Quantum, Light and Stuff

Mike Hewson
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Topic 197584

I thought it would be good to talk* of the properties of light as regards our LIGO detectors. They use light to measure spacetime and in so doing we hope to catch a wave from space. We could do this and yet never leave the Classical Physics Classroom ( well, almost ). But that would be boring. In any case, starting from a quantum mechanics ( QM ) baseline one can derive the classical stuff anyway. I'll try to keep it real by using LIGO apparatus as examples, and I'll nearly always refer to photons even though we know QM applies to much else besides.

Let's begin with a general chat about the mindset to approach quantum mechanics. You probably get a dreaded feeling when you see that phrase. You'd be thinking of horrible mathematics, obtuse statements and professors stroking their beards. You'd be thinking hard brain yards. I hope to disabuse you of that. I have found there is a remarkable simplicity within QM, but it is a weird sort of simplicity because you have to put aside a slab of your everyday accustomed logic. You don't have to throw away all of your common sense, but to progress in understanding of QM you have to yield to propositions for which there is no analogy you directly know of. As usual I will use analogy and metaphor and similies and language tricks, such as I can. However there will still remain a core of disbelief though. It won't feel right. The truth is that everybody gets that with QM, including the very people who invented the subject. So do not panic. All of the architects of what is now called the 'Copenhagen Interpretation' of QM did struggle to grasp intellectually and/or emotionally its machinery and its implications. Please try to put such discomfort aside and, perhaps like reading a sci-fi novel, accept the assumptions & premises as they are. So that you may move on and enjoy the story. :-)

One big reason why QM is uncomfortable is the plain fact that really little things just don't behave like much larger things. We human beings are amongst those much larger things. For instance the 'wave-particle duality' you may have heard of is an attempt to marry experimental findings of the smaller realm with experimental findings of the larger realm. The deeper issue though is that particles and systems for which QM applies really well do not behave like particles or waves. A wave viewpoint or a particle viewpoint upon measurable results are extremes upon a spectrum of interpretations that we may make. They are not bad aspects to use. It's important to understand that they aren't complete though. There are some matters quantum mechanical for which wave or particle perspectives are not what works fully or even at all eg. entanglement and Bell's inequalities. There's probably some even greater weirdness to come .... :-)

So to assist in adjusting to QM I strongly recommend taking a modeller's perspective. That being : one has a logical apparatus - yes, expressed often in mathematical symbology - which produces for a given scenario predictions to then take to the real world and compare with measurement in order to judge the success of the model. So that requires a type of rigor in thinking and a willful suspension of disbelief. This ought be emotionally neutral, or as Richard Feynman sharply put it : if you don't like it then go to another Universe ! :-)

Take a photon for instance. For physics a photon is really a photon model. By that I mean we have an apparatus to simulate/emulate the object that we label as a photon. We crank the handle of that machinery to emit a prediction for an outcome of some meaurement involving that photon. QM requires that you can only really speak firmly of what you measure. For all other non-measurable entities eg. a quantum mechanical state, we use them within the machine as an intellectual cog. We accept the existence of that cog in order to achieve the desired end of the process. So the cog is not 'real' but an artifice that serves a purpose. In physics that purpose is prediction. So that means making statements regarding the future. The general flow of mental work in QM, as with much else in physics, is to :

- define a system to be described.

- define that system's initial conditions at some time.

- progress that system through time using some recipe for change.

- hence arrive at some final system configuration at a later time.

- extract information from that later configuration to get a hard number(s) of interest.

Now there are a few non-negotiable parts of QM. By that I mean you have to accept them to have any hope of following the topic further. Annoying perhaps but true. Let's tackle the first of these head on :

Quantum Versus Classical Probability

These are nearly the same. Alas like many things that are nearly the same one risks the substitution of one for the other, and perhaps without realising that one did that. You may be aware of the classical rule with adding of probabilities for mutually exclusive events. A set of mutually exclusive events is such that if one event in the set occurs then none of the others in the set can occur as well. There are plenty of examples of this each and every day. I could take the main highway to work OR I could go the back way. I could stay in bed today OR I could go to work. Filling up with petrol could cost me [less than $20] OR [$20.00 to $30.00] OR [more than $30.00]. I could go to the casino OR I could go and watch the footy. Etc. That OR is an exclusive or. These are generally 'forky' scenarios where some choice/option is irredeemable.

Take some given horse race. Suppose Mother's Pride has a ( true, not bookies ) probability of winning = 0.2 : this means that hypothetically if I run this particular race ( same conditions, same track, same jockeys, same field of horses ... ) say five times then Mother's Pride will most likely win once out of those five occasions. This is called a priori probability in that it is predictive and based upon presumably good information & knowledge about horses. With the same race there could be Naughty Johnny who has a probability of winning = 0.1, so one could ask the question : “given these probabilities, what is the probability of either Mother's Pride or Naughty Johnny winning?†You know the classic answer. Add the individual probabilities to, in this instance, get 0.3 for that. Only one horse will win** but for that subset of wins – either MP or NJ – it's 3 out of ten.

