Trying to understand electromagnetic waves in quantum theory

[atyy]

the way to teach is now part of the question, and i feel i have as much expertise here as anyone. i think the way to teach is to find progressive approximations to the truth. this is what humans use in languge that express (but simplify) feelings, myths that characterize (but simplify) culture, and even in science that describes (but simplifies) nature. if we took the answer "that's just how it is. period." would we even have started science in the first place? all this means that i believe there is some way to approximate a reasonably good answer to my question that is better than just saying don't ask. and by the way, if you're bothered by me asking, you really don't have to try to be the one to answer.

I am bothered by you asking, because it is fundamentally wrong to think that science is about asking "why".

i guess at this point i at least wonder if i could just accurately say "most scientists have no explanation for exactly how em waves are produced, in the form of waves, from electrons" - or is even this true? is this just the view of the few i have run into here, or is this really the majority view? maybe you guys could at least help me refine this statement, in case a student asks the quite natural question: why waves?

You are confused. There are two "waves". One is the EM wave and the other is the Schroedinger wave equation. The Schroedinger wave equation is at the moment fundamental and cannot be derived. That does not mean it will not be explained by a more fundamental theory in the future. But at present there is no observed phenomenon that is not described by the Schroedinger equation.

As for EM waves, at the classical level, the EM wave can arises from 3 things
(1) a charge produces an electric field
(2) a changing electric field produces a changing magnetic field
(3) a changing magnetic field produces a changing electric field.

But if you want to ask why (1), (2) and (3) are true, you are going down the wrong track, because science is not fundamentally about "why".

 

[bhobba]

I am bothered by you asking, because it is fundamentally wrong to think that science is about asking "why".

Andrew Gleason attended a lecture by Mackey that asked why the Born rule. That lit a fire in him and he came up with an answer. Was Mackey wrong to ask that? Was Gleason wrong to look for the answer? Was it not an advance when he found it?

There is an interesting book - Conceptual Developments of 20th Century Field Theories:
https://www.amazon.com/dp/0521634202/?tag=pfamazon01-20

Its thesis is progress is made in science by asking the right question. Time and time again things were stalled until the right question was asked. Many times that question was - why? - the correct why - but why nonetheless.

Thanks
Bill

 

[zonde]

but i think you may be downplaying the implication of him saying it was incomplete - i believe he meant that determinism could still be discovered when qm was completed.

I think you are right about this. Here is link:
https://www.marxists.org/reference/subject/philosophy/works/ge/einstein.htm
"If it should be possible to move forward to a complete description, it is likely that the laws would represent relations among all the conceptual elements of this description which, per se, have nothing to do with statistics."

 

[atyy]

Andrew Gleason attended a lecture by Mackey that asked why the Born rule. That lit a fire in him and he came up with an answer. Was Mackey wrong to ask that? Was Gleason wrong to look for the answer? Was it not an advance when he found it?

There is an interesting book - Conceptual Developments of 20th Century Field Theories:
https://www.amazon.com/dp/0521634202/?tag=pfamazon01-20

Its thesis is progress is made in science by asking the right question. Time and time again things were stalled until the right question was asked. Many times that question was - why? - the correct why - but why nonetheless.

The reason "why" is not a fundamental question is that if one has the fundamental theory, one will state it in the form of axioms. There will probably be several different sets of axioms that produce the same physical predictions. Choosing between the axioms is only a matter of taste, since they are all part of one coherent framework. This is why it is fundamentally in error to ask why.

Gleason's theorem does not answer a fundamental "why" question. It only shifts the why to "why noncontextuality?"

 

[zonde]

but wasn't einstein's position refuted (in a way that i don't entirely understand) by the Bell inequalities experiemnet

Local realism is not refuted by current experiment. However experiments have come very close to that border and from different sides so that there are a lot of people that consider "safe" to say that local realism is refuted by experiments.

 

[bhobba]

There will probably be several different sets of axioms that produce the same physical predictions.

But the why question can also produce a paradigm shift outside a systems axioms. Why did thermodynamics applied to the black-body problem give physical nonsence?

Thanks
Bill

 

[atyy]

But the why question can also produce a paradigm shift outside a systems axioms. Why did thermodynamics applied to the black-body problem give physical nonsence?

