Explaining the Heisenberg Uncertainty Principle and Electron Orbits

[stevmg]

It is stated that electrons orbit the nuclei of atoms not as particles. By the Heisenberg Uncertainty Principle (whatever that is) one cannot pinpoint their actual location and one cannot track the motion of an electron as it orbits the nucleus.

What is that all about? Please use 10th grade English or College freshman English to explain.

 

[mathman]

The simplest answer is that "orbits" are not an accurate description of electrons in atoms. The only accurate description is in terms of quantum states, without trying to specify location.

 

[Naty1]

Electrons imagined as oribiting point particles around a central nucleus is NOT an accurate atomic model: such a model works for the large scale, like planets, but NOT for subatomic particles. In fact if such a model is used, the electrons would immediatly collapse into the nucleus. Quantum theory, of which Heisenberg uncertainty is a component, is required to explain electrons more as a cloud enveloping a nucleus, a spread out phenomena, rather than a point like particle.

More here:
http://en.wikipedia.org/wiki/Atomic_orbital#Introduction


And here is some info on Heisenberg uncertainty:
http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle

 

[Antiphon]

What is that all about? Please use 10th grade English or College freshman English to explain.

In the atomic realm (very tiny things) objects usually act as though they are waves and not little solid balls. This means an electron more closely resembles ripples in a bucket of water. They are all over the area of the bucket and you can't point to anyone place and say that's where the thing actually is.

 

[alxm]

https://www.physicsforums.com/showthread.php?t=510001" on essentially the same subject is only a few days old (and if you use the search feature, it's been asked many times before).

 

[stevmg]

Electrons imagined as orbiting point particles around a central nucleus is NOT an accurate atomic model: such a model works for the large scale, like planets, but NOT for subatomic particles. In fact if such a model is used, the electrons would immediately collapse into the nucleus. Quantum theory, of which Heisenberg uncertainty is a component, is required to explain electrons more as a cloud enveloping a nucleus, a spread out phenomena, rather than a point like particle.

More here:
http://en.wikipedia.org/wiki/Atomic_orbital#Introduction


And here is some info on Heisenberg uncertainty:
http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle

So, it is stated that an electron is not a particle but a standing wave about a nucleus and the wavelength times a whole number is the circumference of the orbit.

So where is the probability factor? Since there is no particle there is no way to know where such a non existent particle is, just where the whole energy wave is as a whole.

Now is the energy wave up and down from the orbital? (Up meaning away from the nucleus and down meaning towards the nucleus)?

Now, interestingly enough, when we describe in probability and statistics a probability density function, such as the normal curve, there is NO probability of finding a particular point on the x-axis but the probability of being within x = x' $\pm$ $\epsilon$ is the area of the probability function between x = x' - $\epsilon$ and x = x' + $\epsilon$. I see the parallel but don't get the connection.

 

[Naty1]

So where is the probability factor?

Did you read here:

http://en.wikipedia.org/wiki/Atomic_orbital#The_shapes_of_orbitals

That's as good a simple description as you'll likely find.

ok here is another not too complicated:

"The Schrödinger equation details the behaviour of ψ but says nothing of its nature. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful.[12] In 1926, just a few days after Schrödinger's fourth and final paper was published, Max Born successfully interpreted ψ as a quantity related to the probability amplitude, which is equal to the squared magnitude of ψ.[13] Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory— and never reconciled with the Copenhagen interpretation.[14]"

from Wikipedia, "Schrodinger wave equation"

 

[danR]

It is stated that electrons orbit the nuclei of atoms not as particles. By the Heisenberg Uncertainty Principle (whatever that is) one cannot pinpoint their actual location and one cannot track the motion of an electron as it orbits the nucleus.

What is that all about? Please use 10th grade English or College freshman English to explain.

They orbit the nuclei when they are in hyperexcited states. Then they go round and round the nucleus like little planets, except the force keeping them there is electromagnetic, not gravitational.

By 'hyperexcited' I mean they have a lot of energy compared to normal, and that let's them revolve around the nucleus very far away.

But when they are very close, their wave-length (properties) are around the same as the scale of the the close-up region. Now the electrons are behaving more and more like they are waves, and there is much uncertainty about just where a wave's 'position' is.

