Explaining the Heisenberg Uncertainty Principle and Electron Orbits

[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 Schrödinger equation)

http://www.kettering.edu/physics/drussell/Demos/Schrödinger/Schrödinger.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.

 

[Studiot]

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. This is analogous to the standing waves in the vocal tract when a speech sound has some ideal resonance representing an integral number of wavelengths from the glottis to the lips.

Not sure if this is a question or a statement?

The Schroedinger (wave) Equation of quantum mechanics is rather unfortunately named since it does not really describe a wave. It is considerably more than that.

 

[danR]

Not sure if this is a question or a statement?

Exactly?

 

[Studiot]

Sorry I still haven't got whatever it is.

 

[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.

 

[danR]

deleted. irrelevant

 

[danR]

Perhaps it will be simpler if we sweep aside all the preceding, and Studiot please give a brief description of the Electron field and whatever it is made of, and how the various topics of Heisenberg Uncertainty Principle, wave function, and the orbit of the electron around the nucleus, orbitals, etc. relate to each other.

 

[Studiot]

OK I think I understand the question, so forgive me if this is irrelevant.

Let's go back a bit in history.

When it was first mooted that the atom was not an indivisible 'lump' but had a structure of its own all sorts of models were proposed.

The first was the pudding model with all the elementary particles mixed up like a pudding.

This was quickly followed by the first planetary model with little negative balls (electrons) orbiting the positive nucleus (protons) in circular orbits.

At this stage they still thought in terms of little balls and I think they hadn't discovered the neutron, although they knew about the vast difference in mass between the proton and electron.

This model was quickly changed to elliptical orbits like the planets in the solar system.

There was, however, an irresolvable difficulty with this model. This was that known physics demanded that a charged ball, such as an electron was though to be, rotating round and round would radiate energy as electromagnetic waves, eventually losing energy and falling into the nucleus.

This was when the standing wave theory was conceived - to answer this difficulty and also because electrons had by then been shown to exhibit some wavelike properties.

The basic idea was that if the electron was not really a particle, but a (standing) wave that exactly 'fitted' the circumference of the orbit (whatever shape it was) then there would be no charge rotating round and round and no loss of energy.
This theory can be described by the normal 'wave equation' from mechanics, which appears in many places in physics.

Quite independently Schroedinger developed a much more complicated equation about energy (or momentum) which looks, superficially, like the wave equation but with some extra terms.

This was then called the Schroedinger Wave Equation (SWE) and the name has stuck.

The actual subject of the SWE is a mathematical construct, not directly equal to any foregoing physical quantity like energy or momentum etc and is called the wave function (Oh dear)
The square of this was later identified with the probability you mention associated with some region of space.
The plots probabilities of these in 3D are called orbitals.

So yes the (standing) waves in the wave equation are electrical field oscillations, the
orbitals are quite different as you surmised.

Hope this helps.

 

[stevmg]

When we talk about probability of finding an electron at a certain point we have to be clear what we mean. To wit - with a roulette wheel, the probability of landing on any arbitrary number is 1/38. Now, with any quantity that expresses itself as a real number, the probability of landing on anyone point is ZERO. The probability of landing on a range of points (an interval) is not zero but for a single point, ZERO.

Let us go from there. Let us get on the same page or same sheet of music. When we talk about probability of finding an electron at a given location are we talking about a specific point or an interval?

Studio T. I really enjoyed the old series with Martin Clunes "Doc Martin." The medicine as presented was pretty accurate (not 100% so) and my wife and I loved the scenes of Cornwall. I don't know if you mean Cornwall by your "south western England" statement.

 

[jjustinn]

When we talk about probability of finding an electron at a certain point we have to be clear what we mean. To wit - with a roulette wheel, the probability of landing on any arbitrary number is 1/38. Now, with any quantity that expresses itself as a real number, the probability of landing on anyone point is ZERO. The probability of landing on a range of points (an interval) is not zero but for a single point, ZERO.

To be a stickler, you can work it out to have a finite probability for anything to be at a single mathematical point -- the probability density is a 'certain value of infinity' at that one point (i.e. a Dirac delta function), so that when the density is integrated over any interval containing that point (including in the limit, that point itself), the probability of it being in that interval is finite / nonzero (1 if there's no scaling factor).

Of course it's not very...probable...that a physical system would get into such a situation (and thereby having an a totally uncertain momentum -- e.g. it's just as likely to be moving at 0.999c as it is to be standing still).

 

[stevmg]

To be a stickler, you can work it out to have a finite probability for anything to be at a single mathematical point -- the probability density is a 'certain value of infinity' at that one point (i.e. a Dirac delta function), so that when the density is integrated over any interval containing that point (including in the limit, that point itself), the probability of it being in that interval is finite / nonzero (1 if there's no scaling factor).

Of course it's not very...probable...that a physical system would get into such a situation (and thereby having an a totally uncertain momentum -- e.g. it's just as likely to be moving at 0.999c as it is to be standing still).