Yes, if the theory is giving nonsensical answers, or an answer that does not match observation, then there is good motivation to look for a new theory.

The current searches beyond the standard model to solve the measurement problem or the problem of quantum gravity are of that form, where it seems our laws have a cut.

My main problem with the OP is that the "why" is misplaced. First the OP should understand what the current laws of physics are.

 

[jeffery_winkle]

I'm afraid you are really confusing a lot of different concepts with each other. Yes, it is true that any particle, such as a photon or electron can be either viewed as a either a "particle" or a "wave". Yes, it is true that in quantum mechanics, we speak of a "wavefunction" which is related to the probability of detecting a particle. However, those two uses of the word "wave" are not the same thing! Also, another person here mentioned wave packets, which is very misleading. I suspect this was someone trying to visualize something being both a wave and particle at the same time, and they guessed, incorrectly, that a wave packet was, in their mind, "a particle containing waves". That has nothing whatsoever to do with the idea of a particle being both a particle and wave. A better way to think is that, a particle, such a photon or an electron, is a subatomic entity, unlike anything in the macroscopic environment, which under some circumstances is best approximated using the same mathematics that we also use to describe particles, and under different circumstances, is best approximated by the same mathematics that we use to describe waves.When you view it as a particle, there is a probability of detecting the particle at a given point in space and time. It has to do with operators. You could say that it doesn't really exist as what you think of as a particle, at a specific location, unless you are trying to see where it is. Also, you have to understand, that when viewed as a wave, unlike water waves or sound waves, there is no material that it is propagating through, so there is, in fact, nothing "waving".

 

[jeffery_winkle]

The following quote appeared in this discussion thread.

"First and foremost, photons aren't little particles like you're probably imagining. They are packets of energy."

The first sentence is undeniably true but the second sentence is utterly wrong. In fact, it's gibberish. You're not the only person to have said this, in what is supposed to be a scientific context. Just recently I read the latest issue of the Planetary Report, and in the part where they were talking about the LightSail, the author unfortunately said, "Photons are made of energy".

Nothing is "made of energy". In a scientific context, energy is an ability to do work. You could think of it as analogous to money which is an ability to buy things.

Look at the following equations from classical mechanics.

p = mv

F = ma

K = 1/2 mv^2

E = K + U

Yet, non-scientists seem to think that "energy" is some sort of material or substance. In science fiction, they might refer to an "energy being" or an "energy barrier". What's that? A being made of energy? A barrier made of energy? In Eastern mysticism, especially Western popularizations of Eastern Mysticism, such as martial arts, king fu, karate, judo, tia chi, or message, meditation, acupuncture, feng shu, or other New Agey things, they constantly use the word "energy", to refer to something that flowing through you, or whatever.

And then unfortunately, people bring this into a scientific conversation, claiming that photons are "packets of energy" or superstrings are "made of energy".

This is wrong. A photon is not a "packet of energy". Energy is not a material.

 

[Drakkith]

The first sentence is undeniably true but the second sentence is utterly wrong. In fact, it's gibberish. You're not the only person to have said this, in what is supposed to be a scientific context. Just recently I read the latest issue of the Planetary Report, and in the part where they were talking about the LightSail, the author unfortunately said, "Photons are made of energy".

Unfortunately I don't know of a better way of explaining to someone that a photon is not a little particle like they are imaging. I could tell them this, but then I need to tell them what it actually is. If you've got something easier for the average person to understand, please say so.

A photon is not a "packet of energy". Energy is not a material.

My other explanation is that a photon is the quantized interaction of an EM wave. Is that any better?

 

[bhobba]

Nothing is "made of energy". In a scientific context, energy is an ability to do work.

That is not the modern view of energy as the conserved Noether charge from time translational invariance:
http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

It's also the reason things like fields that may seem just mathematical fictions to help in calculations have both momentum and energy which are usually associated with physical objects like particles. Along with certain no go theorems worked out by Wigner its the reason fields are considered 'real' and in that sense are made of energy.

From that we get Einstein's famous equation E=MC^2 that shows mass is indeed energy and its not too far from the truth to say particles are made of energy. Note it doesn't show the converse.

That photons are packets of energy isn't too far from what Quantum Field Theory tells us.

Thanks
Bill

 

[morrobay]

For a specific example here how is refraction described in QM terms ?