 

[stevmg]

This is what I am getting to... everything seems to "beg the question." The electron is BOTH wave and particle simultaneously. Now, as a wave, it is a standing wave in circular (elliptical) format that goes around the nucleus. As a wave, it has a wave length and the wave length is a natural number divided into the circumference of the orbit (so to speak). As a particle, it just orbits the nucleus in an ellipse.

Einstein believed everything was deterministic. The only "probability" was that we, as yet, did not have the tools or the mathematical equations to pinpoint the position/momentum of the electron as a particle. As a wave, there is no specific point where it is because its a wave.

Please comment on what I italicized above.

 

[danR]

This is what I am getting to... everything seems to "beg the question." The electron is BOTH wave and particle simultaneously. Now, as a wave, it is a standing wave in circular (elliptical) format that goes around the nucleus. As a wave, it has a wave length and the wave length is a natural number divided into the circumference of the orbit (so to speak). As a particle, it just orbits the nucleus in an ellipse.

Einstein believed everything was deterministic. The only "probability" was that we, as yet, did not have the tools or the mathematical equations to pinpoint the position/momentum of the electron as a particle. As a wave, there is no specific point where it is because its a wave.

Please comment on what I italicized above.

You've accidentally italicized all of it, but I think you're on the right track. You cannot pinpoint the position/momentum simultaneously because they are aspects of the same thing, and a tool good for measuring momentum is bad for measuring position, and particle-ness measuring devices are commensurately bad for measuring wavs. So when something unclassical like an electron comes along which seems to have wave and particle properties, then it's pretty hopeless.

Everytime a new paper comes out claiming to nail X, someone comes along and says they really nailed Y, and least as far as I can determine from the prose in the popular press.

 

[stevmg]

That assumes that the measuring of the position or the velocity (i.e., momentum) of an electron as it is orbiting impacts on the measurement of the other "parameter." In a theoretical sense, even though we cannot measure both at the same time because of the restrictions of the measuring devices, these two quantities (position and momentum) DO exist and "God" knows what they are.

However, if an electron existed as a standing wave with an integral number of wavelengths to circumnavigate its orbit, where it is as a particle at any point in time is not possible to discern because all points on this orbit are possibilities at the same time and each instant in time thereafter, no matter how close or how far from the original point has an equal likelihood of being where this electron as a mass point would be.

Note the use of "likelihood" as opposed to probability. In statistics a quintessential probability density function is the normal distribution curve with area of 1 under the curve with a continuous variable (call it "x") on the abscissa. The ordinate component is NOT the probability of finding "x" but the likelihood. The probability is the area between x₁ and x₂ divided by 1.

 

[danR]

I believe there is a school of thought that holds an agnostic view that the electron does not have any actual position until it is measured. God can't know what does not exist. 'measured', here, meaning any interaction that requires its position. There is a horrible philosophical debate over whether a conscious entity is entailed by 'measure', and it's way beyond me.

But you wanted grade 10 or 1st year prose anyway.

 

[stevmg]

I believe there is a school of thought that holds an agnostic view that the electron does not have any actual position until it is measured. God can't know what does not exist. 'measured', here, meaning any interaction that requires its position. There is a horrible philosophical debate over whether a conscious entity is entailed by 'measure', and it's way beyond me.

But you wanted grade 10 or 1st year prose anyway.

Hey, let's not get into that... "Does a tree falling in the woods with no one around make a noise?" It may or may not make a sound but it sure thinks to itself, "Aw, sh--!"

What I am stating is that the reason for the unpredictability of the position of an electron in the "electron cloud" about a nucleus is that there is no one point at which it exists. If the electron is a wave then any point of condensation is anywhere on the wave at the same time Make sense? This has nothing to do with the manner of measurement or the effect of measurement on the position/velocity of the electron.

This is 10th grade-freshman college level English which actually is far superior to technical jargon.

BVy the way, I am just taking the info given to me above in the earlier posts about the "standing wave," etc. and applying common sense to it.

 

[danR]

Hey, let's not get into that... "Does a tree falling in the woods with no one around make a noise?" It may or may not make a sound but it sure thinks to itself, "Aw, sh--!"

