# Macroscopic versus microscopic standing waves

• I
Gold Member
I have read the description of electrons as standing waves based on an analogy with a string vibrating at its natural frequencies: thus the different quantum levels are akin to the tones or harmonics of the string, right?

So far, so good, but then I have seen contradictory complementary views:

- One is that Planck introduced an "additional restriction or condition": that the energy levels where the electron oscilates are quantized.
- Other explanations instead do not talk about such additional thing, they just use the above mentioned analogy without seeking a differentiation with the macroscopic case.

I tend to think that the second interpretation is the right one. After all, the basic idea is that, to excite the electron to a higher level, one needs a photon, for example, of the right frequency. Then it can be the fundamental one (f) or the same multiplied by an integer (nf). So one could say that the interaction frequencies are quantized. And then one can convert frequency into energy by multiplying it by the Planck constant, which acts as a conversion factor (E = h n f).

Conclusion: one could equally say that to create a standing wave in a string you need the appropriate quantums of energy or that to create a standing wave in an an atom you need to play it with notes of its harmonic series... Is this right?

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PeterDonis
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I have seen contradictory complementary views
That's probably because you're reading pop science sources instead of textbooks or peer-reviewed papers. We can't know unless you give some actual references.

the basic idea is that, to excite the electron to a higher level, one needs a photon, for example, of the right frequency
Yes.

Then it can be the fundamental one (f) or the same multiplied by an integer (nf).
No. The various light frequencies associated with atomic energy level transitions are not, in general, integer multiples of each other. This is just one way in which the pop science analogy you are working with breaks down.

Is this right?
It's probably not even wrong. Again, we can't tell unless you give some actual references.

Gold Member
Thanks Peter.

I muddled my question with some improvisation, which is never advisable when your knowledge is so fragile...

Yes, of course,

The various light frequencies associated with atomic energy level transitions are not, in general, integer multiples of each other.
At least in the hydrogen atom, I understand that they are given by the Rydberg formula, which gives the spectral line for transitions down to a given level from various upper levels, right?

This is just one way in which the pop science analogy you are working with breaks down.
Still the analogy between an electron in the atom and a standing wave in a string, is a peacefully accepted one, isn't it? I wonder why you are deprecating it, I think it is generally used by very orthodox sources. I came back to it after some recent internet readings, I can concede, but I would swear that I have read it in my physics manuals... If you insisted on making me work, I would look for serious references...

Of course, one should make a judicious use of such "analogy", i.e. not treat it as an "equality". I mean, a macroscopic system is not a microscopic system, so one should expect that an atom does not behave as a string in all respects. But the analogy does not claim so, what it claims is that both systems behave equally only for certain purposes, precisely those represented by tha abstract concept "standing wave", which can thus be applied to both systems.

In the light of this, let me reformulate my question in a simpler way and please forget the rest: is a standing wave in a string a quantized system, i.e. one with which you can only interact through quanta of energy?

yes standing waves in a :macroscopic" string are a resonant system which has good overlap, conceptually, with resonance in atomic transitions.

that is about as far as I would go though with that or any other analogy.

quantum systems are there own thing - all analogies are just crutches until you can develop your mathematical skills enough to shed them.

it's only human to link new knowledge to past conceptual schemes - it won't help you progress though.

PeterDonis
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At least in the hydrogen atom, I understand that they are given by the Rydberg formula
Only as a first approximation. As spectroscopic measurements got more accurate, what appeared to be single lines in Rydberg's time turned out to be multiple closely spaced lines (this was called "fine structure"), and then as measurements got more accurate still, even more detailed splitting of lines was discovered (called "hyperfine structure"). There is no simple formula that describes all this; it's a fairly complicated series of corrections.

Still the analogy between an electron in the atom and a standing wave in a string, is a peacefully accepted one, isn't it?
Only as a very rough analogy. It breaks down pretty quickly when you try to dig into details.

is a standing wave in a string a quantized system, i.e. one with which you can only interact through quanta of energy?
No. There are plenty of ways to interact with a string, whether or not it is in a standing wave state, that involve continuously varying amounts of energy.