Let's go down to Quantum City racetrack where they bet on particles going thru slits :

So here comes the big weirdness of quantum mechanics. IF a particle arrives at the finish line having had a choice of going through one or other of two slits :

- particles only arrive in whole number lots. No fractional particles. Ever.

- IF you did not disturb the particle while going from start to finish THEN the total probability = [probability thru slitA] plus [probability thru slitB] plus [an interference term]

- IF you did disturb the particle then the interference term = ZERO, so the total probability = [probability thru slitA] plus [probability thru slitB].

- the probability obtained ( whether you did or didn't disturb the particle ) then determines the chances of your detector firing off : “I got one!â€

- where the interference term contains an assessment of the differences between the path thru slitA compared to the path thru slitB. This term may increase the probability of a result. It may reduce the probability of a result. Sometimes it doesn't affect it at all.

That was the simplest way that I can put it without loss of crucial content.

An undisturbed flight from start to finish implies interference behaviour ( to be described later ). Disturbance sets that interference to zero. Please do not thrash yourself trying to work out which way the particle went when you were not looking/disturbing. This also has nothing to do with human consciouness and/or anthropocentric concerns***. It is simply true that once the flight of the particle is disturbed - specifically in a way that yields “which way?†information - then all interference aspects disappear. Without fail that interaction changes what you are describing. This stuff occurs whether or not you send particles through the setup singly or in groups. It happens if you take a long time to test this or if it was only done briefly. The behaviour of one particle in the two slit arrangement depends on neither the behaviour nor the presence of others.

Yes, there is no everyday analog. Yes, it does not make sense. Yes, it is a conspiracy of nature. Suck it up.

The QM apparatus handles this as follows. Each path/choice is assigned an amplitude. An amplitude is a number of a special type. For mutually exclusive paths one adds the amplitudes. Then the result of that addition is 'squared' in a particular way to give the probability of the combined outcome. This brings us to the next non-negotiable component of QM :

Complex Numbers

Cheers, Mike.

* I am an irremediable chatterbox .... :-)

** Yes, I know dead heats are sometimes awarded down at the racetrack, but not for this example!

*** The Schroedinger's Cat construct has been thoroughly misunderstood. It was a straw man meant to be destroyed in order to discuss the manner of how small quantum influences might propagate to the attention of big macroscopic blobs like us. It has no place in the learning of QM.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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Quantum, Light and Stuff

Complex Numbers ( Definition )

I have covered these elsewhere, but let's be more thorough this time and focused within the context of QM. You could, by some heroic convolution of the language of mathematics, do QM without mentioning the word 'complex number'. You could pretend that there is no such thing as the square root of minus one. You could grab a QM textbook and mark out, and then suitably substitute something else, all the references to this area of mathematics called complex numbers/analysis. But you still will not escape the need in QM to have a pair of numbers attached to the objects that we describe, with the characteristics of 'magnitude' and 'phase' along with the rules of complex number arithmetic. In what follows please remember that it is not just the choice of objects being described, but that set plus a framework under which they interact.

It is probably best to begin by thinking of each single complex number as a box, and inside the box is a pair of real numbers. But with special rules of combination to be discussed. These are not rules to combine the real numbers together within a given pair, we don't ever do that, but rules to combine the pairs.

A real number is, if you like, any of the ones you are used to in everyday life. Like zero, or minus five, or sqrt[78], or 24.81 etc. They are the ones on the number line that you saw at school :

So by gathering real numbers up in pairs we are doing an accounting trick of sorts. Why would you want to deal with numbers in pairs? There may be lots of reasons in physics and life generally why you would want to have two ( or more ) distinct real numbers, but firmly associated in some form. As noted we'll be thinking of 'magnitude' and 'phase' semantics for QM. Now you probably have dealt with ordered pairs before eg. (2, 3) indicating that there is a 'first' and a 'second' position in the pair. So (2, 3) is not the same as (3, 2) for instance. And you can map these on some two dimensional Cartesian plane :

Quote:
Cartesian : after Rene Descartes, the French chap who put algebra into geometry ie. invented analytic geometry. Mr Isaac Newton just loved his work, reading and re-reading until he got it all in. It was of such a high intellectual calibre that a pope just had to ban his writings. But still, his death could have been by natural causes anyway ... :-0


The Cartesian Plane is the result of forming a Cartesian Product of two sets. It is the set of all ordered pairs of real numbers. So the elements of a cartesian product are pairs ( or n-tuples in higher dimensions ). More generally I have a set A and a set B with their respective elements, and so their cartesian product written as AxB. But we could have anything at all, say S = all sausages and R = all rabbits. Their cartesian product, written as SxR, is the set of all pairs where the first is a sausage and the second is a rabbit. Not a wildly helpful set in the modern world admittedly. However SxR is not the same set as RxS, as sausages and rabbits are not generally interchanged. Usually. Moreover not all sausages are the same amongst themselves necessarily, nor for that matter the rabbits. So even if sausages and rabbits were the same type of thing, each particular sausage/rabbit may not necessarily substitute for another without some difference or error. Evidently with real numbers 2 != 3 etc. While this may seem an unusually arcane point to emphasise*, please read on ....