What I am stating is that the reason for the unpredictability of the position of an electron in the "electron cloud" about a nucleus is that there is no one point at which it exists. If the electron is a wave then any point of condensation is anywhere on the wave at the same time Make sense? This has nothing to do with the manner of measurement or the effect of measurement on the position/velocity of the electron.

This is 10th grade-freshman college level English which actually is far superior to technical jargon.

BVy the way, I am just taking the info given to me above in the earlier posts about the "standing wave," etc. and applying common sense to it.

If I say: 'if the electron is a wave...', then I'm presupposing my manner of measurement, and I'm unable to say anything about a point of condensation, or an 'anywhere' on the wave. I figure all I can do with wave (deliberately omitting a determiner that would imply a count-noun) is talk about its momentum.

 

[stevmg]

I am lost in the details. Don't bother going over it again.

Should I accept the Heisenberg Uncertainty Principle? Is it true? Can certain chemical reactions (Stanley Miller's 1950s experiments with abiotic production of prior only organic compounds made by living organisms later finally redone 60 years later after many failed attempts to reproduce the results) be an extrapolation of this?

 

[danR]

I am lost in the details. Don't bother going over it again.

Should I accept the Heisenberg Uncertainty Principle? Is it true? Can certain chemical reactions (Stanley Miller's 1950s experiments with abiotic production of prior only organic compounds made by living organisms later finally redone 60 years later after many failed attempts to reproduce the results) be an extrapolation of this?

Personally I take it on experience that every time they try to nail down both attributes of some quantum entity, someone says something like: Well they thought they were measuring X, but really they were measuring Y. Or a proxy-X, or an ensemble X, or type-X, not a token-X. That's how my reading of pop science of these things seems to go. So I'm pretty confident about Heisenberg.

It's not whether it's true or not, but whether it's falsifiable (yes), and has it been falsified. Not as far as I know. So after hundreds of tests, the confidence-level is absurdly high. But 'true'? Pretty true.

 

[stevmg]

Personally I take it on experience that every time they try to nail down both attributes of some quantum entity, someone says something like: Well they thought they were measuring X, but really they were measuring Y. Or a proxy-X, or an ensemble X, or type-X, not a token-X. That's how my reading of pop science of these things seems to go. So I'm pretty confident about Heisenberg.

It's not whether it's true or not, but whether it's falsifiable (yes), and has it been falsified. Not as far as I know. So after hundreds of tests, the confidence-level is absurdly high. But 'true'? Pretty true.

Now, to make things clear, one corollary of the HUP is that of predicting the location of an orbiting electron - which Heisenberg says you cannot do with absolute certainty. It also means that from one instant to the next, no matter how close these "instances" are, a particular electron can be anywhere else in the electron cloud around the nucleus.

I think my interpretation as just stated is correct (forget about momentum for now.) Hence, my reliance upon the "wave" theory of an electron as a wave has NO point on it which is an electron.

PS _ I really enjopy thios bantering back and foprth. Not many people I know persoanlly can even discuss this on any level so you are a welcome relief.

 

[danR]

Now, to make things clear, one corollary of the HUP is that of predicting the location of an orbiting electron - which Heisenberg says you cannot do with absolute certainty. It also means that from one instant to the next, no matter how close these "instances" are, a particular electron can be anywhere else in the electron cloud around the nucleus.

I think my interpretation as just stated is correct (forget about momentum for now.) Hence, my reliance upon the "wave" theory of an electron as a wave has NO point on it which is an electron.

PS _ I really enjopy thios bantering back and foprth. Not many people I know persoanlly can even discuss this on any level so you are a welcome relief.

I'm actually a physics idiot. My real field is linguistics. I'm sorry you're not getting a more precise discussion, because you seem to have a better handle on this topic than your opening statement implied.

You may also want to explore the deBroglie/Bohm model of the electron that holds the electron is a real point-particle with a real (but theoretically undeterminable ) position at all times, and its wave-function is more like an actual field, than just a mathematical construct for determining the mere probability of its position at any point. This old idea seems to be having a renaissance, with some smart arguments on both sides, but rather technical and over my head. Those discussions are actually pretty lively at times around PF, and you just have to look around for them. I have no bias one way or the other.

 

[stevmg]

I'm actually a physics idiot. My real field is linguistics. I'm sorry you're not getting a more precise discussion, because you seem to have a better handle on this topic than your opening statement implied.

