Why do electrons occupy quantized energy levels around nuclei?

[Cheman]

Can quantum theory be explained? Is it possible to look at quantum theory and say "this calculation works because in terms of physics..."? (even if the physics does not entirely correlate with classical ideas)

Eg - is it possible to explain in physics terms why electrons can only exist certain distances from nulcei in quantized energy level? Not just "cause the equations say so"!

Thanks.

 

[^_^]

Because the empirical evidence says so.

 

[Kane O'Donnell]

Because the whole universe hasn't collapsed in a massive radiation burst? Is it just me or does someone ask this question every second day on this forum?? I'm not sure people who ask that question actually know what a 'physical' explanation is - hell, I would have a hard time trying to come up with a good definition, if one existed.

There are such things as questions that only exist because English is able to put abstract words into a grammatically correct sentence, like "what is the square root of poetic yoghurt?". I respectfully submit that asking for a 'physical' explanation, which really means a NON-physical explanation (since physicists, the professionals charged with actually DOING physics, give an explanation that IS physical (it is physics after all) but apparently not acceptable), is a question in the latter category.

People asking questions like this should perhaps try instead challenging themselves by asking themselves what a physical explanation IS.

Ok ok, I'll throw a bone here - discrete orbits are a consequence of discrete energy levels which are in fact discrete eigenvalues that arise as a property of certain operators on certain dense subsets of a Hilbert space that defines the state of the dynamical system. The operator describing the Hydrogen atom has countably many discrete energy eigenvalues (as well as a continuous positive energy spectrum), and hence the bound states of an electron are discrete.

Kane O'Donnell

 

[selfAdjoint]

Newton began the fashion of going by prediction rather than explanation (hypothesi non fingo), and physicists have run with that ball ever since. But it's something that doesn't come naturally to most people and those who encounter advanced physics (whether QM or SR) for the first time are understandably confused. They don't deserve to be brushed off. What we need is a sticky on the subject of explanation versus prediction, with a good presentation given and then the thread closed so cranks can't add their two cents.

 

[dextercioby]

Discrete orbits are a consequence of discrete energy levels which are in fact discrete eigenvalues that arise as a property of certain operators on certain dense subsets of a Hilbert space that defines the state of the dynamical system. The operator describing the Hydrogen atom has countably many discrete energy eigenvalues (as well as a continuous positive energy spectrum), and hence the bound states of an electron are discrete.

Kane O'Donnell

You're mixing things and it's not good...Bohr theory (semiclassical) and QM.Yes,discrete orbits are a consequence of the discrete values for the angular momentum of theelectron rounding a nucleus on a circular path/orbit.Yes,discrete energy levels are a consequence of the discrete values for the total angular momentum of the electron rounding the nucleus.
At least that's how i was presented with Bohr's theory in high school and i reasons to believe it's hystorically and logically correct.
Yet i subscribe for the QM part u posted.Possible quantum states of a system are the eigenstates of the Hamiltonian.

Good idea,SelfAdjoint...Put in practice!

 

[Coelum]

Can quantum theory be explained? Is it possible to look at quantum theory and say "this calculation works because in terms of physics..."? (even if the physics does not entirely correlate with classical ideas)

Have you read R. Feynman's QED? It answers your question - at least IMHO - and is a highly recommended reading.

 

[imabug]

Can quantum theory be explained? Is it possible to look at quantum theory and say "this calculation works because in terms of physics..."? (even if the physics does not entirely correlate with classical ideas)

Eg - is it possible to explain in physics terms why electrons can only exist certain distances from nulcei in quantized energy level? Not just "cause the equations say so"!

Thanks.

I would argue that QM is an explanation/model of the world we see (at the small scale) and therefore trying to explain it would be rather redundant. You'd be trying to explain an explanation. You look at the physical phenomena (quantized energy level) and come up with a model to explain it (quantum mechanics). If it fits, then you use the model to predict other physical phenomena and try to find evidence of it. You seem to be trying to go the other way which is only going to make life difficult for you.