 

[bhobba]

For a specific example here how is refraction described in QM terms ?

If that interests you this is the book to get:
https://www.amazon.com/dp/0691125759/?tag=pfamazon01-20

There are also video lectures:
http://www.vega.org.uk/video/subseries/8

Note, as you have probably guessed by now, as you go from popularisations, to beginning texts, to intermediate texts, then to advanced texts, what you are told at the start sometimes isn't true eg the myths I posted at the start. This is no exception, but we all have to start somewhere.

Thanks
Bill

 

[zonde]

The reason "why" is not a fundamental question is that if one has the fundamental theory, one will state it in the form of axioms. There will probably be several different sets of axioms that produce the same physical predictions. Choosing between the axioms is only a matter of taste, since they are all part of one coherent framework. This is why it is fundamentally in error to ask why.

Atty, are you sure about this? That fundamental [physics] theory is stated in the form of axioms? I think you are talking about math theory not physics theory.
In my opinion fundamental physics theory has to introduce new physical concept, say like "quanta". And then using this concept make things easy, consistent, interconnected in different ways (whatever that can be considered as an explanation).

 

[bhobba]

Atty, are you sure about this? That fundamental [physics] theory is stated in the form of axioms? I think you are talking about math theory not physics theory.

I don't agree with Atty on the why thing, but on that point he is correct. It's likely you just haven't seen the more advanced treatments that do this.

In my opinion fundamental physics theory has to introduce new physical concept, say like "quanta". And then using this concept make things easy, consistent, interconnected in different ways (whatever that can be considered as an explanation).

The physics of an axiomatic treatment lies in the axioms specifying how it maps to actual situations. For example in the Kolmogorov axioms of probability theory it speaks about events. Its application requires mapping that to things like the face of a dice when it is thrown. Although its not a book on physics, Feller explains the situation in the early chapters of his classic on probability:
https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

Thanks
Bill

 

[zonde]

The physics of an axiomatic treatment lies in the axioms specifying how it maps to actual situations. For example in the Kolmogorov axioms of probability theory it speaks about events. Its application requires mapping that to things like the face of a dice when it is thrown.

In axiomatic system mapping to physical reality is done trough undefined terms i.e. you give description for undefined terms.
In your example "event" is undefined term and it is described as face of the dice.

Besides I woud like to emphasize that I was speaking about fundamental theory of physics (as opposed to phenomenological theory).

 

[bhobba]

In axiomatic system mapping to physical reality is done trough undefined terms i.e. you give description for undefined terms. In your example "event" is undefined term and it is described as face of the dice.

An event is not undefined any more than a line or point in Euclidean geometry is undefined. To start with you simply use what a dictionary says and its exact meaning develops with experience in applying it.

Besides I woud like to emphasize that I was speaking about fundamental theory of physics (as opposed to phenomenological theory).

I have zero idea how to tell the difference. For example if Primary State Diffusion is correct then QM is simply phenomenological, but if Copenhagen is correct its fundamental.

Thanks
Bill

 

[zonde]

An event is not undefined any more than a line or point in Euclidean geometry is undefined. To start with you simply use what a dictionary says and its exact meaning develops with experience in applying it.

Look up this page http://en.wikipedia.org/wiki/Primitive_notion

 

[bhobba]

Look up this page http://en.wikipedia.org/wiki/Primitive_notion

You are thinking in terms of pure not applied math/physics.

The axiomatic systems of those areas use things that are part of the models they define - things like event, points, point particle, observation etc etc. Its part of what makes it applied rather than pure math.

An example would be Euclid's axioms and Hilbert's axioms of Euclidean Geometry.

Look them up - its easy to see the difference

Euclid's Axioms:
https://plus.maths.org/content/maths-minute-euclids-axioms

Hilbert's Axioms:
http://en.wikipedia.org/wiki/Hilbert's_axioms

In QM the two axioms in Ballentine are:

1. Given a Hermitian Operator O its eigenvalues are the possible outcomes of an observation.

2. There exists a positive operator of unit trace P such that the expected outcome of observation O is trace (PO). P is defined as the state of the system.

That is axioms like those defined by Euclid. Here observation is the key term and we all have a fairly good idea from normal usage what it means, but like probability it develops further with more experience.