Now we invent a new number ( which is not a real number ) that we call the imaginary number. I suppose you could say that there is only one imaginary number. That number is typically given the symbol i ( often j in electrical engineering ). It is defined by the equation :

i * i = - 1

........ so it is the square root of minus one. There is no deeper definition worth discussing here. If you assume that defining quality - of when multiplied by itself you get minus one - then much follows. We can thus have multiples of i :

Quote:
It's not a big point but : some authors say there is only one imaginary number but that has as many real number multipliers as you please ( infinite ). Others say all real number multiples of i are also called imaginary numbers. Of no operational consequence provided the rules of combining complex numbers are well understood. Potato/potato, tomato/tomato ..... :-) :-)


Thus we have a real number line/set and an imaginary number line/set as distinctly defined entities. You get the punchline : combine the two as ordered pairs where the first number is real and the second number is real and is the multiplier of i. That second real number is called the imaginary part of the complex number. There is a variety of equivalent notations for complex numbers :

(x, iy)

OR

x + i * y

.. this is an ordered pair format and you could use it on the Argand Plane, which is distinct from the Cartesian Plane :

... where one plonks the imaginary number line at right angle to the real number line, intersecting at the zero point for both. While I use '*' as indicating multiply, I could write iy just as well. The '+' however is not addition in any sense at all. We do not add the real part of a complex number to the imaginary part of the same complex number. The '+' is used here in the sense of “associate†or “attached†or “the numbers on either side of me form a single complex numberâ€.

(r, theta)

OR

r * exp[i * theta]

... this is the polar form, related to the work of Abraham de Moivre and Leonard Euler ( more on that later ) :

... so one may view the very same point on the Argand plane in either of two ways. An ordered pair of numbers measured in orthogonal/rectangular sense, or an ordered pair of numbers in a radius/angle sense. With conversions using trigonometry. You could also think of it as a vector going from the plane's origin to the point. We will deal with the operations on complex numbers next ie. how do you combine complex numbers to produce another complex number ?

Cheers, Mike

* Hinting at the later topic of particle interchanging/fungibility here. It may or may not be the case in QM that you can swap particles with/out there being a difference. What is obvious for sausages and rabbits ain't so in the Quantum Paddock.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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Complex Numbers ( The Messy

Complex Numbers ( The Messy Bits )

We need operational definitions of ways to combine complex numbers. Since a complex number 'contains' real numbers then we ultimately rely upon the rules of real number arithmetic to form the rules of complex number arithmetic. Start with any two complex numbers :

z1 = x1 + iy1

z2 = x2 + iy2

Recall that '+' does not indicate an operation, just the association of the parts of a complex number.

Addition :

z1 + z2 = (x1 + x2) + i (y1 + y2)

Here '+' means the operation of complex number addition being defined as that which is on the right of the equals sign, whereas '+' is to remind you of real number addition.

Subtraction :

z1 - z2 = (x1 - x2) + i (y1 - y2)

Likewise '-' declares complex number subtraction, with '-' being real number subtraction.

Quote:

Now one can write :

x - iy

as long as you understand that it really means :

x + (-1)iy

OR

x + i(-y)


Whew! Sorry to be so particular about such symbology, but there is considerable semantic overloading of the notation happening. Which may trap the unwary, but I won't make those distinctions from now on. By extension one can go to longer and more difficult expressions, which one can evaluate by coming back to stepwise evaluation. Parentheses do their normal job of partitioning & ordering the work.

Multiplication and Division

z1 * z2 = (x1 + iy1) * (x2 + iy2)

= x1 * x2 + x1*iy2 + iy1*x2 + iy1*iy2

= (x1 * x2 + i*i*y1*y2) + i*(x1*y2 + y1*x2)

= (x1 * x2 – y1 * y2) + i(x1*y2 + y1*x2)

which is true though messy. But there is a neater way provided you have the radius/angle format* :

z1 = r1 * exp[i*theta1]

z2 = r2 * exp[i*theta2]

then

z1 * z2 = r1 * r2 * exp[i*(theta1 + theta2)]

or for that matter :

z1 / z 2 = (r1 / r2) * exp[i*(theta1 – theta2)]

... I'm not going to bother you with the horrible expression for complex number division in orthogonal/rectangular format. Mind you addition/subtraction of complex numbers has no evident simplicity with the radius/angle representation either. Let's gather this all up.