 

[Kane O'Donnell]

You're mixing things and it's not good...Bohr theory (semiclassical) and QM

Um. I don't understand what you mean by this. I wasn't referring to Bohr theory at all.

I've been trying over the last half hour or so to write a short post on how quantised 'orbits' arise from QM, but it has been difficult to make it concise without sacrificing accuracy and I find that unsatisfactory in the context of this thread. However, the main problem is of course deciding what an 'orbit' is going to be characterised by - for example, you have suggested angular momentum, and it is indeed possible to show that the bound states (E < 0) in a Coulomb potential have a discrete angular momentum spectrum. This is by no means referring to the Bohr model - obviously in the Bohr model there is no question of what an orbit means.

What I said in the previous post was that discrete energy spectrum => discrete orbits. It's probably a bit loose. What I mean, technically, is that the operator $$\hat{L}^2$$ shares a non-trivial set of eigenvectors with the Hamiltonian $$\hat{H}$$. All of the eigenfunctions corresponding to the eigenvalues in the bound spectrum of $$\hat{H}$$ (E < 0) have periodic angular boundary conditions imposed, and this gives rise to a discrete angular momentum spectrum.

Cheman - I must apologise for being rather rude. Obviously I want everyone to want to learn about QM, but I find it very frustrating when someone asks for a 'physical' explanation, when by definition 'physical' means 'according to physics' which in this case really does mean according to quantum mechanics, and even Feynman, Lord of the Physical Explanation, didn't try to give a better explanation for such a thing. The de Broglie model of course does explain quantised orbits, but QM says that such orbits aren't really physical anyway, so basically using the Bohr model/de Broglie model to explain such a thing would be giving a 'physical' explanation to an essentially unphysical phenomena!

I strongly agree with selfAdjoint on the explanation vs prediction matter, although I have a feeling it is such a subtle point that the crackpots would have plenty of time to rush in before the presenter could finish. I recommend reading this:

http://www.fotuva.org/online/science.htm

although Feynman is well known for his strong views on the scientific method

Cheerio!

Kane O'Donnell

 

[QuantumTheory]

Um. I don't understand what you mean by this. I wasn't referring to Bohr theory at all.

I've been trying over the last half hour or so to write a short post on how quantised 'orbits' arise from QM, but it has been difficult to make it concise without sacrificing accuracy and I find that unsatisfactory in the context of this thread. However, the main problem is of course deciding what an 'orbit' is going to be characterised by - for example, you have suggested angular momentum, and it is indeed possible to show that the bound states (E < 0) in a Coulomb potential have a discrete angular momentum spectrum. This is by no means referring to the Bohr model - obviously in the Bohr model there is no question of what an orbit means.

What I said in the previous post was that discrete energy spectrum => discrete orbits. It's probably a bit loose. What I mean, technically, is that the operator $$\hat{L}^2$$ shares a non-trivial set of eigenvectors with the Hamiltonian $$\hat{H}$$. All of the eigenfunctions corresponding to the eigenvalues in the bound spectrum of $$\hat{H}$$ (E < 0) have periodic angular boundary conditions imposed, and this gives rise to a discrete angular momentum spectrum.

Cheman - I must apologise for being rather rude. Obviously I want everyone to want to learn about QM, but I find it very frustrating when someone asks for a 'physical' explanation, when by definition 'physical' means 'according to physics' which in this case really does mean according to quantum mechanics, and even Feynman, Lord of the Physical Explanation, didn't try to give a better explanation for such a thing. The de Broglie model of course does explain quantised orbits, but QM says that such orbits aren't really physical anyway, so basically using the Bohr model/de Broglie model to explain such a thing would be giving a 'physical' explanation to an essentially unphysical phenomena!

I strongly agree with selfAdjoint on the explanation vs prediction matter, although I have a feeling it is such a subtle point that the crackpots would have plenty of time to rush in before the presenter could finish. I recommend reading this:

http://www.fotuva.org/online/science.htm

although Feynman is well known for his strong views on the scientific method

Cheerio!

