Bak to klasikal fysiks part 2.
( Kapital-i, in 'I, myself', now spels Il as in isle or aisle.
Leter y spels sym for seem or seam and partys for parties.
Leter w spels swn for soon. )
Links to sektions:
Relativity is bais'd on the limiting maximum velosity of the spyd of lIt, c for constant ( in konventional speling ). Quantum theory is bais'd on a limiting minimum of enerjy transfer in 'lumps' or 'quanta'. Max Planck invok'd this nyr infinitesimal quantity, h, the 'quantum' in relation to a problem with infinitly kontinuus wavs of radiation, thrown up by the theory of ovens or blak boxes.
LIt had byn shown to hav wav-lIk efekts of interferens paterns,
such as ar syn in water wavs. But introdusing lIt quanta was lIk saying
wavs ar fundamentaly mayd up of partikls. It is as if uon wer to chanj
uon's belyf, that water wavs form a kontinuus flow, to konsidering them as
ryly mor lIk dwns or wavs of sand, that on very klos inspektion ar
mayd up of tIny grains.
However, this simpl analojy gloses over the dyp puzls enkounter'd on an
altogether smaler skail of fysikal fenomena.
Einstein twk up Planck's idea in the form of 'lIt quanta' or fotons, to explain the 'foto-elektrik efekt'. It was found that bombarding the surfas of metal, with sertain byms of lIt, disloj'd elektrons from its surfas. But the efekt didnt depend on the intensity or brItnes of the lIt yus'd. If the lIt was ultra-violet, no mater how dim the bym, it stil suksyded in noking off elektrons.
Einstein explain'd that hIer frequensys of lIt, or mor jeneraly
elektro-magnetik radiation, such as violet or ultra-violet lIt wer, in
efekt, faster 'bulets' or quanta of lIt enerjy. Diming this mor
enerjetik lIt only ment that fwer bulets wer being fIr'd. But wen they
hit an elektron, bound to a metal atom on the surfas of a metal plait, they
wud stil disloj it.
In kontrast, it didnt mater how brIt yu mayd longer wav, red or infra-red
lIt; in other words, it didnt mater how many red bulets wer fIr'd, they
wer al tw low enerjy to ionis the metal surfas.
An analojy to the foto-elektrik efekt mIt be tw walkers, walking into som-uon. They ar both going at the saim spyd. But uon
walker is taking long slow strIds and so not yusing much enerjy. He hardly
disturbs yu, pasing by. But the other walker is taking short fast steps, and
noks you out of the way, busling by.
Thys tw walkers going the saim spyd kompar to lIt spyd as being
konstant. The velosity of lIt equals its wav-lenth tIms its frequensy. (
Or, v = l x f. Texts normaly yus Greek smal leters lambda for wav-lenth
and nu for frequensy. ) Red lIt is just as fast as violet but the former
maks up for its lower frequensy with longer way-lenth. Red lIt's longer
wav-lenth is lIk the longer strIds of the walker making up for a lower
frequensy of steps.
Einstein's 1905 paper on the foto-elektrik efekt is sum'd up in the
formula, E = hf. Enerjy, E, equals Planck's konstant, h, tIms the
radiation frequensy, f ( usualy denoted by the Greek smal leter nu ).
The formula E = hf sujests a kompromis betwyn the wav and
partikl theorys, of mater in sub-atomik fysiks, mesur'd as kontinuus and
diskryt quantitys. The quantum, h, is a minimum diskryt enerjy quantity, of
wich kontinuus quantitys of wav-lenths must always be an exakt multipl.
The frequensy, f, is nown in the klasikal fysiks of kontinuus
sirkular motion as the tIm rait of chanj of an angl -- kal it angl Q. (
Again the typikal Greek smal leter theta is not jeneraly availabl for yus.
)
Wav motion is related to sirkular motion, wich is konveniently expres'd in polar ko-ordinats.
Wen I had to draw grafs at skwl, a long tIm ago, they wer always rektangular shaip'd. That is to say in rektilinear or Cartesian ko-ordinats. Yu hav an x-axis wich is the horizontal ordinat and a y-axis wich is the vertikal ordinat, both mesur'd from a zero starting point or orijin, by konvention, from the botom left-hand korner of yor graf paper.
