Statistics Help

In particle physics, sought after events that confirm many aspects of the theory of elementary particles can be relatively rare.  Determining if signals recorded in detectors are from these events, rather than minor blips in the plethora of background events,  requires the application of sophisticated statistical analysis.  If anyone out there can recommend a good primer in statistical methods that covers Monte Carlo methods, Markov chains, and Bayesian statistics, please let me know.

Small project; refinishing block front dresser.

Project news... not actual woodworking this time, just refinishing. Several weeks ago I picked up a piece of Kindel vintage furniture, from somewhere between the late early and late 40's I guess. It looked forlorn and beat up standing outside a gas station/antique store when I cycled by it, but I recognized the quality and the name Kindel, Grand rapids confirmed my estimate. So I paid the $75 asking price and drove down later to pick it up. Got in the basement and went to work on it. 

Removed blotchy dark coating  and factory lacquer finish. Polished the brasses and applied several coats of new lacquer to the wood. The piece had been dusted or polished with silicon bearing cleaner, waxes, or polishes. This prevents the new finish from adhering so I had to use a special additive in the new lacquer. It helped but the results are still not as good as they would have been otherwise. So DO NOT USE anything on your furniture or other surface that eventually may be refinished unless IT SAYS SILICON FREE. [ i.e it's guilty until proven innocent]

The dresser  went from before to after pretty well, I think. Judge for yourself. It now sits in the alcove of our bedroom.

Of Men (and Women) and Molecules

Here's a thought about the parallels between physics and people. Kinetic molecular theory is used to model the behavior of large assemblies of gas molecules. It correctly describes the average energy, how the assembly will respond to energy inputs, being compressed, allowed to expand, etc. It can tell how many molecules will have a given energy. It cannot tell you much about the location, speed, or direction of an individual. Seems similar to knowing about populations not telling you much about individual people. I like to consider people as individuals rather than as members of populations.

Real Magic

It is easy to understand the appeal of fantasy literature. Magical words, hidden kingdoms, supernatural creatures, are interesting and exciting. As a fan of Harry Potter, and much more so, of Middle Earth, I have no difficulty in seeing why so many people find fantasy worlds so attractive,....but....

I took a walk in the woods today and I came upon a stretch of forest, not magic,  but magical. Trees twenty times a tall man’s height, each tree a forest of soaring limbs in it’s own right, crowned by a sea of leaves, translucent green, bright yellow, shining gold, and brilliant scarlet, all against a perfect blue sky, waving wildly, yet gracefully in the full strength of the wind from which they sheltered the forest floor.

The real world is magic enough.  All you need do is take the time to see it. Go..... go and find the magic.

Scouts

Boy scouts if done right, is one of the best things that can happen to a boy between 11 and 18.

Here's are pictures from a recent outing to a local cave.

On DeBroglie Wave length

An interesting take on the De Broglie wavelength. This is adapted from Ch1 Sect 5 of Quantum Mechanics of Particles and Fields by Arthur March, available from Dover Publications.

Consider a particle as an object of great or even infinite length along the x axis. Coordinates for this particles frame are x’,y’,z’,t’. Assume it oscillates say in the y direction with period tau’. Assume it is now moving in the x direction with respect to a set of coordinates K ( x,y,z, t) at velocity v. For simplicity we set t=0 when t’=0 at x=x’=0. Because the times in the particle frame and K differ, different positions, x, will see different phases of the oscillation. In particular let’s find a position, x*, in K where at time t=0 the phase is the same at x =0. This means that the time in the particle frame at this location must be tau’, i.e. a full period, ahead or behind t’=0. Let’s assume a period head i.e. t’ at x=x* and t=0 =tau’

Using the Lorentz transformation we have

t=0=gamma(tau’+vx’/c^2).                             [gamma =1/sqrt(1-v^2/c^2) ].

This means that x’ corresponding to t=o and x=x*

is given by vx’/c^2 = -tau’  or x’ = -tau’ c^2/v.

From this we can, using the Lorentz transformation for x,  find x*

x* = gamma (x’+vt’) = gamma (-c^2tau’/v +v tau’)

=gamma v tau’(-c^2/v^2 +1)  = gamma v tau’   v (1-c^2/v^2)

taking a factor of c^2/v^2 out of the parentheses we have

= gamma tau’  c^2/v(v^2/c^2- 1)     note that the term in () is   –1/gamma^2

we have x* = (- c^2/v) tau’/gamma.