Kane O'Donnell

Ok, I'm 16, and am learning calculus. I am deeply interested in space and quantum mechanics. I know a lot about derivitives, integrals, and some of vector calculus.

I really don't appreciate you trying to show off on how 'skilled' you are in physics. I don't care if your some genius, you don't have to show off, and it really upsets me.

This person asked a honest question, I come here with only a medium amount of knowledge about calculus, yet you try and show off so not a lot of people can understand it.

I like this forum a lot, especially this section, I like it when I can understand what I read, or when people actually try and post something constructive rather than explain the world with equations.

Now, I wanted to ask you a question. What does the ^ mean above some of the (constants?) varibles, you wrote?

I'm familiar with that as the square sign, I've seen it in advanced vector calculus books, but never understood it.

Thanks

 

[quantumdude]

Now, I wanted to ask you a question. What does the ^ mean above some of the (constants?) varibles, you wrote?

The hat denotes an operator (as opposed to an eigenvalue or a function).

For instance, in the equation $\hat{p}|p'>=p'|p'>$, $\hat{p}$ is the momentum operator, while $p'$ is the eigenvalue of the momentum operator that corresponds to the eigenket $|p'>$.

In short, the 'hatted' p acts on vectors in momentum space, and the 'unhatted' p' is just a number.

 

[Kane O'Donnell]

*sigh*

Look, what I wrote wasn't showing off. It's the answer to the question. The point is, it's not in *any way* a trivial thing to answer "why" using Quantum Mechanics, it takes work.

I'm glad that at 16 you know a lot about calculus. When I was 16 I didn't know anything about calculus. However, there is a *lot* of maths, and I really mean that, between the mathematical framework of quantum mechanics and calculus, and I can't just reduce an answer to "you integrate this, then substitute this and differentiate there". That doesn't give a *reason*. It's just the legwork - the reasons come from the properties of the underlying framework, and it takes a bit to get them out. I'm not the one who should be apologising to you for that.

Physicists *do* explain the world using equations - you have to come to grips with this. Explaining a phenomenon means having a set of rules that describe as many aspects of the phenomenon as possible, and it has to be that way - how else can you explain things? Metaphor? Misleading. Pictures? Not in 4D, buddy, and they don't usually contain enough information. Words? Well, words represent a way of getting a thought in our head out into the 'world beyond' and vice versa, so that isn't going to help if we have to define the words themselves.

At the end of the day, we have the scientific method, and to effect it, we need a consistent, precise and concise way of representing all our models and explanations. These requirements are fulfilled by mathematics. I would not dare to say that that is all there is to maths - maths is something in itself, but as a tool, it's what us Aussies put in the category of "bloody useful".

I am still learning! Everyone who seriously studies physics is always learning, but at a certain point, the only way to go forward is to throw yourself into a very mathematical world. I really hope that one day I can burst out the other side and have as much clarity and confidence in my viewpoint as Feynman or Einstein or Dirac, but I'm nowhere near there and until then, mathematics has to be my guide.

Regards,


Kane O'Donnell

 

[Louis Cypher]

If what we are looking at has no tangible evidence than could we not be mistaken at least in some of the finer points of quantum mechanics, we have set up experiments for quantum mechanics for years, which have both bemused and amazed us but, sometime I do rather feel like we're looking for an answer without asking the right questions. indetreminacy throws a rather grey palour over QM, but it's also what makes it so interesting.

I think we need to take a more scientific approach, I know we have no direct proof, but a little more subjectiveness, would help, after all it's all very well saying there is an infinite number of 'ghost' quark pairs formed between the quarks, but what does that really mean, there could easily be nothing but the quarks themselves, I just feel that perhaps we need to start looking a bit more criticaly at some of the things we hold to be true, or somewhat factual, and start questioning the foundations of something that Einstein and Schrödinger both understood was not the whole answer; Qm is our best guess for now, however it's far from the truth, Schrödinger and Einstein realized they probably would not see proof of it's downfall in their lifetime and by downfall I mean progress to somehting closer to the truth, but will we ever see direct proof in the future and if we don't what does QM scientific value? if we don't start looking for ways to question our firmly established belief, we may well find that one day we are perched upon a stack of cards; are we still just stumbling around in the dark looking for answers which aren't there, I don't know but I'm more willing to ask questions, than speculate on speculation to find answers.