But an equaly important tIp of ko-ordinat system is polar ko-ordinats. Ther orijin is the 'pol' or senter of a sirkl. Uon, of ther tw ordinats, is the radius, wich swyps round lIk the minut hand of a klok. ( Tho, the konvention is that the radius turns anti-klokwIs. ) The other ordinat is the angl Q that the radius swyps out, lIk an angl betwyn tw lIns of longitud from pol to equator, if yu ar only sying a sfyr tw-dimensionaly from the north or the south pol.
The tw ko-ordinat systems may be match'd so that the x-axis stands at '3 o'klok' and the y-axis stands at '12 o'klok', so to spyk, on the sirkular 'fais' of the polar ko-ordinat system. This alows uon to match or synkronis the tw ways of drawing the saim graf. Supos the radius starts off at 3 o'klok and turns anti-klokwIs to the vertikal '12 o'klok position. In other words, the radius movs a nInty degry angl from an x-axis position to a y-axis position.
Plas'd besId this kombin'd graf, yu kan hav an ordinary rektilinear graf, wich folows or maps the chanjing vertikal hIt of the radius, wer it tuches the sirkumferens, abov or below the horizontal diameter, and traises a path up to the krest of a smwth hil.
The asent's shaip is match'd by the desent from the krest bak to the 'surfas', level with the orijin. This is the haf wav-lenth point, koresponding to the radius having rych'd the '9 o'klok' position on the polar 'klok fais': that is, its negativ x-axis.
Then, the ordinary rektilinear graf's trais desends into the trof of the waiv, wich is as dyp as the krest is hI. ( Krest and trof korespond to the ful hIt of the positiv and negativ y-axis positions on the polar sirkl, at '12 o'klok' and 'six o'klok' respektivly. ) Finaly, the rektilinear trais clIms out of the trof to the surfas to komplyt the wav-lenth, wich is equivalent to the radius komplytion of a ful sirkl and returning to the positiv x-axis or '3 o' klok' position.
The wav-lenth, l, such as from krest to krest or trof to trof, on the rektilinear x-axis is the saim lenth, tw pi tIms r, as the sirkumferens, C, of the sirkl in the polar system: C = 2πr = l. As such, the wav-lenth is a spatial wav, mesur'd by the sirkumferens distans.
But it is also posibl to hav a wav-lenth in tIm, kal'd a period, P. This has byn implisit in komparing the polar system to a 'klok'. Given kloks with 24 hours on the dial, uon revolution of the hour hand marks out a period of revolution of the erth on its axis. The usual 12-hour kloks mark out a haf period of revolution.
The number of revolutions a radius komplyts usualy gos by the leter, n.
On the koresponding rektilinear graf, the wav-lenth, is repyted n tIms
lIk a serys of perfektly uniform ripls. This kan be don for wav-lenths
in spas or in tIm. Hens, the distans, s, kover'd by n wav-lenths is: s =
ln = 2πrn.
And the tIm, t, over n temporal wav-lenths or periods is: t = Pn.
Distans, in a strait lIn, travel'd, kontinuusly, over tIm, equals
velosity or spyd in a strait lIn. Long ago, skwl mayd me familiar with
this. But I dont remember lerning the related formulas for rotary motion, as
distinkt from linear motion.
The distans kover'd in sirkular motion is given by the number, n, of
revolutions thru a komplyt sirkl, hws angl is 360 degrys or 2 pi. This
is kal'd the 'angular distans', say, Q. So, Q = 2πn.
Therfor, s = Qr. Or, the linear distans equals the angular distans tIms
the radius.
Similarly, for ( linear ) velosity, v, and angular velosity, w.
Angular velosity equals angular distans divided by tIm: w = Q / t =
( 2πn / Pn ) = 2π / P.
And v = s / t = ( 2πrn / Pn ) = wr.
( In turn, a further similar relation exists betwyn akseleration and angular
akseleration. )
But wy'v syn, abov, that velosity also equals wav-lenth tIms frequensy: v = l x f.