If you get a hold of Varadarajan - Geometry of Quantum Theory you will find something along the lines of Hilbert where everything is put in terms of undefined terms. For example - from page 12 - An observable is defined as a homomorphism.

Thanks
Bill

 

[zonde]

You are thinking in terms of pure not applied math/physics.

...

I understand what you say (I think) but my wish for consistent use of therms does not seem make this discussion very meaningful.

 

[monesh]

hello everyone! so, i asked the original question here and i may have since found part of the answer on my own: what makes the em radiation emit in waves is SIMPLY the oscillation of the electron FROM THE NUCLEUS, in its flexible em bond that bounces it like a weight on a spring when disturbed. The disturbance emits the em radiation, the bouncing makes it emit in waves. This was discovered by John William Strutt, aka Lord Rayleigh, in 1904. MAYBE combining this with Uncertainy means that the waves that are emitted from probable electron positions create probable photon positions, or probability waves. This second part is my own idea and is most probably wrong, but at least it puts me in the ball park, conceptually of what to analyze and improve on next. I must say, looking back on your initial answers, you were too quick to overcomplicate the question, not really listen to the actual question, send me on a wild goose chase of unrelated reading, mystify any possible answer as a matter of esoteric math or unreachable philosophy (all the nonsense about never asking why in science), which, in summary, is exactly what impedes the growth of scientific literacy with the general public or even with other disciplines (my background is in philosophy, although I became a middle school teacher). but, on the other hand, upon returning and seeing that you continued talking about the subject at length, adding more voices, and even debating with each other, well that was pretty inspiring. that seems more like the best part of science. but anyway, back to my original question - so, what do you think of my (possible) answer?

 

[Drakkith]

- so, what do you think of my (possible) answer?

It is absolutely wrong. An example that immediately comes to mind of why is the fact that an electron accelerated in a static electric field will produce EM waves despite the fact that there is no nucleus and nothing is oscillating. So no, the EM radiation is not emitted in waves because the electron is bouncing around like it is on a little spring.

(Edited for clarification)

This second part is my own idea and is most probably wrong, but at least it puts me in the ball park, conceptually of what to analyze and improve on next. I must say, looking back on your initial answers, you were too quick to overcomplicate the question, not really listen to the actual question, send me on a wild goose chase of unrelated reading, mystify any possible answer as a matter of esoteric math or unreachable philosophy (all the nonsense about never asking why in science), which, in summary, is exactly what impedes the growth of scientific literacy with the general public or even with other disciplines (my background is in philosophy, although I became a middle school teacher).

On the contrary, the answers that were provided to you were exactly what you wanted and asked for. It's far more likely you just don't have any idea what you're talking about and your criticism is both confusing and insulting to those of us who tried to help you.

By the way, here is your original question:

What is a wavelength? I mean, if electrons and photons can be described as moving in waves, and if the frequency of photon waves determine whether radio/visible light/x-rays/etc. are emitted, but if quantum theory says that these waves are actually just waves of probability, why are some somes parts of the wave more probable and others less probable for the appearance of a photon or an electron, and why do different probabilities turn into x-rays while others into color, etc?

Your above answer doesn't even address ANY of your original questions.

 

[monesh]

let's say I'm wrong. why are you so shrill and hostile about it - you are the one being insulting to me. it's ok to be wrong in science you know. and don't you think you're being too dismissive and hasty? furthermore, i think you're actually wrong, or how else do you explain this: "Rayleigh also knew that an atom is made up of a positively-charged nucleus surrounded by negatively-charged electrons. (As we know from Bohr, this is essentially correct.) What would happen if you were to somehow pull one of these electrons away from the nucleus? Assuming you didn’t pull too hard before you let go, the nucleus would pull the electron back in, and the electron would oscillate around the nucleus like a mass on a spring...Of course, the electron was already (to a good approximation) orbiting in a circle around the nucleus, and it doesn’t stop orbiting after we perturb it. But because it keeps overcorrecting for the perturbation, the electron yo-yos back and forth between two elliptical orbits...Rayleigh’s insight was that a propagating electromagnetic field–that is to say, light–pushes and pulls at the electron in the exact way necessary to make it wobble back and forth. Of course, there’s a price to pay. Wobbling the electron costs energy, which is taken out of the electromagnetic field, causing the incoming light to be absorbed by the atom and disappear." Now, can you give me a calm, relevant, scientific response without being hostile, saying I'm "absolutely" wrong, or kicking me out?