 

[Chronos]

QM is conceptually difficult [As is GR] because it does not obey the 'common sense' rules we are accustomed to dealing with in our slow moving, low energy, macroscopic reference frame. It only makes sense mathematically and that is the only way [AFAIK] it can be expressed intelligibly.

 

[Kane O'Donnell]

Qm is our best guess for now, however it's far from the truth

How far do you claim it is? Quantum Electrodynamics is the most accurate theory physics has ever had, they're up to 12 significant figures now aren't they? How can you say that a theory that agrees so well with experiment is 'far from the truth'?

It is more accurate to say that whilse QM has excellent predictive powers, we would like an underlying theory that perhaps shows why the postulates of QM arise.

Of course, there may not *be* such a theory - maybe the commutator postulate is simply a property of the universe. I wouldn't want that to be true of course because that leaves the value of hbar unexplained, but if it's the case, tough. Most high-energy physicists are looking for a theory that has the least possible number or arbitrary numbers in it, or even better, none. The whole "Einstein was against QM" argument has been heard over and over, and the fact remains that QM's predictive power is a testimony to the fact that *it* at least is very likely a partial truth*. This is more than can be said for string theory or loop quantum gravity.

Besides, I hardly think it's correct to say that up until now we haven't had a 'scientific' approach to QM - it's one of the most thoroughly investigated fields in Physics and indeed all of Science, and the modern theories have passed every experiment, which is one of the problems facing people who are looking for deeper theories.

Regards,


Kane O'Donnell

(*The usual disclaimer applies to any association of a theory with truth. I strongly suspect QM won't be destroyed utterly, but there are of course no guarantees. )

 

[Mike2]

How far do you claim it is? Quantum Electrodynamics is the most accurate theory physics has ever had, they're up to 12 significant figures now aren't they? How can you say that a theory that agrees so well with experiment is 'far from the truth'?

It is more accurate to say that whilse QM has excellent predictive powers, we would like an underlying theory that perhaps shows why the postulates of QM arise.

Of course, there may not *be* such a theory - maybe the commutator postulate is simply a property of the universe. I wouldn't want that to be true of course because that leaves the value of hbar unexplained, but if it's the case, tough.

Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?

 

[ZapperZ]

Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?

Hey, if there are people who make claims based on their own imaginary understanding of QM, then QM can certainly incorporate imaginary numbers into its theory.

Zz.

 

[Modey3]

"Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?"

Hah, that is funny. The incorporation of " i " into exponential functions represents a wave through the Euler realtionship with orthagonal real and imaginary parts. The answer comes from Diff EQ. I know its a difficult concept to grasp, but it works mathematically- which is the basis of all logic.

I think the real question of QM is why can't an electron exist in one place, why are electrons "spread out" over real-space in discrete wave like patterns? The result of the Double-Slit Experiment still fascinates me to this day. The answer lies there. How can an electron pass through both slits at the SAME time? The only answer that works is that electrons are not the billiard balls that we want them to be.

Anyways, I'm new to this forum. So I would like to introduce myself. I am a Materials Science & Engineering grad student, but I'm a physicist at heart. I have an interest in Condensed Matter Physics and its parent QM. I honestly don't think we as human beings will know why QM works.

Modey3

 

[cartuz]

"Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?"

Hah, that is funny. The incorporation of " i " into exponential functions represents a wave through the Euler realtionship with orthagonal real and imaginary parts. The answer comes from Diff EQ. I know its a difficult concept to grasp, but it works mathematically- which is the basis of all logic.