You may also want to explore the deBroglie/Bohm model of the electron that holds the electron is a real point-particle with a real (but theoretically undeterminable ) position at all times, and its wave-function is more like an actual field, than just a mathematical construct for determining the mere probability of its position at any point. This old idea seems to be having a renaissance, with some smart arguments on both sides, but rather technical and over my head. Those discussions are actually pretty lively at times around PF, and you just have to look around for them. I have no bias one way or the other.

Not quite a physics idiot that you state - just a non-physicist who has an interest in these matters. If you think about it, if the electron were a point particle, then the probability of finding it at any arbitrary point on the orbiting spheroid or ellipsoid would be zero. As I have mentioned above using the normal distribution curve as an example, with the x-axis being the continuous variable, then the y-axis is the likelihood, not the probability, of such a value.

I am not a physicist either, just a medical doctor with an interest in these matters.

That's why the concept of a standing wave makes sense, because a wave has zero probability of a particular point but a total probability of one (1) and is continuous. It is everywhere simultaneously equally likely, or, at least a smooth relationship between the different points on the spheroid/ellipsoid.

There is a neat problem from topology that states there are at least two points on the globe in which the pair barometric pressure and atmospheric temperature are the same - due to continuity.

 

[danR]

Not quite a physics idiot that you state - just a non-physicist who has an interest in these matters. If you think about it, if the electron were a point particle, then the probability of finding it at any arbitrary point on the orbiting spheroid or ellipsoid would be zero. As I have mentioned above using the normal distribution curve as an example, with the x-axis being the continuous variable, then the y-axis is the likelihood, not the probability, of such a value.

I am not a physicist either, just a medical doctor with an interest in these matters.

That's why the concept of a standing wave makes sense, because a wave has zero probability of a particular point but a total probability of one (1) and is continuous. It is everywhere simultaneously equally likely, or, at least a smooth relationship between the different points on the spheroid/ellipsoid.

There is a neat problem from topology that states there are at least two points on the globe in which the pair barometric pressure and atmospheric temperature are the same - due to continuity.

I use stats terms loosely, and oddly the term 'likelihood' is absent from the major Wiki entries (though Google returns pages missing it anyway) on wave function.

I'm not sure it's safe to compare two scalar correlations with point and momentum correlations.

 

[Studiot]

I seem to be known here as an old fuddy duddy because I am always recommending - wait for it - shock, horror - books.

If you are prepared to go further after that confession then I comend to to two volumes that are couched in the terms you seek but discuss the question(s) you raise.

Both are a series of essays by past and present great thinkers.


Firstly

A Question of Physics

Conversations in Physics and Biology conducted by Paul Buckley and David Peat
with
Heisenberg, Dirac, Rosenfeld, Penrose, Wheeler,Weizacker, Prigogine, Rosen,Pattee, Somorjai and Bohm

Routledge

Secondly

On Space and Time

Essays by Connees, Heller, Majid, Penrose, Taylor, Polkinghorne

Edited by Majid

Cambridge University Press

You may recognise some (or even all) of these. The essay by Majid is particularly pertineent to this discussion as he explaind why (he thinks) it cannot be resolved in the terms currently presented.

Essentially, Majid (Professor of Maths at Queen Mary, London) is proposing the problem arises because we are trying to apply continuum maths to a quantised or discretised universe.

You will need to read the whole essay to fully appreciate his point.

go well

 

[stevmg]

I use stats terms loosely, and oddly the term 'likelihood' is absent from the major Wiki entries (though Google returns pages missing it anyway) on wave function.

I'm not sure it's safe to compare two scalar correlations with point and momentum correlations.

You don't sound like an ignoramus in the least!. I haven't even wrapped my brain around trhe momentum of a "wavicle"

I am still from the old school where momentum was mass * velocity

If you have a point in space and it has mass and is moving at a velocity (velocity is not scalar) then you have instantaneous momentum. This momentum is forever changing because of the circular or near circular path the point mass is following because the direction is changing. In a mechanical universe, this could go forever as the mere fact of a circular orbit does not bleed energy from the particle (energy is a scalar quantity). Now, with an electron, the circular motion does bleed energy off the electron - these are two different worlds and the energy will bleed off.