 

[zonde]

let's say I'm wrong. why are you so shrill and hostile about it - you are the one being insulting to me. it's ok to be wrong in science you know. and don't you think you're being too dismissive and hasty? furthermore, i think you're actually wrong, or how else do you explain this: "Rayleigh also knew that an atom is made up of a positively-charged nucleus surrounded by negatively-charged electrons. (As we know from Bohr, this is essentially correct.) What would happen if you were to somehow pull one of these electrons away from the nucleus? Assuming you didn’t pull too hard before you let go, the nucleus would pull the electron back in, and the electron would oscillate around the nucleus like a mass on a spring...Of course, the electron was already (to a good approximation) orbiting in a circle around the nucleus, and it doesn’t stop orbiting after we perturb it. But because it keeps overcorrecting for the perturbation, the electron yo-yos back and forth between two elliptical orbits...Rayleigh’s insight was that a propagating electromagnetic field–that is to say, light–pushes and pulls at the electron in the exact way necessary to make it wobble back and forth. Of course, there’s a price to pay. Wobbling the electron costs energy, which is taken out of the electromagnetic field, causing the incoming light to be absorbed by the atom and disappear."

It's ok to be wrong in science but it's not ok to be ignorant and inconsistent.

 

[Drakkith]

Now, can you give me a calm, relevant, scientific response without being hostile, saying I'm "absolutely" wrong, or kicking me out?

Sure. It ignores everything about quantum physics and isn't even consistent with classical EM theory. Your answer addresses the question "why is EM radiation emitted in waves", which was answered in the 1800's by James Clerk Maxwell and is covered by his 4 equations dealing with electromagnetic waves. That's why.

 

[Delta2]

I think the OP has a justified confusion about the probability waves that correspond to solutions of Schrodinger's equation in QM that describes the time evolution of the wave function of a particle, and the EM-waves that correspond to solutions of the classical four Maxwell's equations that describe the time evolution of the EM field.

Those two kind of waves have nothing in common, except that they might concern the same particle, that is an electron. The EM-waves generated by an accelerating electron, have nothing to do with the probability waves that correspond to the wave function of the electron, when the electron is free, electron inside a box, or electron under the coulomb potential of a nucleus (different wave functions for each case).

The EM-wave is real (as much as the EM field is considered to be real), however the probability waves are not that real (though this is a debate). Probability waves and the wave function are just a mathematical tool that makes the wonder of QM to happen. Though in a threat i had posted long ago, i seemed to get answers by some sci-advisors that probability waves may correspond to waves in the fermionic field of a particle.

 

[atyy]

Just to add a bit to Delta²'s point that there are two different concepts:
(1) the wave function from which probability can be calculated, and
(2) the electromagnetic field configuration which is a classical concept that can be taken into quantum theory.

One way to see that there are two waves is to first consider the ordinary quantum mechanics of one particle. There we have position $x$, and the wave function $\psi(x)$ allows us to calculate the probability of observing a particle at each position.

When we go to the electromagnetic field, the analogue of position is the electromagnetic field configuration $A(x)$, and the analogue of the wave function is the wave functional $\psi(A(x))$.

 

[zonde]

I agree with Delta^2 point. I had the same feeling that OP is mixing up wave of probability amplitude and EM-wave.

But certainly it's easy to mix them up if you look at photons. Probability wave and EM-wave have the same wavelength for given beam of light (that means they have synchronous phase). And they both propagate at c.

 

[atyy]

But certainly it's easy to mix them up if you look at photons. Probability wave and EM-wave have the same wavelength for given beam of light (that means they have synchronous phase). And they both propagate at c.

Technically, the wave function doesn't travel at the speed of light, because it doesn't travel any "where", as it is a wave in Hilbert space, not a wave in ordinary space.

However, for single photons, it does make approximate sense to talk about the wave function of a photon traveling at the speed of light (search for "photon wave function").

 

[thierrykauf]

oh great - i got a response - thank you so much! and ok, i can picture a wave packet. but i guess my question is why is anything' waving'? before i read quantum theory, i thought there was something physically oscillating, like an electron, that emitted a line , like a line of photons, in a wave - like water from a garden hose being moved up and down. but in quantum theory, the wave is said to be a probability wave of where the electron or photon is most likely to be. ok, but why are probability paths still shaped like a wave? what is 'oscillating' the probability?