- if you are going to add/subtract then x + iy is very convenient. It works in the Argand Plane like adding two vectors head-to-tail by adding/subtracting their corresponding components :

- if you are going to multiply/divide then r * exp[i* theta] is far easier. It works in the Argand Plane by scaling the radii/magnitudes against each other and summing/differencing the angles :

- you could use the above formulae to come down to this one :

exp[i*theta] = cos(theta) + i * sin(theta)

Quote:

I ought throw in Richard Feynman's favorite one :

exp[i * PI] + 1 = 0

ie. as cos(PI) = -1 and sin(PI) = 0, this relates the numbers 0, 1, PI, i and e ( natural logarithm base ) all in the one equation.


In QM when there are a number of different ways that things can happen we will add the QM amplitudes** from each. When things can happen in some sequence we will be multiplying QM amplitudes. ( To be discussed in detail later )

So complex numbers have the right sort of behaviours that suit QM calculations. The Argand Plane is descriptive device without any physical reality. You could consider that each thing we are describing - photons, electrons, paths etc - has their own private Argand Plane upon which QM amplitudes play out. We will discover that any single angle there cannot be measured, though differences between angles have an effect on results ( the interference stuff ). A radius is also not directly measurable, but it's value may be indirectly inferred via it's impact on the statistics of groups of objects aka square it to get a probability.

Next up we will discuss something which is optional but very highly advised IMHO, slightly negotiable perhaps ! I was never taught it on my first time around*** through QM and only saw it much later via Richard Feynman's Big Red Lecture Books**** . That is :

The Dirac Notation

Cheers, Mike.

* This exponential format came out of Euler & de Moivre and believe it or not actually corresponds/connects to the 'ordinary' real domain for the natural exponent function !
You are probably aware that :

exp[C + D] = exp[C] * exp[D]

this also works if complex numbers are used as the argument of the exponent ie.

exp[C + iD] = exp[C] * exp[iD]

** Especially note that 'amplitude' != 'magnitude' as they might be in other contexts. A QM amplititude is an entire complex number with two parts. It's magnitude is, we will find, the radius part when in the radius/angle format or equivalently the third Pythagorean side in orthogonal/rectangular format.

*** They really only showed me some advanced methods on how to solve specific partial differential equations and use some snappy integral theorems. Which I later successfully vomited back up. It was applied maths with some quantum language splattered about. Here I'm trying to describe QM in a way I wish I'd been taught ! :-)

*** The Feynman Lectures On Physics. What a treat it would have been to meet the guy or sit in on a lecture or whatever. If you are even vaguely interested in physics you will never regret getting a copy. Three volumes originally. I have an original set of the large floppy paperbacks which I bought after asking the QM lecturer ( above ) some questions about the non-applied-mathematics-methods aspects of QM. He said : "Read Feynman". So I did. Here's my upbeat review ( ~ 2006 ) from Audible.com for the recorded versions :

Quote:

"The Voice Of A Giant"

Close your eyes, you're in a CalTech lecture theatre in 1962, sounds of chalk on board, chairs moving, chuckles and coughs from the audience. A distinctive accent narrates, like a tour guide, a path through the core of an apparent maze - quantum physics. He pauses, thinks, re-states an important point, and you can imagine him assessing - can they understand me? You ARE there, and this IS Dick Feynman at a peak, one of many, passing on the baton to the next generation. It is an engaging world view, full of common sense and an acute sense of good approximation. He even tells you what he doesn't know. His analysis of the central mystery of quantum mechanics - lumpy bullets, interfering waves, happening with real photons and electrons, giving counts & distributions which cannot be understood from our macroscopic experience - is a stunning display of not only his intelligence but his riveting personal style. A classic dissertation. I'd bought and read the hardcopy of his edited lectures, all of them, in the late 70's. This audio presentation was much better, like a full color movie compared to a pencil sketch. Wow! Even if he'd never won a Nobel, or touched quantum physics, he would still rate on the short list of all time greats. You can now hear why. This spoken record will stand for centuries, without staleness, a Principia from a modern day Newton.


Evidently I have caught the Feynman Bug a long time ago. :-)

( edit ) You could also derive the trignometric double angle formulae from the above equations, eg.

sin(A + B) = sinA * cosB + cosA * sinB

and

cos(A + B) = cosA * cosB - sinA * sinB

from

z1 * z2 = (x1 * x2 – y1 * y2) + i(x1*y2 + y1*x2)

= 1 * 1 * exp[i*(theta1 + theta2)]

= cos(theta) + i * sin(theta)

etc. Or conversely use the double angle formulae as part of proof of the above.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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The Dirac Notation Paul

The Dirac Notation

Paul Dirac was a shy quirky genius. A very private fellow, man of few words. He really liked Mickey Mouse and Cher though. For his doctoral thesis he collected together the latest ideas from de Broglie, Einstein, Shroedinger, Heisenberg, Bohm, Bohr plus older classical concepts from Hamilton, Poisson and Lagrange. He gave cohesion to QM at a time ( say 1924 thru 1927 ) when the modern 'Copenhagen' formulation of was a sort of condensing mist.