I haven't got a clue what or why this is so. That's why the concept of a standing wave is so appealing as I can picture that and being a doctor, I know from the physics of blood pressure in the lower extremities what a standing wave is. Also, by visualizing a standing wave, I clearly see that there is no point that defines the wave and the whole thing is the electron and there is no one point that defines the entire object. That's about as far as my poor brain can take it.

I may be wrong but that's my story and I am sticking to it.

 

[danR]

You don't sound like an ignoramus in the least!. I haven't even wrapped my brain around trhe momentum of a "wavicle"

I am still from the old school where momentum was mass * velocity

If you have a point in space and it has mass and is moving at a velocity (velocity is not scalar) then you have instantaneous momentum. This momentum is forever changing because of the circular or near circular path the point mass is following because the direction is changing. In a mechanical universe, this could go forever as the mere fact of a circular orbit does not bleed energy from the particle (energy is a scalar quantity). Now, with an electron, the circular motion does bleed energy off the electron - these are two different worlds and the energy will bleed off.

I haven't got a clue what or why this is so. That's why the concept of a standing wave is so appealing as I can picture that and being a doctor, I know from the physics of blood pressure in the lower extremities what a standing wave is. Also, by visualizing a standing wave, I clearly see that there is no point that defines the wave and the whole thing is the electron and there is no one point that defines the entire object. That's about as far as my poor brain can take it.

I may be wrong but that's my story and I am sticking to it.

I cannot see any sort of wave, moving or standing, as scalar, as its components will be vectored at some deeper level of analysis.

I understand the wave's energy component cannot bleed off (and the electron captured in the nucleus) because it requires a disruption of the orbital resonance over a potential hill. As the orbital resonances (almost by definition) are quantized, this cannot happen.

Remember that a macroscopic system of resonance, standing wave, vibration mode, or whatever we want to call it, the energy resides in an unquantized jumble of discrete particles, that can lose energy through matter escaping, phonon, kinetic energy, EM loss, any old way.

I see the electron bound to the nucleus with a quantized photon exchange particle(s) of the EM field whose energy is also quantized. One wavicle, one photon, one resonance, one potential hill, that cannot be scaled without an extra energy push from somewhere. Without some assist, the electron will sit in its potential hole until the universe dies as a sea of completely entropized radiation.

 

[stevmg]

I cannot see any sort of wave, moving or standing, as scalar, as its components will be vectored at some deeper level of analysis.

I understand the wave's energy component cannot bleed off (and the electron captured in the nucleus) because it requires a disruption of the orbital resonance over a potential hill. As the orbital resonances (almost by definition) are quantized, this cannot happen.

Remember that a macroscopic system of resonance, standing wave, vibration mode, or whatever we want to call it, the energy resides in an unquantized jumble of discrete particles, that can lose energy through matter escaping, phonon, kinetic energy, EM loss, any old way.

I see the electron bound to the nucleus with a quantized photon exchange particle(s) of the EM field whose energy is also quantized. One wavicle, one photon, one resonance, one potential hill, that cannot be scaled without an extra energy push from somewhere. Without some assist, the electron will sit in its potential hole until the universe dies as a sea of completely entropized radiation.

? Beyond me...

 

[danR]

? Beyond me...

My prose?

OK, a standing wave in a bathtub still has particles (molecules) that are describing circular motion. 'standing' is something of a misnomer, the wave is going nowhere, but something is still 'waving' around.

A standing wave of this sort has all manner of ways to bleed off energy, and usually does.

The electron has only itself. It can't bleed off anything except a complete, quantized, EM item. Otherwise there could be a continuum of orbital positions in an atom. That would be the equivalent of an infinite gradation in the resonance modes of a drumhead. That doesn't make sense to me. You have an integral number of nodes in any resonance pattern, and if a new resonance occurs, you change the pattern or number of nodes, but you can't have a fractional node. You either have one or you don't.

The atom, as I see it, throws off entire nodes, and uses photons to do so. That is my story of the thing, and I'm sticking with it.

 

[Studiot]

I'm sorry I intruded on what is clearly a private conversation.

 

[danR]

I'm sorry I intruded on what is clearly a private conversation.

Heck (I'm not allowed to say something else), this conversation cries out for more intrusion.