I think the real question of QM is why can't an electron exist in one place, why are electrons "spread out" over real-space in discrete wave like patterns? The result of the Double-Slit Experiment still fascinates me to this day. The answer lies there. How can an electron pass through both slits at the SAME time? The only answer that works is that electrons are not the billiard balls that we want them to be.

Anyways, I'm new to this forum. So I would like to introduce myself. I am a Materials Science & Engineering grad student, but I'm a physicist at heart. I have an interest in Condensed Matter Physics and its parent QM. I honestly don't think we as human beings will know why QM works.

Modey3

Can I write my opinion? I understand double-slit experiment as the pattern of classical test particles is in the gravitational background of random gravitational fields and waves. It is the total classical interpretation with classical random fields and waves. That interpretation I named Stochastic Gravitational Interpretation. We do not surprised why we can see the interference at the surface of the water, for example. Double-slit experiment is analog to this pattern. Quantum property is not the property of the particles. It is the property of environment (background) with resonant property of particles.
Quantum Property=Background Property+Resonant Property of Particles
You can read about this interpretation in T.F.Kamalov, How to Complete the Quantum-Mechanical Description?// In the book “Quantum Theory: Reconsideration of Foundation-2”, Vaxjo, Sweden, June 1-7, 2003, p. 315-322, E-print arXiv: quant-ph/0212139.
http://xxx.lanl.gov/abs/quant-ph/0212139

 

[quantumdude]

Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?

1. Arguments from incredulity are never valid.

2. Anything that can be done with imaginary numbers can also be done without them.

3. Imaginary numbers enjoy the same ontological status as real numbers. You are just getting confused by the unfortunate naming scheme that we are stuck with.

Please see the following thread: Imaginary numbers that was just posted today. It has some information that will help you understand that complex numbers are a natural, well-defined extension of the real numbers.

 

[Kane O'Donnell]

Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?

What's wrong with a theory incorporating imaginary numbers? They're *extremely* well understood mathematically. Your argument is just like saying the number $$\pi$$ doesn't exist because you can't write it down. Well, saying that you can't write it down means that it isn't a *rational* number, but it is a well-defined *concept* as well as being an element of the irrationals and hence the real numbers.

So are we allowed to use pi? No? Oh well, can't find the area of a circle anymore.

Seriously, the issue that needs fixing with QM is where all the constants come from.

Besides, the fact that the numbers can be imaginary is a very important feature of the mathematical framework. For example, since an arbitrary phase factor multiplied by the wavefunction doesn't change the probability distribution, an action by symmetries on the underlying Hilbert space can be 'symmetic' modulo the set of complex numbers with modulus one. This allows us to represent a symmetry as a unitary transformation (the existence of such a unitary transformation is a theorem of Wigner).

Imaginary numbers are not a problem with QM. I will accept that they give QM a distinctly different flavour to say, classical mechanics, but I don't see how that affects it's viability.

Besides, I don't claim that QM is the ultimate truth, I just claim that it isn't *so* wrong that some new theory is going to totally blow it out of the water. It's a possibility, but QM's experimental success suggests it is at least *mostly* right.

Regards,

Kane

 

[Kane O'Donnell]

Modey3:

Hi :) Just wanted to point out that mathematics isn't the basis of all logic, it's sort of the other way around :D

Cheerio!

Kane

 

[cartuz]

About imaginary numbers in Quantum Mechanic. Sorry, but I can not see the problem here. In Special Relativity Theory you can see the systems of coordinates with axis ict. Here c is light velocity. It is forth direction which orthogonal to 3-dimentinal space. Imaginary axis (time axis) is mean that time axis is orthogonal to 3-dimentinal spaces only, I think.