Unless your interest is only in generalities, then one needs to have a specific/firm representation in mathematics if one is going to take a particular physical system and predict it's future in detail. That is : actual numbers need to be emitted from theory in order to take to the lab and see what is what, and what is not. So for that purpose one can't escape the need for a concrete mathematical formulation. Dirac had a clever idea though. Without taking away from that need, one could use a compact notation which didn't drag all the messy specific representational details right throughout one's analyses.

QM States

Now this is a 'ket' :

|mike>

which symbolically represents the state of a QM system called 'mike'. This might be me, a photon, your lunchbag, anything really. There is no mathematical form used here to firmly specify 'mike', so we have a label in ket form obscuring any underlying detail. But from now on you may talk of 'mike' as a QM object in the theory to be handled properly by QM logic, keeping any messy maths in abeyance. Thus I can manipulate this 'mike' ket in my discussions and provided I adhere to QM rules of operations I have saved alot of writing, error and can focus on essentials.

For instance you could express 'mike' as a linear combination of two distinct QM states :

|mike> = a1 * |asleep> + a2 * |awake>

where :

|asleep> = that QM state where you are certain to find 'mike' sleeping ( if you measured )

|awake> = that QM state where you are certain to find 'mike' awake ( if you measured )

... evidently there is some uncertainty here, in detail a probabilistic situation. At least while we hold off on measuring, 'mike' exists in a mix of two states. The 'a1' and 'a2' are QM amplitudes ie. complex numbers encapsulating magnitude and phase information. They allocate fractions ( in the QM sense ) of the 'asleep' and 'awake' states within the mix. Now suppose the set of states

{ |asleep>, |awake> }

... was complete in that it listed all the possible distinct outcomes of measurement, no more and no less. Then I have entirely specified 'mike' as a QM state by listing these base states and the specific amplitudes to go with each of them in the linear combination above. IF for some system I know a complete set of amplitudes with respect to a given basis set then I know all about that system. Now and forever. I could calculate any quantity of measurable interest at all. So that QM recipe for 'mike' is now fully defined in terms of those base states. Of course I've been mathematically vague about the exact detail of the base states here, but the point is that even if with such a higher level abstraction I can still discuss the generalities of how 'mike' either is now, or might evolve in time later. But as mentioned, when the rubber has to meet the road you still need to evaluate via some concrete mathematical form.

This is a 'bra' :

... in this case meaning the amplitude for finding my system in a state of 'asleep' given that I began with the state 'asleep'.

I could have :

... meaning what amplitude do I have for my system being found 'asleep' given that it started in the state 'awake'. Ditto . You can spot the language gag : a 'bra' placed to the left of a 'ket' becomes a 'bra-ket'. I fold the two adjacent '|' glyphs into the one :

a

is not a

but a

So I've snuck in the idea of state here. If you study or calculate in QM then specifying states and the way they change in time is the key. For example if I have a single particle in free space – meaning unaffected by forces, or equivalently all potentials are zero – then it will continue in that state indefinitely. Gee, does that remind you of something classical ? Sure does -> The Principle of Inertia. If I do have a non-zero potential then the rate of change of that potential with distance ( = force ) is proportional to the rate of change of ( the expectation value of ) the object's momentum with time. Now that wouldn't be F = dp/dt would it? Yes, indeedy! So it does make both notational, mathematical and finally physical sense to write :

d (|mike>)/dt

... or the rate of change of the state 'mike' with time. In older parlance |mike> would be designated as a 'wave function'. A bra applied to a ket then corresponds to a complex valued 'functional', meaning a function upon the set of wave functions that produces a complex number as it's output.

One especial problem with the term 'wave function' is that for all but the most simplest of systems the mathematical form of the function does not actually describe a wave at all. Take a function describing two electrons bound to a Helium nucleus. That function would depend upon two position vectors ( one for each electron ) and time. You can't legitimately separate these out into two 'wave' functions for the reason that there is correlation of the position of one electron with the position of the other. What you could do, well approximately at least, is calculate the expectation value ( a 'weighted mean', more later ) of the position of the first electron and find that it is usually on the other side of the nucleus with respect to the expectation value of the second electron. One never 'sees' either as waves, even as a classical analog. Do you then call those electrons “particles with correlated positions� Maybe. Except that if you try to determine that idea by some measurement - say hit one of the electrons with a photon to deduce where it was – then by said disturbance you will no longer have the original system. Game over ...... Sad Panda.

The upshot is that states are theoretical abstractions serving a computational purpose. One evolves the states via recipes for change aka Schroedinger's Equation(s). You ask questions of the state by applying mathematical operators ( see later ) to them, so producing the all important amplitude values for each possible outcome. Then you can compute a 'first moment' or average of the amplitudes, plus a 'second moment' or variance if you like. That gives, say, the corresponding classical concepts of 'expected position' or 'expected energy' with the variance values becoming the uncertainties of measurement.