Regarding the books. I'm going blind and cannot read further than my present requirements.

 

[stevmg]

danR & \Studiot -

Hey, no problem. I just reached the limit of my poor brain's ability to understand. Everyone is welcome to these "conversations." Really.

 

[Studiot]

Essentially, Majid (Professor of Maths at Queen Mary, London) is proposing the problem arises because we are trying to apply continuum maths to a quantised or discretised universe.

I did state the essence of the explanation. Perhaps you missed it?

 

[stevmg]

I did state the essence of the explanation. Perhaps you missed it?

I haven't got a clue as to what this refers to. It is totally beyond me. I am "tapped out."

 

[Studiot]

I'm not sure what 'tapped out' means but I assume that since you have responded you are interested in expanding the discussion.

Should I accept the Heisenberg Uncertainty Principle? Is it true?

You referred to the uncertainty principle at the outset and subsequently. I will elaborate on the nature of a continuum later so just bear with the flow for a moment.

Now most of our mathematics is based on the ideas of continuity and a continuum. Most proofs rely on this underlying idea and to our everyday experience nature appears continuous. Most of the results in mathematics have only been proved to be valid in a continuum and many are only valid there. In particular the Schroedinger equation is only provably valid in a continuum.

So it is not suprising that physicists using mathematics to describe the physical world want to apply continuum mathematics.

There is today an emerging branch of mathematics called discrete mathematics or colourfully concrete mathematics, but it is yet in the cradle.

The ancient Greeks produced what are known as Zeno's paradoxes when they were in a similar situation applying their system of mathematics to a situation it was inadequate for.

Essentially Zeno's paradox are variations on this:
An arrow can never reach it target because before it can reach its target it must cover half the distance. Before it can cover the remaining distance it must cover half that distance and so on. No matter how close it gets there will always be half the remaining distance to cover.

Now we have a resolution of this in today's mathematics but have encountered new issues instead.

By a continuum I am talking about the idea of completeness.

Think about the integer numbers 1, 2, 3, 4... they go on to infinity.

But between any two in the list we both know there are more numbers, not on the list.

The first offering to fill this gap are called the rational numbers ie fractions 1/2, 3/4, etc

But then we find that there are yet more numbers that cannot be expressed as a fraction. Numbers such as $\pi$ etc.

Continuity
There is a proof, which I won't bore you with, that once we introduce all of these to our list, the list is complete. That is there are no more numbers to discover. If we draw a line to represent these numbers (we call this the real number line) it is continuous ie we draw it without taking our pencil off the paper. You cannot get from one number to the next without passing through other numbers and every number has a 'neighbour' or another number next to it. There is nothing else between numbers.

We say that the real line is continuous. Most of current mathematics is base on using this underlying logic.

Now there is a saying that "God gave us the integers, all else is the work of man".
But if the universe is not like this ie not continuous then we need new mathematics to deal with it, like the Greeks needed new mathematics to deal with Zeno.

 

[stevmg]

I'm not sure what 'tapped out' means but I assume that since you have responded you are interested in expanding the discussion.



You referred to the uncertainty principle at the outset and subsequently. I will elaborate on the nature of a continuum later so just bear with the flow for a moment.

Now most of our mathematics is based on the ideas of continuity and a continuum. Most proofs rely on this underlying idea and to our everyday experience nature appears continuous. Most of the results in mathematics have only been proved to be valid in a continuum and many are only valid there. In particular the Schroedinger equation is only provably valid in a continuum.

So it is not suprising that physicists using mathematics to describe the physical world want to apply continuum mathematics.

There is today an emerging branch of mathematics called discrete mathematics or colourfully concrete mathematics, but it is yet in the cradle.

The ancient Greeks produced what are known as Zeno's paradoxes when they were in a similar situation applying their system of mathematics to a situation it was inadequate for.

Essentially Zeno's paradox are variations on this:
An arrow can never reach it target because before it can reach its target it must cover half the distance. Before it can cover the remaining distance it must cover half that distance and so on. No matter how close it gets there will always be half the remaining distance to cover.

Now we have a resolution of this in today's mathematics but have encountered new issues instead.

By a continuum I am talking about the idea of completeness.