 

[ZapperZ]

Can I write my opinion? I understand double-slit experiment as the pattern of classical test particles is in the gravitational background of random gravitational fields and waves. It is the total classical interpretation with classical random fields and waves. That interpretation I named Stochastic Gravitational Interpretation. We do not surprised why we can see the interference at the surface of the water, for example. Double-slit experiment is analog to this pattern. Quantum property is not the property of the particles. It is the property of environment (background) with resonant property of particles.
Quantum Property=Background Property+Resonant Property of Particles
You can read about this interpretation in T.F.Kamalov, How to Complete the Quantum-Mechanical Description?// In the book “Quantum Theory: Reconsideration of Foundation-2”, Vaxjo, Sweden, June 1-7, 2003, p. 315-322, E-print arXiv: quant-ph/0212139.
http://xxx.lanl.gov/abs/quant-ph/0212139

If you are going to use a "fringe" science as an explanation, you need to show peer-reviewed citation rather than someone's book or e-print arxiv article. Otherwise, this "opinion" of yours has not been verified by your peers to even be considered to have any validity, and it will go into the TD wasteland.

Zz.

 

[jtbell]

In Special Relativity Theory you can see the systems of coordinates with axis ict.

Which nobody uses nowadays, as far as I know. The $$ict$$ fourth dimension was an early attempt to make Einstein's space-time look more like a four-dimensional Euclidian space by letting us write the invariant space-time interval as $$s^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 = x^2 + y^2 + z^2 + (ict)^2 = x^2 + y^2 + z^2 - c^2 t^2$$.

Nowadays we prefer to acknowledge that space-time isn't really Euclidian by showing that "-" sign in the last version explicitly and defining the space-time interval as

$$s^2 = x_0^2 - x_1^2 - x_2^2 - x_3^2 = c^2 t^2 - x^2 - y^2 - z^2$$

(which is of course the negative of the previous definition, but we do it consistently so it doesn't matter)

Similarly the invariant mass (squared) is

$$(mc^2)^2 = E^2 - p_x^2 - p_y^2 - p_z^2$$

and in general the invariant product of two four-vectors is

$$a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3$$

We could in principle reformulate quantum mechanics without $$i$$ by postulating that a particle's probability density is determined by two real wave functions that correspond to the real and imaginary parts of the $$\psi$$ that we all know and love. We might call them $$\psi$$, the wave function, and $$\phi$$, the "wave co-function". Then we would define the probability density as

$$P(x) = \psi^2 (x) + \phi^2 (x)$$

and instead of a single Schrödinger equation we would have two coupled differential equations which in one dimension would look like

$$-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi = -\hbar\frac{\partial\phi}{\partial t}$$

$$-\frac{\hbar^2}{2m}\frac{\partial^2\phi}{\partial x^2} + V\phi = \hbar\frac{\partial\psi}{\partial t}$$

I suspect that we don't do this because the complex-number formalism makes things so much more convenient here than in relativity, and physicists were already familiar with using complex numbers in describing waves, simple harmonic motion, etc.

 

[ZapperZ]

The problem here isn't the "imaginary number". The problem here is that the person making the point has a history of having only a superficial knowledge of physics and, as is obvious here, mathematics. As Tom has pointed out, there is NO difference in terms of mathematical "status" of real and imaginary number. Anyone who has done any amount of complex algebra can already see this. The name "imaginary" number doesn't mean "I-made-it-up-all-in-my-head" number. It is silly to think that if I use $$sin (\omega t)$$, then I'm OK, but once I use $$e^{i\omega t}$$, then my theory no longer has any meaning just because I have a complex number there!

Again, imagination without knowledge is ignorance waiting to happen. Except in this case, it has already happened.

Zz.

 

[Mike2]

1. Arguments from incredulity are never valid.

2. Anything that can be done with imaginary numbers can also be done without them.

3. Imaginary numbers enjoy the same ontological status as real numbers. You are just getting confused by the unfortunate naming scheme that we are stuck with.

Please see the following thread: Imaginary numbers that was just posted today. It has some information that will help you understand that complex numbers are a natural, well-defined extension of the real numbers.

I read your post on how counting numbers lead to whole number which leads to integers which leads to rational numbers which leads to irrational number which leads to complex numbers. The last step is non-sequitur.