Something is waving because if it didn;t. scattering experiments would not yield what we observe, and atom's stability would receive no explanation. The wavefunction wasnt supposed to be a wave, and it is not . Only it;s complex conjugate squared is observable. The wavefunction, exponential of the action in hbar units was arrived out after the classical action angle formalism had emerged out of the study of classical equations of motion. The understanding that a particle is a wave-packet and is thus as localized as a wave. that is not at all if its momentum or wavelength is known, and that being a wave it simultaneously ges through every possible path and interferes with itself leaving behind the contribution of all the paths which interfered constructively, which were paths of sinilar action, i.e. paths for which the action varied little, that is was extremal, explained the notion of trajectory as a classical large hbar description. The wavefunction beig complex, cannot be an ordinary wave, for waves are real, and obey a second order in time.equation, which the Schrodinger equation is not.

 

[MacroMeasurer]

hmmm...well, thank you again for your time. but let me ask this: is my garden hose and water stream wave a good analogy of how we thought em waves were being emitted from oscillating electrons BEFORE quantum theory came along? or would it have been a bad analogy even back then?

If we "leave out QM," and you were asking about radio waves versus visible light,
maybe you are asking about the classical relationship between frequency and energy?

I think it might be more helpful to use a rope in that thought experiment
(and potentially even classroom demonstration).
You can make a wave by lifting and lowering one end of a rope, right?
You can double the frequency by raising and lowering your end of the rope twice as fast.
Meanwhile, you will realize you are working twice as hard waving the rope up and down.
That's why frequency is linked to energy: high frequency waves are an embodiment of more work.

It's more obvious that the wave in a rope comes from work you yourself are doing than with a hose,
where the water pressure seems to be doing a lot of work and you seem to just stand there.
Also, it would make for a much less messy classroom demonstration.

 

[meopemuk]

But certainly it's easy to mix them up if you look at photons. Probability wave and EM-wave have the same wavelength for given beam of light (that means they have synchronous phase). And they both propagate at c.

When you do the double-slit experiment with photons (or light waves), which one of the two waves is doing the interference? Is it the photon wavefunction, which interferes with itself on the two slits? or it is the electomagnetic wave, which does the trick? or they both do the interference simultaneously?

Eugene.

 

[bhobba]

When you do the double-slit experiment with photons (or light waves), which one of the two waves is doing the interference?

I may have posted it before, but here is the correct explanation of the double slit:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Its got nothing to do with waves.

The wave-particle duality is a myth - as explained in this forums FAQ:
https://www.physicsforums.com/threads/is-light-a-wave-or-a-particle.511178/

Thanks
Bill

 

[meopemuk]

Hi Bill,

So, your explanation says that the light interference picture is a quantum effect related to photon wavefunctions. And that light is just a flow of particles - photons - obeying the Rules of Quantum Mechanics. By the way, here I agree with you. Are you then willing to go further and say that description of light by means of classical "electromagnetic waves" is redundant and doesn't add anything to our understanding of this phenomenon?

Eugene.

 

[bhobba]

So, your explanation says that the light interference picture is a quantum effect related to photon wavefunctions. And that light is just a flow of particles - photons - obeying the Rules of Quantum Mechanics. By the way, here I agree with you. Are you then willing to go further and say that description of light by means of classical "electromagnetic waves" is redundant and doesn't add anything to our understanding of this phenomenon?

No - and obviously so. Classically it's described by EM fields as per Maxwell's equations. In QM its described by a quantum state. They are entirely different concepts.

Note: The explanation I gave of the double slit is problematical when applied to photons because the QM description of photons is itself problematical - one must go to Quantum Field Theory. When discussing the double slit best to stick to electrons.

But if you are willing to delve into that level of 'hurt' here are the details:
http://arxiv.org/pdf/1103.0100v1.pdf

At the beginning level best to stick with the link I originally gave and just accept it has issues with photons. Physics is unfortunately sometimes like that. Although often used at the start, the wave particle duality is wrong, and the first link I gave is a much better explanation - but even it isn't correct - the real answer being quite advanced and 'hard'.

Thanks
Bill