Interpreting QM states in terms of classical observables is neither an esoteric nor historical point. With every QM system under study you do have to bring matters back to the classical domain because that is where WE are. QM is quite anthropocentric in that regard. All experiments have the requirement to revert ultimately to a human perceptible form of output. That does not IMHO imply an issue with human consciousness per se, merely an issue with regression/amplification of effects to the human scale.

Don't panic if this post seems a bit unreal. It's a different paradigm from classical mechanics for sure. For instance when modelling an artillery shell under Newton et al, you start with the assumption of dynamical variables like position and momentum having definite values. That is : the object called an arty shell actually possesses such quantities. QM doesn't even admit that. You have to probabilistically extract those values from an underlying state/schemata which is not directly measurable. Position and momentum information appear as derived qualities which need to be determined by a mathematical process. There will be uncertainty. I'll fill out many of the pieces later.

Speaking of the transition from microscopic to macroscopic : next up and definitely non-negotiable is

Planck's Constant

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

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Planck's Constant Planck

Planck's Constant

Planck was a sad figure. He lost his wife tragically when young, both daughters died giving birth, one soldier son died in WW1, the other shot as a traitor in WW2. He is famous for saying no and thus silencing a dictator when discussing the topic of race, and yet he remained alive to tell of it. He was a classicist at heart and never really accepted QM.

Thermodynamics was his forte, that giving rise to the number named after him. He was a contemporary and a foil to Boltzmann, at a time when atoms were merely an hypothesis under vigorous debate. To reconcile theory with observations regarding radiation from equilibrated bodies he was obliged to assume that smoothness was out and quanta were in. That is, energy exchanges at a given wavelength/frequency of light were lumpy/discrete. Einstein later said – to explain the photoelectric effect – that it was the photons themselves that were lumpy. De Broglie later extended that to all particles. So these are two simple but legendary equations :

E = hv

… relating the frequency (v) of a photon to it's energy ( E ).

p = h/w

… relating the QM wavelength (w) of anything to it's momentum ( p ).

I want to convince you that h is not merely some constant of proportionality, but has rather deeper meaning. It is the indicator of the scale at which the character of physical behaviours change substantially in this Universe. I'll jump ahead a bit and reveal a result of solving Schroedinger's equation for an object with well defined energy. That means the object is sitting ( or in constant motion ) there undisturbed without any input or output of energy, in energy equilibrium amongst it's parts, not transforming to something else etc. Just existing without change of state. While it is hard to find one of these objects, as the Universe always intrudes upon things despite how we define them, it serves as a useful thing to discuss. The QM amplitude for such an object has a constant magnitude, but the phase varies with time according to a frequency given by

E/h

So imagine your coffee cup* as such a system/object. It's total energy – rest mass equivalent included – is enormous compared to Plancks' constant. In MKS units where h ~ 10^[-33] and assuming a coffee cup rest mass of 100g then the ratio of E to h is about 10^[50]. That is 10^[50] cycles per second. So the phase value – a number b/w zero and 2 * PI – goes around like a clock hand, over and over again at the absolutely insane frequency of 10^[50] circuits per second. One single cycle per 10^[-50]th of a second. Wow! We don't measure that phase or it's change directly, but assuming QM theory is correct ( it is well backed by experiment ) across all time scales, then that is the implication.

QM does allow us to compare phases though, and in doing so produce 'ripples' that signal the presence of interference. Do we have any hope of doing that here for coffee cups? Well I don't suggest that we try to do a two slit experiment with coffee cups. I'd predict a lot of pieces of crockery for no gain. But why? In the two slit setup the distance between the two slits is the determinant of the overall pattern on the far screen. To produce variation in the number of coffee cups arising as a function of position on the screen, there needs to be a phase difference between alternate routes to a given screen point.

I won't derive in detail, but quote that to differentiate between a peak and a trough of some interference pattern viewed on the screen, the wavelength of the coffee cup needs to be of about the same scale as the inter-slit gap. Or if you like : how far does the coffee cup travel in 10^[-50]th of a second? To be even more ridiculous : a coffee cup at near light speed isn't going very far in that absurdly small time interval either. We have no practical hope of providing a meaningful slit separation. Even if we had the very smallest slit separation we could ever make, then on the screen the distance b/w peaks and troughs would then be insensible to measurement. Our devices would straddle very many peaks & troughs and only return an average of the peak and trough values. No ripples whatsoever. That peak/trough average turns out to be the very value you'd get if you did a purely classical analysis.

However 'buckyballs' have been subject to this. These are sphere-like molecular cages formed by linking numerous carbon atoms, in a pattern similar to the stitching lines on a soccer ball. In a famous experiment using objects with 60-ish carbon atoms, then interference fluctuations are revealed entirely consistent with QM.

Thus in the classical realm of human scales we see none of this QM stuff. You need special gadgets to do the experiments that amplify these effects from QM scale back to ours. If you did a two-slit with machine gun bullets say, there would be no safe place to stand ( interference minimum ) at the screen ! :-)

Next up :

Light Polarisation

Cheers, Mike.