Since the phase at x* is the same as at x=0 the distance between the two can be viewed as an apparent wavelength, lambda.  So lambda apparent = (c^2/v)  tau’/gamma. Since frequency nu = 1/tau we have  lambda= (c^2/v) /nu or

lambda nu = (c^2/v)  so that (c^2/v)  can be thought of as the phase velocity.

Since E = gamma mc^2 = h nu

and  momentum, p = gamma mv= mc^2  (v/c^2) =  h nu (v/c^2)

p = h nu / (c^2 /v)= h nu/ lambda nu = h/ lambda or p = h/ lambda,  the De Broglie relation.

Thus when a particle exhibits wave nature it is because the particle is intrinsically oscillating in some sense,   which determines tau’, and it has motion relative to an observer the velocity of which determines, with tau’, the observed wavelength. Note that this also makes the De Broglie wavelength something of an artifact of special relativity, i.e. the fact that the clocks in the two frames are not synchronized.

Summer Report 4 Fruit Trees

A dry May combined with sunny summer with more than adequate rain plus some other factor have led to a banner year for fruit trees around here. Our pear tree and our apple trees, which receive essentially no care from us, have produced abundant yields this summer.

Summer Report 3 Red Sox Game

Went to a Red Sox game. Despite it being a Tuesday night with a pronounced threat of rain, the Sox being in last place, the opposing team being the uninspired Indians, the stands were nearly full and the fans were having a great time. Some of them even paid attention to the game, between trips to the food and drink stands. An American summertime tradition is still alive and well in Boston.

Democracy in A Small Town

 In New England towns, as opposed to cities, major decisions are made by vote at town meetings. Any resident who is a registered voter has a direct say in the matter. On Tuesday I got to see a real example of direct democracy.  Because of the rural nature of our town, about 700 year round homes spread over 40 hilly square miles,  companies such as Verizon and Comcast can't be bothered to run fiber optic cable or even DSL to most residences, and a large portion of our town is saddled with poor internet and cell service. In response to this and a few other developments, the town was considering issuing bonds to help construct a fiber optic network. Well over 300 of the town's 1200 voters showed up at the town meeting to consider the matter. Most people were pretty well convinced it would be the right thing to do and it was almost a foregone conclusion the bonding would pass. Still all who wanted to, mostly opponents to the measure,  were given the opportunity to speak, and the meeting in a crowded, hot and humid town hall went on for over 90 minutes. Since it was a bonding issue, a two thirds majority was required for passage and the measure passed easily. Regardless of he outcome it was very rewarding to see such clearcut example of democracy in action.

Summer Report 2 Ididaride

For the last several years my birthday gift to myself is to take a day off from whatever project I'm working on and go for a long bike ride. For the last two years I have been eyeing the Ididaride, a 75 mile fundraising ride run by the Adirondack Mountain Club. This year I decided to do it,.... along with about 500 other people. The ride starts in North Creek near Gore Mountain and goes mainly on state highways through Speculator and Indian Lake and back to North Creek. There was some nice scenery and a few spots with great views of the southern Adirondacks. The ride includes about 4500 feet of elevation gain. The last ten miles are a long downhill followed by a longer flat stretch along the upper Hudson River.

I was among the older riders there and I had not done a ride of over 40 miles in nearly a year and had not ridden much at all this year, so I was a little concerned that I might end up finshing late enough to cause the organizers concern. My usual strategy on these long group rides is start slow and conserve energy so I can finish a little less slow. I also enjoy passing those inexperienced riders who expend too much energy early on and fade at the end.

Well, it seems that no inexperienced riders sign up for 75 miles in the Adirondacks, so I had to content myself with not being passed too often at the end, mostly because there weren't too many folks behind me anyway. Actually I was pretty pleased, spending about six hours in the saddle and getting back long before the last riders, despite having to change out a flat about 62 miles into the ride. I think I would have been faster if I had stayed more hydrated in the first 50 miles. Five water bottles full are not enough when it hits 93 in the sun. I might do it again next year. If I do I'll take some pictures.

On the drive home I was reminded of how beautiful NY Routes 372 and 67 are. If you are ever in the area of Cambridge, New York give them a try.

Some Thoughts about the Uncertainty Principle

Here are some thoughts about a way to make the uncertainty principle more intuitive.

In classical mechanics, action is the integral over time of L [ i.e.T-V]. But let’s call integral of H [ that is  L+ 2V or simply T+V] over time as “total action” . I think that it is total action that is quantized in integral quantities of h [Planck’s constant] but I am not sure how V fits in.  I think the rest energy must be included in all this.