And v = s / t = ln / Pn = l / P. Therfor, the frequensy is inversly equal to the period: f = 1/P. Twelv hours on the klok fais akount for half the period of revolution of the erth. Therfor, the ( angular ) frequensy is 2 turns per period.
A radius swyp wich starts at '3' on the polar 'klok' produses a so-kal'd sIn
wav from its equivalent starting point in a separat graf of rektilinear
ko-ordinats at ther orijin, wer x- and y-axis kros.
Starting the radius swyp at '12' on the polar klok, starts a so-kal'd kosIn wav from that ful
hIt ( the maximum amplitud ) on the rektilinear y-axis, direktly abov the
orijin.
The sIn wav starts at zero amplitud.The kosIn wav starts at ful
amplitud or the krest of the wav. If you draw them both on the saim graf,
the x-axis always marks the sIn wav 90 degrys or uon-haf pi ahed of the
kosIn wav. This is just as the polar graf shows the 3 o'klok position is
90 degrys ahed of the 12 o'klok position, as the radius kounter-klokwIs
swyp is mayd to go by mathematikal konvention.
This nInty degrys out of step betwyn the sIn and kosIn wav is kal'd
a diferens of feiz angl ( kal it angl q ). Supos that the
sIn and the kosIn wavs hav both gon the saim lenth of wav,
koresponding to an angl of sIz Q in the polar system. Given that the feiz,
q is 90 degrys, then sIn ( Q + q ) = kosIn ( Q ).
This folows from skwl trigonometry of sIn Q equals the rIt angl'd
triangl sIds' ratio of oposit over hypotenus. KosIn Q is ajasent over
hypotenus ( triangl sIds ). In the polar system, a radius is konsider'd as
the hypotenus of triangls it forms by droping a vertex ( for a y-axis
ryding ), from the point the radius tuches the sirkumferens, onto the
sirkl's horizontal diameter ( for an x-axis ryding ). Uon kan chek on uon's
kalkulator, for exampl, if Q = 0 degrys and q = 90 degrys, then sIn ( 0 +
90 ) = kosIn ( 0 ); or, sIn ( 30 + 90 ) = kos ( 30 ).
Julian Barbour yuses the standard introduktion, to the many mysterys of wav mekaniks or quantum mekaniks, with the dubl slit experiment. ShIn a bym of lIt thru a singl slit onto a wal. Most of the lIt wil go strait thru the slit and form the densest target area on the wal. The rest of the lIt wil get mor or les deflekted from the ejes of the slit and skater about the densest area.
Replasing uon slit with tw slits, klos together, produses a surprisingly
diferent piktur. On the wal, a serys of bars form, the midl bars being
most densly lit. Ther is an absens of lIt strIks betwyn the bars.
The singl slit experiment kud hav byn explain'd as either a partikl or
wav aktivity of lIt. But the dubl slit experiment has to be explain'd in
terms of lIt being in the form of both partikls and wavs.
The bars of lIt ar karakteristik of interferens paterns found in water
wavs. Wen tw radiating sirkls of wavs, such as from tw stouns drop'd in
a pond, kolId, the krests of the two rings may re-infors ych other kreating
hIer krests. The trofs may re-infors ych others' depth. Wen uon ring's
krest ko-insIds with the other ring's trof they nutralis ych other to
surfas level.
The snag is that even wen uon foton at a tIm is sent at the dubl slit, the foton hits on the wal bild up the saim patern, as if they wer sent in a stedy strym that wav'd into ych other. The basik paradox of quantum theory's wav-partikl duality of lIt ( and elektrons etc ) is that a singl partikl kan interfyr with itself lIk a wav.
The dubl slit experiment's patern of foton hits is given a probabilistik
prediktion by Schrödinger's equation of the wav funktion ( denoted by the
Greek leter psi ). The wav funktion that givs the best interferens
efekts, in the experiment ar so-kal'd momentum 'eigen-states' ( German for 'proper' or 'karakteristik' staits ).
Eigen-staits of position or momentum ar the only
uons that kan be mesur'd with komplyt or unit probability of matching
prediktion.