Think about the integer numbers 1, 2, 3, 4... they go on to infinity.

But between any two in the list we both know there are more numbers, not on the list.

The first offering to fill this gap are called the rational numbers ie fractions 1/2, 3/4, etc

But then we find that there are yet more numbers that cannot be expressed as a fraction. Numbers such as $\pi$ etc.

Continuity
There is a proof, which I won't bore you with, that once we introduce all of these to our list, the list is complete. That is there are no more numbers to discover. If we draw a line to represent these numbers (we call this the real number line) it is continuous ie we draw it without taking our pencil off the paper. You cannot get from one number to the next without passing through other numbers and every number has a 'neighbour' or another number next to it. There is nothing else between numbers.

We say that the real line is continuous. Most of current mathematics is base on using this underlying logic.

Now there is a saying that "God gave us the integers, all else is the work of man".
But if the universe is not like this ie not continuous then we need new mathematics to deal with it, like the Greeks needed new mathematics to deal with Zeno.

Absolutely beautiful! Thank you. I am very familiar with natural numbers, integers, fractions, and irrationals (including the transcendentals). I am also familiar with the Dedekind cut which divides the set of rationals into two sets, which are distinct, yet there being no defined number for, say, the lower bound (if we go that way.) Thus, if the two sets were "joined" one would return to a continuum. I am sure you are, too. I also understand how when applying math to integers it is oft necessary to truncate or round off (not the same thing) to obtain the integers.

I have never understood but could rotely go through the proof of "uncountably infinite set" as compared to the "countably infinite set." The proof seemed like smoke and mirrors to me.

I also never understood but could rotely go through the barometric pressure, temperature proof that on the globe, if one followed from any given point on a great circle and continued about it - any great circle, one could always find a second distinct point (with one exception) that has the same barometric pressure/temperature pair measurements. The exception, of course being if one, perchance, started at a maximal point or minimal point of pressure/temperature, this being due to the continuity of physical phenomena.

The interesting thing to note here is that a computer mimics the real world by use of a very, very large set of discrete numbers. The computer is not continuous. Likewise, it is conceivable that the universe is made up of countably infinite (therefore discrete) "elements" which appear continuous but are not. Countably infinite sets are sort of continuous, because no matter how small one makes an interval about a given value, there are elements of the set contained within the interval.

Am I on the same page as you?

"Tapped out" means no further stretching of the brain is available to me. Are you American or British because "tapped out" has a meaning here. I couldn't pick up any words like "colour" or the like to give me any clue. There are subtle differences between American English and non-American English which can cause some confusion.

P.S. - I love Zeno's paradox. Just shows that logic is man made but may have nothing to do with the "real world." At least that poor slob at the receiving end of the arrow would find that out pretty quick. Just like a court room where lawyers, or barristers or whatever make black white and white black.

 

[Studiot]

Studio T is my small company in the South West of England.

 

[jjustinn]

It is stated that electrons orbit the nuclei of atoms not as particles. By the Heisenberg Uncertainty Principle (whatever that is) one cannot pinpoint their actual location and one cannot track the motion of an electron as it orbits the nucleus.

What is that all about? Please use 10th grade English or College freshman English to explain.

Here's a great visual example of what particles look like in standard quantum mechanics (e.g. the Schrodinger equation)

http://www.kettering.edu/physics/drussell/Demos/schrodinger/schrodinger.html

Note that the 'darkness' / 'solidity' of the particles in these animations represent the *probability* of finding the particle at that point if you looked (and in principle, there's a vanishingly small probability that you could find it in the 'white' areas as well).

 

[danR]

[more pertinently] These are all assertions:

Perhaps I should back up a little: the wave function, and the 'standing' wave of the electron, are not exactly the same thing, and someone can clarify or correct where I may be off.

The former determines the likelihood of the electron's position, the latter is a/the node of the orbital determined by the oscillating electric and magnetic components.​

These assertions are essentially accurate, a first approximation, wrong, meaningless, reparable, thoroughly discreditable, or some other option.

Throughout the course of this post, and perhaps implicit in the wording of its title 'Electron Field', the OP may have conflated the two notions of wave function and of standing wave--the actual electromagnetic field of the electron. Perhaps he is essentially correct, perhaps they are aspects of the same thing. I don't know.