* Get it out and then sit and look at it. Try to imagine the phase on it's private Argand plane spinning around in front of you.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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Light Polarisation Let us

Light Polarisation

Let us examine some LIGO stuff now and in the next post take a QM viewpoint. You may be aware that the laser light traveling in the interferometer ( IFO ) arms hits mirrors at either end and in doing so the photons circulate for many round trips before exiting, or absorption etc. Those photons absorbed by the mirrors tend to heat them up ie. add to the energies of lattice vibrations within the mirror's substance. This changes the shape of the mirror and thus it's focusing behaviour. That is very important. Each mirror is manufactured to a particular shape knowing that will change with heating during interferometer operation, so this heat/shape issue can be partly mitigated. This is called a 'thermal lens' ie. a temperature distribution dependent shape. However that is not enough because one can't predict the exact conditions of use.

Now there is a stage called the Power Recycling Cavity ( PRC ) sited between the main laser and before the photons enter the arm sections. Think of it like a holding bay or buffer that attempts to keep the main arms constantly supplied with photons. So the main laser 'pumps up' the PRC and the arms draw optical power off that ( in an energy flow sense ). Recall that photons possess both energy and momentum so their power is a very real thing to speak of. Of particular interest are the Input Test Mass mirrors, one for each arm, which are sited at the corner station. They are in vacuum and form surfaces for the photons to bounce against.

To assist the circulation of photons some of the mirrors have a Thermal Compensation System ( TCS ). Generally if alignment is good the main laser beam hits the centre of the mirrors. In doing so some energy in the beam is transferred to the mirror and it heats up. That is photons are absorbed and adds to the motion of the atoms in the mirror substance. This makes the centre of the mirror hotter than the rim and changes the shape of the mirror ever so slightly. This is a bad thing and it primarily affects the control* of the PRC. Here's a graphic from the LLO ( Livingston ) E-log demonstrating mirror distortion from this heating :

… the scale on the right indicates that the color coding represents nanometers. A nanometer is one billionth of a metre. We are looking at a sort of contour map of the distortion of the wavefront from 'pure'. Now as the laser is of 1024 nm wavelength, then some parts of the wavefront ( comparing centre to rim ) are out by up to ~ ¼ of a wavelength. Which is a quarter cycle ( = PI/2 or 90 degrees ) of phase error. This is too much. How to combat this?

The approach taken is to get another laser and shine it in onto the mirror at an arm's input. The photons from this laser can also be absorbed and heat the mirror substance too. Specifically if you shine this laser in an annular pattern it will heat the outer rim of the mirror. So combined with the main beam heating the centre, the whole mirror will get hotter, but doing so close to evenly and thus the 'good' shape is maintained. At least that is the intent.

Most of the TCS is outside of the vacuum space. There is a periscope to pipe the TCS laser photons in, that comes within ~ a metre of the input test masses ( ITM ie. at the corner station ). It doesn't interrupt the main beam of course but sits off to the side and shines on the mirror in a pattern determined by the optics on the outside of the vacuum. As mentioned, even though the ITM mirrors are designed to reflect suitably when 'hot', in practice the precise shape will be a 'moving target' and thus requires it's own active control system to adjust the TCS laser power as time goes by.

Now for one reason & another, revving a laser up and down in power has it's own problems. One better idea is to have a constant stable laser ( maybe adjusted occasionally ) and normally emitting photons at a nice ~ fixed rate, but use some other method to transmit varying power downstream. One way to vary the transmission of light is to use polarisers**. These allow some photons through but not others and can serve to 'throttle' the transmitted power. There are many other uses of polarised light but this example serves to concentrate our thinking.

Cheers, Mike.

* It would seem there is a modulation of the main laser beam before it goes in, to the effect that 'sidebands' are produced. These 'ripples' in the beam's intensity are picked up by the control devices for the PRC and serve to guide adjustments to the cavity to keep it on the 'sweet spot' for best operation. Pretty complex and I think I've missed something important here in my research ….. we'll see ! :-)

** This is only one component of a fairly complex system, and I'm describing some of the bits that I understand ( there's plenty that I don't ). For instance there is a heater placed around the rim of the mirror, by contact directly putting in heat energy. It seems there are inputs to help decide whether the TCS is responsible for some given mirror shape change, or maybe is it some other IFO component causing that? Not simple. The TCS has evolved – using lessons as obtained during IFO operations & testing over the years – to be a pretty sophisticated gadget now. Not described here is how the TCS senses the wavefront/mirror shape and thus suitably adjusts ( one needs a detailed model of how mirrors distort too ). Of course one can analyse with just classical optics, but I reckon thinking with QM is more funner! :-) :-)

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Alex
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@ Mike: Might be a little bit

@ Mike:
Might be a little bit out of topic, but: to get gravitational waves from any direction, isn't there a third arm, a Z-arm, necessary?