Do all measurements of E require some finite time and all measurements of require some finite distance? If so, can they be thought of as measurements of total action?

In some ways E= h nu is getting it backwards, or at least upside down. What does frequency mean for a particle? On the other hand looking at it as

E tau = h where tau =1/nu is more intuitive. Tau is simply the characteristic time [or period] it takes for a particle of energy E to accumulate one h worth of increase in total action.

Similarly p lambda = h indicates that a particle of momentum p must travel a distance lambda to accumulate one h of increase in total action.

If total action is truly quantized, then there is no measurable change until total action changes by h.

 Heisenberg’s uncertainty principal can be explained as follows: Suppose one measures the change in total action over a time interval, t,  less than tau. Say sometime during that interval, t,  the total action changes by one h.  The only things you know are that sometime during t total action changed by h. From this you conclude that E could be as high as h/t. On the other hand if you think about it a bit you realize that your interval t could have started after a large fraction of tau had passed since the last increase in total action by h, so that most of the accumulation of energy time leading to an addition of h in total action occurred prior to your beginning your measurement. Therefore the energy could be much lower( near zero?). Thus you are uncertain about E by h/t.

Now suppose your measurement takes place over a period, t’,  several times tau, lets say  t’ = 7 tau for example. Depending on exactly when you begin your measurement vis a vis  when a period begins [ i.e. the time when the last increase in total action by h occurred prior to the measurement]  you will measure either 6 h or 7h as the change in total action. So you know the energy is between 6h/t’ and 7h/t’  but since t’ is 7 x t , you have reduced your uncertainty by a factor of 7.

Similarly, for momentum if you measure the particle’s total action over a length , L, shorter than lambda , you may detect a change in total action of h. Then you can conclude that the momentum may be as high as h/L. However, most of the accumulation of momentum times distance since the last change in total action may have occurred in the particle path before your measurement, so p could be much lower (near zero?) so your uncertainty in p is h/L. IF you measure over L’ = 7 lambda  you will measure between 6h and 7h change in total action, and the uncertainty becomes h/L’ or 1/7 of h/L.

Summer Report 1 Trip to Britain

It's been a very busy summer. We were in Britain starting June 23 and returning July 16 with our nearby neighbors. We visited Salisbury and it's famous cathedral; spent a few hours in Gloucester; three days in the Shopshire towns of Church Stretton and Ludlow; seven days on a canal boat; a few hours in  Chester; six days in the Scottish Highlands and a morning in Oxford.

 The Cathedral in Salisbury with it's 400 foot tall spire

 My wife and our neighbors above the carding mill valley in Shropshire

 The Carding Mill Valley, a truly lovely place.

 My wife and my neighbor did all the steering. He did all the tricky bits.

  A bank in Whitchurch.

 The aqueducts that carry the canal above steep sided valleys were a highlight of the trip. Built around 1800, the canal were out done by railroads about 40 years later. The higher structure carries a railroad. Both still in use about 200 years after they were built. Not much we do today will last that long.

 The breakfast club waiting for toast.

The view from the Ferry from Oban to the Isle of Mull. The flat topped mountain in the far distance is Ben Nevis, the highest mountain in the United Kingdom. While only 4400 feet high the base is near sea level so it;'s a pretty good walk. My neighbor and I had to to walk through a couple of snow fields on our way to the top. Not bad for a low mountain in July.

 A couple of views from Pitlochry, a town on the southern edge of the highlands.

Weather could have been better in Scotland, but still a great trip.

More on scouting

The new boys in our troop spent two days at a Council run camporee and had a great time. The event was well organized and there was plenty to do, more than any one could do in fact. The only downside was that  many units were not there, probably because they did not kow how much fun they wwould be missing, and probably more so, because of the overwhelming time commitments demanded by sports.

Among the many activities, the boys learned to build a small bridge with no rope or fasteners and got to throw hatchets at targets ( pieces of logs).

Physics: Action and Wave Function

I believe action, S,  is defined as integral of KE - PE over time or integral L over time. There appears to be an intimate relationship between the wave function and the action. A sample wave function can be A exp[(-h/i)S(omega t -kx)] or something similar. A function of this type satisfies Schroedinger's equation in that the second spatial derivative  gives KE and the first time derivative gives total E as long as E =hbar [ Plancks constant /2 pi] omega.

Since the probability of a system being in a state is proportional to the wave function* dot  wavefunction, the question is why the probability density is related so closely to the action. L can be thought of as two times the work input from the field into the particles KE in going from some reference point to the current position.