPRESENTATION of the Hamiltonian - in other words, breaking the whole down into the bits that it is made of like white light is broken down into the component wavelength colours.

 I've got as far as the commutator proof. The Hamiltonian commutes with the other projection operator Q.

Flatford Mill: group velocity

Something caused these circular ripples at Flatford Mill, scene of The Haywain. What was noticeable is that the wave group stayed together and did not disperse from each other as they spread out. Looking it up https://en.wikipedia.org/wiki/Group_velocity that must mean that there were waves of only one frequency within the group. It suggests that the group velocity must equal the phase velocity.

Pushing down versus pushing sideways

For future reference in estimation challenges like the can crusher, I have done pushing directly downwards with my right hand versus pushing horizontally against a wall. It is easier to get my weight behind it pushing downwards, reaching 150N which is twice the 75N from pushing sideways.

Ingenious water level meter

I found this meter on the side of a water tank in rural Essex. There must be a float in the tank that is connected to the red and white level indicator by a wire. The odd thing is that the higher the water level, the lower down you have to look for the reading. What is the relative weight of  the float and the indicator? The float must be less dense than water but have enough mass to keep the wire taut. The indicator must have enough weight to keep the wire taut too. I'm guessing that they will be the same weight. If the indicator were heavier than the float, the float would stay suspended when the water level falls.

Can crusher

I have been wanting for some time to analyse this camp site can crushed but was missing one vital piece of information - the force of my arm pushing down. So I used the Newtonscales and got 150N.

From the photographs, I put 150N 30cm from the pivot. The centre of the can is maybe 3cm from the pivot. Crushing force x 3 = 150 x 30 by Principle of Moments so crushing force = 1500N. It has multiplied my arm force by 10 so this type of lever is called a force multiplier.

Ear plugs and frequency

These wax ear plugs come with a note detailing the protection by frequency. It is good to note that the average noise reduction is bigger at higher frequencies. In class I have found that not all notes played with the same amplitude are heard as the same volume. Notes around 4000 Hz sound loudest and most likely to cause damage. APV stands for Assumed Protection Value. https://esselacoustics.com/news/assumed-protection-value-apv/ You'll notice that in this data APV = average noise reduction - standard deviation. This is explained in the link. What you actually hear with the ear plugs in is EXTERNAL NOISE - APV = EXTERNAL NOISE + STANDARD DEVIATION - AVERAGE NOISE REDUCTION. In other words, the uncertainty caused by the spread of the data across the population means that it is assumed that you will get less protection than average. More on standard deviation later.

Storm Callum and the jet stream

It has rained all day today. The Weather Eye column in The Times newspaper carries a very effective explanation of how the jet stream winds intensified the storm. The jet stream is a narrow fast moving band of air that is roughly at the boundary between the troposphere and the stratosphere. I blogged about it in February having ascertained that trans-Atlantic aeroplanes over here were probably just into the stratosphere, 9 km up or just higher than Everest. The Polar jet stream has cold air north of it and warm air south of it. In his newspaper article, Paul Simons explains that Storm Callum crossed from the warm side to the cold side. In doing so, one side of the storm was touching the faster centre of the stream when the other side wasn't being pushed as fast. The difference made the storm spin faster, like spinning a Waltzer at the fair.

Glacial deposits at St Bees

I had a good look at the collapsing cliffs south of the beach car park at St Bees. Quite a lot of the pebbles trapped within it are angular, suggesting that they didn't travel far in rivers or the sea which tend to round them by bashing them together.So that's evidence that these stones were transported by ice. That last Ice Age was about 12000 years ago so there hasn't been time for it to form into a sedimentary rock. A lot of the pebbles are local sandstone but some have come further.

Update on waves hitting the beach

Having posted about waves on the beach at Blackpool, I had another look at St Bees. Once again it seemed to me that the wavelength increased as the waves came up the beach. I thought that the theory was that waves slow down in shallow water. This is why they bend as they come onto the beach. So I was thinking that maybe it is because the shallower bottom spreads out the material more - less depth so more width because water is incompressible. However, although I know that a water wave is a circular motion and not pure transverse, surely it is a movement of energy not material. It should travel over the water, not be a movement of water. I've tried to make observations and then work out explanations on this blog so will continue to watch waves to see if the initial observation really is correct.

Rayleigh scattering and Mie scattering

This is the view from Whinlatter across to Helvellyn in the distance. The Sun is coming in from the top right so any light coming from the middle and left of the picture is secondary. We'd say that the green is being reflected from the fields but the air isn't reflecting light. We say that the gases in the atmosphere scatter light. The light hits molecules, is absorbed and is then re-radiated in all directions. There is more than one type of scattering. Rayleigh scattering https://en.wikipedia.org/wiki/Rayleigh_scattering is from particles much smaller than the wavelength. Blue is scattered most by molecules in the air. However, look at the mist below the ridge line in the distance. The water vapour looks white. That is because it is due to a different type of scattering called Mie scattering where light is scattered by particles about the same size as the wavelength. All wavelengths are scattered equally in the visible range and so the mist looks white.

Solar panels at Silloth mill

A new solar farm has been put in by Silloth dock. https://www.solarpowerportal.co.uk/news/abp_and_custom_solar_complete_250kwp_solar_farm_at_port_of_silloth 960 panels. Looking at the picture in the article they look to be max 2 square metres so 1920 square metres. At about 700 kWh per square metre solar irradiance that gives over 1 million kWh per annum, 4 times the figure in the article so some wrong assumptions on my part!

Rainbow at a narrow angle

Look at this rainbow taken near St Asaphs. It has an angle of nearly 45 degrees to the horizon. Last week I blogged about one that was nearly vertical. The smaller the angle, the higher the Sun must be in the sky so the nearer we are to the middle of the day. The anti-solar point is a long way below the horizon. I hope to do some work on the angles in circles to see if I can work out the height of the Sun.

Wave velocity in Maryport Aquarium

The wave tank at Maryport Aquarium is about 4 metres long. The waves are made through a complex set of rocks so it is quite hard to say exactly when a wave starts but a wave takes about 4 seconds to reach the far end so must be travelling at about 1 metre per second.

Diffraction in Silloth harbour

The north-west wind was whipping big waves through the gap into the outer harbour at Silloth. They are plane waves with wavefronts at 90 degrees to the entrance walls. If you look on the left hand side of the picture, those plane waves have bent through 90 degrees and travelled along the inner wall. This is diffraction. I teach that the gap has to be the size of the wavelength for it to be really effective but the gap here is two or three times the wavelength.

Pepper's Ghost at Maryport Aquarium

Look at the fish tank on the left. It looks as though there are two illuminated panels in the back of the tank. In fact they are reflections of two panels that have over-exposed on the right of the picture. This form of partial reflection in glass is a good way of getting images to appear in remote places. If there were a screen on the right of this image, all you would see would be the reflected panels and if the originals were laterally inverted, you could have the information appear in the tank. This method was used for special effects in Victorian theatres and was known as Pepper's Ghost.https://en.wikipedia.org/wiki/Pepper%27s_ghost

Looking down from Blackpool Tower

This is looking down from the observation deck on Blackpool Tower. It is 120 metres down to the ground. Assuming no air resistance, an object dropped would follow the equation s = 1/2 g.t^2. So time squared = 120x2/10 = 24. Time to hit the ground = 4.9 seconds.

Dendochronology at Whinlatter

As a physicist I am very interested in time measurements so this display at Whinlatter caught my eye. There is something pleasing about the way growth is able to mark laps of the Sun. The rings are also different thicknesses showing that some growing seasons were better than others but this doesn't affect the counting of the years.