It turns out that the wav funktion of a partikl with a definit momentum
( the momentum eigen-stait ) is tw super-pos'd plain wavs out of feiz by a
quarter of a wav-lenth. By definition, sIn and kosIn wavs ar out of
feiz by 90 degrys or a quarter of a wav-lenth ( as was diskus'd in the
previus sektion ).
The quantum mekanikal wav funktion is a komplex funktion. Roger Penrose, in
'The Emperor's New Mind', explains how komplex numbers ar yus'd in this
funktion.
SufIs it to say, a horizontal x-axis kud represent the direktion of the tw wavs. Bakward or forward direktion is related to wich of the tw wavs koms first. A y-axis kud giv a bak-ground dimension to the sIn and kosIn wavs, turning them from just undulating lIns into planar wavs. A z-axis kud mesur the hIt or amplitud of thys wavs. But the y and z axes ar tryted as komposit or komplex numbers, wich ar order'd pairs of numbers. Thys represent tw intensitys. The sum of ther squars is the 'probability density' of the ( komplex ) wav funktion, psi, of the x-variabl. Notably, this givs the probability that a trial mesurment wil find a partikl at x.
A partikl has a definit momentum bekaus its wav funktion has a regular
and definit wav-lenth. At the saim tIm, the partikl's position is
komplytly indefinit. Its probability density is uniform thru-out spas,
bekaus the sum of the squars of tw sinusoidal wavs, uon-quarter wav-lenth
out of feiz, is always uon, given that they hav unit amplitud.
This koms from Pythagoras' theorem in trigonometrik form: sin²Q + cos²Q =
1.
Fourier show'd that ading or super-posing harmonik wavs of diferent wav-lenths kan produs any kurv, even down to a spIk, that karakterises the position eigen-stait. ( Mathematiks imitats natur's wav-partikl duality. The saim wav patern kan be regarded as super-pos'd wavs of diferent wav-lenths or super-pos'd spIks, with diferent ko-eficients. )
Thys ar the extryms betwyn komplyt and nul information we kan hav
betwyn komplementary pairs of quantitys, such as momentum and position, or
enerjy and tIm. Heisenberg's unsertainty relation mesurs the extent, that
mor akuratly mesuring uon of thys pairs of quantitys, is always at the
expens of presisly mesuring the other.
The experimenter kan mesur uon or other of the komplementary pairs, al implisit in the wav funktion. So a lak of komplyt nolej is off-set by a ranj of chois as to wat kan be nown.
The dubl slit experiment kan be konsider'd in terms of tw similar plain lIt wayvs super-pos'd or merj'd at a slIt angl ( of fIv degrys ) to ech other. At rIt angls to the mid-lIn of the fIv-degry angl, Barbour's bwk shows, as a komputer-jenerated probability density, the result, in a konsertina-lIk serys of lIt rijes, koresponding to the lIt 'frinjes' that show the interferens efekt on the wal.
This result relats to William Hamilton's klasikal work in optiks. His wav theory show'd that regular wav paterns reprodus lIt rays, without partikls, yet koresponding to the older partikl theory of lIt, and explaining mor than it kud.
Hamilton found an analojy to this in Newtonian dynamiks, for only uon valu of enerjy alow'd. Depending on an equation, his 'prinsipal funktion' has a varying intensity, lIk the 'mist', at ych point of konfiguration spas. This equation is lIk that for his wav optiks, but in multi-dimensional konfiguration spas, insted of the ordinary thry dimensions. Wen the intensity forms regular wav paterns, ther respektiv familys of paths, at rIt angls to ther krests, ar Newtonian dynamiks' historys, having the saim enerjy.
This path-making property of regular wav paterns has given them the naim 'semi-klasikal', wich is also the naim of a fysisists' program, to show how aparent paths in tIm may 'funel' out of a tImles jeometrik struktur under-lIing the univers.
The Einstein, Podolsky, Rosen paradox was a thot-experiment desIn'd to
revyl that quantum mekaniks, kompar'd to klasikal mekaniks, was inkomplyt
in the skop of its explanations. Tw partikls, such as fotons shering
polarisation, or elektrons hws total spin was konserv'd, wud be
korelated. Chanjing the stait of uon wud hav a korelativ efekt on the
other.
