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. 

Visit to Basking Ridge

I'll be in Basking Ridge on Thursday 6/18. If interested in getting together, send me a comment with your email address.

BACK INTO SCOUTING

Our boy scout troop in Ashfield had dwindled down to two scouts. We now have four or five new members and a few of us adult leaders are working to revitalize the troop.

Spent the last weekend in a training program to refresh my knowledge about being a scout leader. It was very gratifying to see that about thirty adults from rural Massachusetts were willing to spend a weekend learning how to help boys develop into useful young men. I was impressed with the friendliness, ability, and dedication of just about everyone there.

Not So Random Thought

When woodworking an instant of inattention or clumsiness can undo days of careful workmanship.

Five minutes of binge eating can undo a week of discipline.

 A few moments of hate or anger can wipe out years of care and love.

Physics Non Progress Report

Been doing a lot of reading of papers written at the inception of quantum mechanics from a collection " Sources of Quantum Mechanics, published by Dover. It leads off with a paper by Einstein concerning how the proper spectrum of radiation from a gas in thermal equilibrium can be derived from quantum absorption and emission. Right now I am in the midst of a paper by Bohr which seems to be an attempt to relate quantum emissions to classical electrodynamics/mechanics of charged oscillators. Trying to relate the classical emissions expected from charged oscillators to emissions, and absorption of quanta from transitions between the quantized "stationary" energy states is a tall order. I expect to have to read this paper more than once to see if Bohr achieves anything like this. I need to get through this and a few more papers before reaching Heisenberg's major paper on some of the fundamental "principles" of quantum mechanics. Wish me luck, I'll need it.

A Bit of an Obituary for a Small Creature

Yesterday our 20 year old cat died. She was officially my son's cat but she lived with us for the last nearly four years. She was quirky and  at first pretty anti-social, but after several months of adjustment she became very affectionate. Whenever she could, she would spend time in our laps or sleeping on us.

She argued with her drinking fountain,  made loud outbursts in the middle of the night, shouted for us to come upstairs and keep her company, needed special food, and lot's of attention, including frequent cleaning of the aforementioned water fountain. However all this was an okay price to pay for her warm affection, her loud purring, and the appreciation she showed for any attention we gave her. Life is simpler and quieter around here. I do miss her.

Miscellaneous Mid April Report

Haven't posted much lately because not too much blog worthy to report and our internet service makes posting painful. Not much snow or precipitation of any kind for the last several weeks. We are having a reasonably early melt out of the snow and the lack of precipitation is holding any flooding to very low levels.

Visited some family in North Carolina in march. Good to see some nice folks.

On the flights back and forth I got to see some of the development around Pittsburgh and Raleigh. It looks like sprawl is still in season, despite all its ills. More reliance on automobiles for everything, nothing within walking distance, kids needing appointments to find other kids to play with, large lots that only the landscapers get to see or use, etc.

Our local scout troop is in a rebuilding mode and we got a nice outing in, using snow shoes and a sledge, while the snow was still deep and firm.

A small project

I recently completed a present for my cousin's fiftieth birthday. I turned a pair of mahogany candlesticks for her, making use of some of what I learned making the night stands. I then decided to make a case for the candlesticks, the globes to be mounted in the candles sticks (wood and burning candles are not a great combination) and the candles. The case ended up taking more time than the candlesticks.  My hand cut half blind dovetail technique can still use more practice.

Here are some pictures.

Some Physics Questions

Looking at the title of this blog, many of you must be wondering when the the physics will appear.  I still do spend time on physics, but mostly wrestling with fundamental issues. For example: if, as Einstein  indicates in his "Meaning of Relativity" and other writings, that spacetime is really the interval  between events, what are the implications of this with regard to the propagation of fields? Does energy/matter curve spacetime, or actually create it it? Is this why spacetime is "denser" near masses?

Why the different signs for space and time; energy and momentum? What is the reason for the the imaginary component of spacetime?  How does it relate to velocity vs displacement for rotating bodies?

Of Ice and Men

While clearing some of the walk way I noticed the following:

Moving a large block of ice is nearly impossible. The surrounding pieces  reinforce it and give it the strength to resist large forces. However, if you chip away at a piece and separate it from the supporting pieces around it, you can clear it away pretty easily.   It's pretty much the same with an individual person and groups of people, especially mobs.

Coping with Winter

Inspired by my good friend Tom, I took to raking snow off my roof yesterday. Not being as tough as Tom I 1) wore socks (and boots) 2) wore long sleeves and a jacket and 3) had to climb up on the porch roof. I cleared an area about 12x25. The snow is about 2 feet deep rising to about 3 feet at the ridge.

After raking down this pretty dense stuff and having to leave about half of it, I was glad I reinforced the rafters under this roof three years ago.

Thoughts from a winter forest

I was snowshoeing the other day and there were strong gusts. I spent a lot of time listening to wind in the trees. The poet in me thought he heard the trees breaking their shackles of ice, bending to the will of the wind. The physicist in me thought the bigger trees were experiencing about ten thousand foot pounds of torque. The scout leader in me thought how embarrassing it would be if I had to be rescued from under a fallen tree.

Project Report: Second Night Stand completed

I think I got it close to right this time. Nature made the nicely figured wood for the drawer front and an English foundry cast the draw pulls, but I'll take credit for the legs.

The two night stands took about 2 months, with lot's of travel, shoveling, and snowshoeing interfering. My hand dovetailing has gotten pretty good. Proves that with enough practice even the untalented may succeed.