I'm not arguing from incredulity. I'm arguing from inconsistency. For there simply is no number the square of which equals a negative number. The square of a positive number gives a positive number, and the square of a negative number gives a positive number. So the square root of a negative number existing is not consistent with the definition of multiplying a positive by a positive, or multiplying a negative by a negative. Both multiplications are positive, so the square root of a negative does not exist.

Yet we have the most accurately confirmed theory of physics based on it. If you can get rid of the imaginary number in the equations of QM or QFT, then by all means do so.

Thank you.

 

[selfAdjoint]

I'm not arguing from incredulity. I'm arguing from inconsistency. For there simply is no number the square of which equals a negative number.

Dead wrong. Numbers on the real line all have positive squares, but i is on the line at right angles to the real line. Complex numbers are vectors in the plane with one real and one "transverse" (Gauss) component. But those are vectors not numbers, you protest. They are numbers because they satisfy all the rules of arithmetic: addition, multiplication, subtraction and division, commutative in both additive and multiplicative operations and with a distributive law, they behave just like rational or real numbers and better. Every root of every polynomial with complex coefficients is a complex number; you can't say the corresponding thing for rationals or reals.

 

[humanino]

I read [...]

For there simply is no number the square of which equals a negative number.

Did not you notice when you read, that every motivation for the next constructions is always providing solution to problems that previously seemed impossible to solve ? There is not much more, but that is already a great deal.

People used to say "but there is no way that, if you add a (strictly positive) quantity A(>0) to any quantity B, you get a result which is zero". Alas, if B is negative it actually works. Simply those people would not accept negative quantities as element of reality, because you never have a negative number of coins in your pocket. They could argue as long as they want, it does neither make negative numbers less useful, nor less in contact with reality.

When Fourier was writing heretic formulae, he was actually able to find physical quantities (namely he was studying the propagation of heat). It took several centuries, and the construction of what is now called distributions, to rigorously built the framework in which Fourier's calculus were justified.

So you can argue against compex numbers, but you only succeed to display your misunderstanding.

And a beautiful word such as "non-sequitur" is not an argument

 

[quantumdude]

I read your post on how counting numbers lead to whole number which leads to integers which leads to rational numbers which leads to irrational number which leads to complex numbers. The last step is non-sequitur.

No, it isn't. Mathematical objects are defined by their properties. As long as the properties don't lead to any inconsistencies in the mathematical system, there is not problem. Note, by "inconsistency" I mean "inconsistency within the formal system", not "inconsistency with Mike2's preconceived notion of consistent".

I'm not arguing from incredulity.

You most certainly did argue from incredulity. Look again at the post of yours that I quoted. It is a textbook example of an argument from incredulity.

I'm arguing from inconsistency. For there simply is no number the square of which equals a negative number.

Actually, there is. It's called 'i', and its square is -1.

The square of a positive number gives a positive number, and the square of a negative number gives a positive number. So the square root of a negative number existing is not consistent with the definition of multiplying a positive by a positive, or multiplying a negative by a negative. Both multiplications are positive, so the square root of a negative does not exist.

It is readily seen and acknowledged that the rule that allows us to multiply under a radical does not hold when imaginary numbers are admitted. This is only a problem if one is determined to keep that rule.

Yet we have the most accurately confirmed theory of physics based on it. If you can get rid of the imaginary number in the equations of QM or QFT, then by all means do so.

Why? It is mathematically perfectly well defined, and it is extremely powerful. Of course, we could dump imaginary numbers in favor the group of 2x2 matrices that is isomorphic (under matrix multiplication) to the group of complex numbers under multiplication. But that's a foolish idea, because we would not be able to readily use such things as contour integration or conformal mapping. You are advising that we throw away all of this simply because you do not understand how complex numbers fit into mathematics as a whole. That's bad advice.

 

[cire]

Oh come on... the square root of a negative number? How could a theory incorporating imaginary numbers be anything except an effective theory of something deeper?

without the i in the shcrodinger equation the propability amplitudes tend to 0 as time increases!
what is wrong in having complex numbers? i is a number with beautiful properties and having them there have a lot of sense,