For exampl, a system of tw elektrons, with a total of zero angular momentum,
implIs that if uon elektron has spin up, the other must hav an equal and
oposit spin down.
The EPR paradox was that quantum mekaniks predikted that if yu mov'd thys korelated partikls far apart, then chanj'd the stait of uon, ther wud be an instant konservativ respons from the separated partikl. But Einstein's special relativity forbids any signal pasing, at mor than lIt spyd, from uon partikl to influens the lawful ajustment of the other partikl.
Bell's theorem show'd how quantum korelations must surpas any relations atributabl to klasikal kauses. Roger Penrose has further refin'd thys distinktions, especialy in his sekond popular bwk, 'Shadows Of The Mind'.
The EPR tym belyv'd the law of lokal kauses wud be up-held against quantum korelation. But by the 1980's, Alain Aspect's experiments wer proving a super-luminal konektion betwyn korelated partikls. This did not myn signals kud be sent faster than lIt; it did not violat the foundations of special relativity. But it did myn the experimenter kud bring about a nown, faster-than-lIt chanj on a distant partikl, by a sertain chanj on its korelated partikl.
With the help of half a dozen diagrams, Julian Barbour givs ryders a fyl for quantum korelations and 'entanglment' in the simplest posibl tw-partikl system. Tw partikls moving on a lIn kombin ther uon-dimensional konfiguration spases to mak a tw-dimensional Q.
The wav funktion's valu, for a singl partikl or a duet, as hyr, varys with tIm at ych point in Q, wich karys information about both partikls, as to ther positions or other quantitys. Thys prediktions ar komprehensiv, if often mutualy exklusiv, and refer to the system rather than its parts.
To find the relativ probability of konfigurations of the tw-partikl system at som point, representativ averajes ar found from the mid-points of a grid on Q. The probability density givs the relativ numbers of thys representativ konfigurations lIkly to be found by repyted trial mesurments.
Barbour lIken'd this proses of prediktion and mesurment, to giving the predikted numbers of konfigurations a proportionat number of tags in a bag and then drawing them out at random, as a trial konfirmation of the predikted proportions.
Thys konfigurations ar, in efekt, rank'd by ther greiter or leser
probabilitys, wich is how the Schrödinger equation konfigurs atomik and
molekular struktur from the konfiguration spas of al possibilitys.
( In the simplest 'platonia' or relativ konfiguration spas of Triangl land
-- diskus'd on the first web paj of this reviw -- a probability ranking may
be establish'd thru best matching al posibl triangls, ych represented by
a point in ther platonia. )
Mesuring the tw-partikl quantum system for, say, the position of uon of
the partikls reduses the tw-dimensional grid on Q to uon dimension. This
so-kal'd 'kolaps of the wav funktion' yilds the only posibl positions
of the unmesur'd partikl, as relativ probabilitys of being som-wer on
the remaining grid lIn.
Hugh Everett asum'd that the wav funktion is 'the basik fysikal entity'. Its unimajinably huj numbers of posibilitys ar taken to konstitut 'many worlds'. Tho we ar only konsius of uon world being rylIs'd, this dos not nesesarily myn that is al ther is. Everett defended this posibility by the linearity or super-position prinsipl of wav mekaniks. Wavs kan split and kombin, to kreat interferens efekts, but they remain themselvs, esentialy unafekted by it.
Akording to Barbour, 'To saiv the apyranses, we do not hav to kreat a unyk history: we nyd only explain wI ther syms to be a unyk history. That was Everett's in-sIt.'
Barbour sys the esens of things in platonia, the relativ konfiguration
spas, or a komplytly relativis'd version of Schrödinger's Q. This jeometrik
land-skaip, of al posibl konfigurations of reality, gathers, lIk a mor or
les dens mist, the wav funktion's probability density. Folowing Boltzmann,
Barbour asums: only the probabl is experiens'd.
Richard Lung.
Barbour on quantum kosmolojy.
To index of simpler spelt pages.