A visit to College of the Holy Cross

Went to Worcester ( the recently crowned snowiest city in the US) Friday to visit Sarah V at the College of the Holy Cross. Holy Cross is on a steep hill overlooking Worcester. It has some lovely buildings including a beautiful library and main building with a very impressive, traditional looking hallway. And of course it has Sarah.

Winter Report

Up to a balmy 2 degrees now with gusts only about 30 mph.  Last Thursday morning it was -6 F during a brief morning walk with 25 mph winds. There are many nice things about -6: the ice is too cold to be slippery, there are no mosquitoes,..... and I forget the rest.

Earlier in the week I looked at the results of my shoveling and concluded they were near nil ( except the porch roofs are still standing):

Before

After

After After

But it is lovely, ......

a local driveway:

Music review

 Went to a concert by the Pioneer Valley Symphony last night. The soloist was a 14 year old violinist, Ilana Zaks. She played the Mozart Concerto No. 3 as well as I have ever heard it played, including recording of the masters; and the same for her playing Sarasate's Zigeunerweisen, gypsy music. If you ever have the opportunity to hear her, take advantage of it.

The concert also included a wonderful overture to Armida, by Haydn, and Beethoven's Second Symphony. I was once again reminded that classical music is one of mankind's greatest achievements and that Beethoven transcends even the greatness of classical music.

Please attend concerts when you can. It will add beauty to your life and you will be helping to preserve a great pillar of civilization.

Training for The Winter Olympics

Yesterday I started training for what is sure to be the next big Winter Olympics Event: porch roof shoveling. While it's a little less entertaining than snowshoeing it certainly provides opportunities for exercise. I figure between the 18" of snow and the 3 inches of ice I must have moved about a ton of stuff.

Here are the before and afters:

Maybe we should read those things

It's amazing what reading instructions can do. We had noticed that Natalie, the upstairs cat in this house, was reluctant to use her drinking fountain (doesn't every cat have their own drinking fountain?) so I decided to order new filters. Looking through the instruction book for the item number, I saw a) that the pump could be disassembled for cleaning , b) and this should be done every couple of weeks. Realizing that a year was more than a couple of weeks( I am quick that way) I disassembled the pump and found enough growths to provide at least one Ph.D thesis in bacteriology, and perhaps a middling length career.

Winter is truly here

It hit -6 this morning but is now a toasty 10 F. It certainly is the white season here: 14'' on Monday and another 5 or 6 last night and this morning; and cold enough for it to all stick around. Here are some photos from some recent snow falls  

the backyard with the snow shoe track, 

some pictures of the forest just north of our house, 

and a shot of the best mode of transportation for breaking trail on 20" of new snow.

More Physics

Today being a snow day ( kind of irrelevant when you are retired), and since I had to wait for some too thick shellac to dry on my nightstand, I had time for more of the physics problems that Mr. Gilmore of Ridge sent me. I have to admit that, contrary to my normal philosophy of in rotational motion viz. angular velocity omega is everything - nearly, I used the velocity to find perigees for orbits, given the  radius and velocity at the apogee. However, you can only use  the total velocity x radius  at the apogees and perigee to calculate the conserved quantity, angular momentum, because the velocity has non tangential ( i.e some radial ) components elsewhere in the orbits. Thus, if you need  to know the velocity at other points you can find the total speed using conservation of energy given the radius, but to find the direction you would find the  tangential or angular velocity using conservation of angular momentum and  omega r^2 is the conserved quantity.

I am now waiting for Mr. Gilmore to let me know if my latest efforts yielded correct answers.

Snow Report and a nightstand nears completion

We dodged the big one last week, getting only 5 inches compared to 30" elsewhere, but we are getting our fair share today. About  14" and near zero temperatures equal a real dose of winter. Hope to get out and snowshoe some tomorrow... after some shoveling.

In the mean time, I am creeping up on completing one of two mahogany nightstands.  Here are a couple of pictures; sideviews. "Just" need to rub out the last coat of finish and make the drawer ( nine pieces including four hand cut dovetail joints- thus the quotation marks around the just.)

Some Physics

 Mike Gilmore, a former colleague has been kind enough to send me a few physics problems to work out. The harmonic motion problems helped me remind myself that in harmonic motion omega is everything, or nearly.  The orbital motion problems were a bit further removed from my usual thoughts but omega is nearly as important in them.  In circular motion the acceleration is equal to omega ( the angular velocity -radians per second) squared times the amplitude and the direction of the acceleration is opposite the displacement. Any other motion where the acceleration can be expressed as  some constant times the amplitude, i.e. a = const x Displacement and is in the direction opposite to the displacement yields an angular velocity, omega,  = square root of the constant . The frequency simply equals
omega /2pi and the period = 1/frequency = 2 pi/omega.

Orbital  motion introduces gravitational potential energy and conservation of angular momentum into the mix, but if the motion is circular omega is still a useful quantity, and omega Rsquared is conserved.

A visit from Daniel

Daniel, our younger son,  was in town working on a paper at Tufts. We visited with him at the Richters outside Boston, and Chris, our older son, and his wife, Tracy, brought our grandchildren. We all had a very good time. 

A Shapely Leg ( at least I think it is.)

Finally got to the point where I can start applying finish to a leg to see what it will finally look like. I had to chop mortises into the front and inside of the top of the leg and then bevel three sides before I could apply the first coat of shellac. This is the rear view of the worst of the 8 legs.

Black Mountain NC

Been away visiting family and friends in North Carolina. 

Black Mountain is a town full of shops selling local arts and crafts including some really fine work. It has some good restaurants and is surrounded by some impressive mountains. Well worth a visit on the way to Ashville, the Smokies, and Blue Ridge Parkway.