Mike Hewson
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Indeed you are right Alex !

Indeed you are right Alex ! :-)

Two arms form at most a plane and one can't disambiguate signal directions above and below that plane, plus other degeneracies. Specifically the signal strength formed by the differential arm delay becomes a photon flux count* at the 'dark' port. Gravitational waves have two polarisations, here noted as '+' and 'X', and there is much symmetry in the IFO response ( DARM ) with the direction of an incident wave :

This is why there is a network of IFO's worldwide. I guess that's rather easier than making a 4km z-arm which would need to be vertical for the current terrestrial arrangements.

The plan is that timing differences of the arrival of an identifiable signal are compared b/w the IFO's that detected such a signal. This adds to any directional clues obtained per IFO. By identifiable I mean there is a point(s) on the waveform of the signal that remains a common feature(s) across the detections from different IFO's ( as distinct from random fuzz ). Here's a graphic which very roughly outlines the strategy :

Naturally the more near simultaneous detections of a given wave from different IFO's, the more pairs of IFO's can be considered to assist in localising the source in the sky. There is considerable analogy here with the way GPS satellites contribute to increasing location precision when more signals are ( near ) simultaneously received.

Cheers, Mike.

* Or more accurately the IFO control signals required to keep the dark port dark are the proxy for the displacement that would have occurred in the absence of said corrections.

( edit ) A given IFO pair, on timing grounds alone, will define a cone to intercept the celestial sphere. The central axis of the cone is the extension of the line b/w the two IFO's ( the 'known fixed distance' as above ). An angle defined on the above diagram is the ( semi ) opening angle of the cone. That cone on intersection yields a lesser circle on the celestial sphere. Two distinct IFO pairs then give two circles which ought intersect in at most two points on the celestial sphere, if not there is a consistency problem with the detections to be resolved. Three pairs ought nail a unique point in right ascension and declination. However there will be errors and margins, so the lesser circles are in fact annuli and their intersections will be roughly quadrilateral areas on the celestial sphere. A final statement of location therefore includes those uncertainties .....

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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Thinking more about this

Thinking more about this differential timing aspect ( ie. notwithstanding any directional clues from single IFO's )....

#1 : I reckon the three distinct IFO pairs have to come from a set of four different IFO's. I'll have to check that, but I'm thinking that a set of three IFO's will give some redundancy in the solutions from the pairings. That could be useful as a consistency check however.

#2 : Suppose the solution circles on the sky did not intersect, or there was considerable unexplained closest angular separation of the annuli. Leaving for the moment the speed of light varying from expected ( ! ) then you would suspect :

- the timings and hence c * t distances were out. For this to prevent intersection of the circles then the error would have to go long ie. that which closes a cone's vertex angle.

- the inter IFO distances were in error.

... as the cone angle is deduced via their ratio ( cosine ). Interesting! :-)

Cheers, Mike.

( edit ) Correction : '..... ought intersect in at most two points .....' that being either one or two points.

( edit ) The 'known fixed distance' goes straight through the Earth's substance from one IFO to another. Here view the body of the Earth as mere scaffolding to hold the IFO's in a fixed relationship ....

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Alex
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Mike, thank you very much

Mike,

thank you very much for the explanation. Your first reply is (for me as a non scientist) clear enough to understand while your addition some hrs ago are a bit more abstract.

I would like to add another question: I took a look into the LISA technology package http://www.esa.int/spaceinimages/Images/2008/06/LISA_Technology_Package. A very nice graphic, but the sensors there seem to be two dimensional only (two identical test masses, each one a 46 mm cube composed of Gold:Platinum alloy), so what is the worth of this experiment?

Alexander

Edit: it could work with a rotating sensor, at least rotating in two directions. But how could a communication interlink be established then?

Mike Hewson
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Sorry for being obtuse Alex,

Sorry for being obtuse Alex, I often think out loud. :-)

#1 : Three non-co-linear points ( interferometer locations ) will define a plane. Any detected source will have the same timing delays as that from a distinct position mirrored across the plane. So I reckon you need a fourth IFO to resolve that :

#2 : You could get an inconsistent set of timings from an IFO grouping. That means the deduced angles to a sky source don't intersect, even if measurement margins are accounted for. So one looks back to the calculation that led to a particular angle ( range thereof ) being deduced :

From first principles : we did a cosine with that, so that's a ratio of lengths. So why might the lengths be out? Some rough ideas :

- we got the differential timing b/w interferometers wrong ( very likely )

- we weren't matching the correct point(s) on the waveform ( very likely too )

- we got the distance b/w interferometers wrong ( much less likely )

- the speed of gravitational waves is not the speed of light ( wow !! )

As for LISA : the original full project was for three widely separated craft in space. The funding for that got nobbled, however the project continues in a cut-down form for the meantime. The LISA Pathfinder is a single craft only, so it is really a proof-of-concept or machinery trial.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

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