That "probability density" referred to in the above quote is what I mean by "likelihood." The standard normal curve or normal pdf (and there are also other probability density functions for continuous variables with the total area under the curve as "1") has an ordinate (y-axis) value for every x value from -infinity to +infinity. That quantity is NOT a finite probability. The finite probability at any point on the continuum is still zero. The likelihood is not zero but the "likelihood." They are related but not the same thing. Thus, by such reasoning, the probability of finding an electron at any point in the fuzzy ellipsoid orbit that surrounds the nucleus for that electron is zero. One must state an interval to obtain a finite probability.

This, above, what I have gotten into, is standard probability and statistics theory, not necessarily physics theory. My original question was "is there a finite probability of an electron being at a given point in the fuzzy ellipsoid orbit" and I wanted an answer to that. From what I have gleaned from the above answers, the answer is "no" and "probability" and "likelihood" are being conflated. My point is that these two measurements are related but are NOT the same thing.

 

[Studiot]

My original question was "is there a finite probability of an electron being at a given point in the fuzzy ellipsoid orbit" and I wanted an answer to that.

Hello Steve, I think your original wider question has prompted an interesting discussion.

Do you now wish to revamp this original?

 

[stevmg]

Studio T...

It appears that everything else including the kitchen sink has been brought into this, therefore I restated my original question as you noted. What I am getting at is that basic question - are we dealing with a fuzzy ellipsoid "orbit" with varying "likelihoods" (using strict definitions from probability and statistics which do not conflate "likelihood" with "probability") of occurrence of the electron at a given point at a given point in time? So, actually, the question, as stated, remains unanswered so far on this thread as the various contributors seem to have skirted that issue.

So, all I can do is restate the question (which is noted above.) The answer to that question will be a starting point for further questions but we have to get basics out of the way before we jump into more complexities.

 

[Studiot]

I'm sorry I don't recall skirting the issue. I did mention it in post42.
So far I have tried to keep this non mathematical but I realize that there is a good deal of statistics involved in some branches of medicine these days. My own daughter should be graduating medicine this year.

Medical statistics is usually about hypothesis testing, sampling and observational data.

Quantum statistics is different. I do not have time to post right now but will try to do something later today.

go well

 

[stevmg]

It's not medical versus quantum statistics.

The concept may be in the breakdown of "points" in the continuum (or actually, lack of continuum)

In 3D space one can conceive of an ellipsoid which is fuzzy and all the points are represented in it - in a sense a continuum.

Maybe with quantum measurements such points do not exist and that all the points in this ellipsoid are discrete and countable and not infinite in number, thus not a continuum. Then, probability of existing at any of the given points is represented by some calculable number, like the roulette wheel example.

Bottom line, if the the ellipsoid is continuus, then the probability of any point being "the one" is zero. Only if the ellipsoid is not continuus but made up of discrete separate points can probability of any given point have a value.

In other words, the ordinate of the normal curve equation is not the probability of the abscissa being the value.

Likelihood is related to probability but is not the same thing.

Steve G

 

[Studiot]

Steve, please don't try to jump the gun. I have been thinking about a suitable form of presentation and will post as promised.

Meanwhile try this.

Visualise a plucked string such as a guitar string.

We know, and can see, the standing wave pattern that is developed.

Now ask yourself this.

If I take a fine needle and poke into the region of vibration what is the variation of probability of hitting the string along the length of the string?

I will address this and also show you that for a quantum situation suprisingly the probability curve is the other way round.

Incidentally 'fuzzy' has nothing to do with quantum probability.

go well

 

[stevmg]

Studio T

Incidentally 'fuzzy' has nothing to do with quantum probability.

Absolutely! That I get.

You are doing fine and I am trying to steer you to how I am thinking so that you can bring it along to what you are trying to say.

 

[Studiot]

So here is the first installment.

In the top sketch I have drawn a 3D representation of our twanged string going away from you from A towards B.

You can see that it is vibrating (up and down ) in first harmonic mode, so there is a node (zero vertical displacement) at A, D and B and an antinode at C and again further along between D and B.

I have also shown our test needle at various points.

At A applied on x-axis at the level of the node.

At C above the x-axis (mean level of string) but within the vibration envelope.

Now since the infinitesimal (point) section of string at A is stationary it is always at A and there is 100% probability that we will hit the string without needle, whatever time we poke it.
A similar situation occurs at D.

But at C the string has furthest to go up and down so spends the least time at any single value (point) of level.
So a random poke with our needle will have to be very lucky to encounter the string ie the probability of encountering the string is a minumum at C.

In the second sketch I have just shown a 2D graph of probability along the x-axis of our needle hitting the string, between A and D

This can be seen to be a cup shaped curve with the minimum half way at C as expected.

This idea of the probability of encountering the string if we suddenly poke in a probe in an ordinary oscillator is equivalent to the idea of the probability of encountering or finding a particle at any particular point in a quantum oscillator, which is for the next installment.

And BTW I'm in Apple County, not Cornwall, sorry - Though I wouldn't mind Padstow.

 

[stevmg]

And BTW I'm in Apple County, not Cornwall, sorry - Though I wouldn't mind Padstow.

Beautiful!

Although I haven't got a clue what Apple County or Padstow are.

Steve G