If you limit yourself only to interactions with the string that take it from a standing wave state with wavelength ##\lambda_1## to a standing wave state with a different wavelength ##\lambda_2##, then yes, those interactions involve different discrete amounts of energy. But these interactions still do not involve single "quanta" of energy in any sense that is similar to the interactions that shift electrons in atoms from one energy level to another. A string is a macroscopic system with a huge number of atoms in it, something like ##10^{23}## of them. You should not expect such a system to have dynamics that are very similar to the dynamics of a system with only one atom in it.

jtbell
Mentor
one could equally say that to create a standing wave in a string you need the appropriate quantums of energy or that to create a standing wave in an an atom you need to play it with notes of its harmonic series
A vibrating string is not a very good analogy to an atom, because a string is one-dimensional whereas an atom is three-dimensional. A better analogy for an atom would be standing waves of sound in a spherical cavity, for which the resonant frequencies are not integer multiples of a fundamental frequency. Note "better" not "perfect"... I'm hedging my bets here. Nevertheless, the mathematical description for both of these uses functions called "spherical harmonics."

Gold Member
Thanks to all. I fully agree with your warnings about relying too much on analogies and the need to work also on the math route. However, when properly used, metaphors are a powerful pedagogical tool (precisely because they build on past knowledge and familiar ground). And using "properly" just means making the concept more abstract, by discarding what differences are irrelevant for the purpose at hand and focusing on the relevant ones.

In this case, thanks, Peter, for the reference to hyperfine structure, I am familiar with that, but it just happens that this complication is simply "more of the same", without altering the validity of the analogy as such. This simply entails that for standing waves in an atom the formula for finding the "tones" or "harmonics" maybe highly more complex than in a string (or may even "as of today" not be a formula, as you point out, but a set of progressive corrections) but whatever it is, it would still be ruled by the same basic guidance: we are talking about a standing wave.

Thanks also jtbell for pointing to spherical harmonics. Yes, it was precisely when I read about spherical standing waves that I became more interested in the analogy (even if in the end I got a little confused when I wrote the OP).

As to Peter's warning that a 1D string (or a 2D drum or a 3D sound cavity) are made of a huge amount of elements (atoms), whereas an atom is still 1 atom, I buy that. That precisely can be one of the aspects that should be removed from the "superabstract" concept of standing wave for it to be useful in the atom domain: just as a 1D string has characteristics that a 3D system does not have, but we still call what happens in both systems a standing wave ("SW"), I believe that certainly the atom SW will present differences versus the multi-atom SWs, but still we can benefit from the similarities.

That is precisely my point: that the concept is there, it is an accepted one, but it is not a very exploited one and we may not be benefiting from its advantages, at least from a didactic point of view.

Now let me re-formulate the idea in a better way based on the received contributions. So the analogy works in the two directions:

- If you want to interact with any standing wave (even for instance a simple macroscopic one like a 1D string), you need "quanta" of energy represented by h times the different tones or harmonic frequencies of the string.
- The atom also being (due to its internal forces) a confined space where the electron oscilates as a standing wave, its tones are its absorption/emission lines. And its energy levels can be tracked based on that, because the emitted frequencies represent differences between such levels.

Non-analogous aspects:

-The easy thing about a string is that here the overtones are simply integer multiples of the fundamental one, while in 2D or 3D systems like the atom itself formulas for predicting the frequencies can get fairly complex.
- Macroscopic systems may have virtually infinite elements and hence oscillation modes and frequencies, unlike the atom.

Better now?

PeterDonis
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it just happens that this complication is simply "more of the same", without altering the validity of the analogy as such
Yes, it does, because if the description in terms of spherical harmonics were correct, the fine and hyperfine structures would not exist. (Not to mention other effects such as the Lamb shift.) Those structures cannot be described by spherical harmonics alone; they cannot be described in terms of the "standing wave" analogy, because that analogy cannot explain why there are multiple closely spaced lines instead of one. To explain that, you have to discard the belief that the electron energy levels are describable in terms of "standing waves".

the concept is there, it is an accepted one, but it is not a very exploited one
What is your basis for this claim? The field of quantum mechanics from the time of Bohr's original model of the atom up through the development of quantum field theory was basically an exploration of all the implications of a "standing wave" model, and a discovery of its limitations--the ways in which it failed to predict new experimental effects that were discovered as measurements became more accurate. The "standing wave" analogy is not used today because it has been superseded, not because it hasn't been explored.

If you want to interact with any standing wave (even for instance a simple macroscopic one like a 1D string), you need "quanta" of energy represented by h times the different tones or harmonic frequencies of the string.
No, this is wrong. Please read my post #5 again, carefully.

The atom also being (due to its internal forces) a confined space where the electron oscilates as a standing wave, its tones are its absorption/emission lines.
This is wrong as well. See above.

in fact as far as analogies go and if I have to use one I prefer the analogy of a ringing bell over standing waves on a string.

Gold Member
PeterDonis: all the physics books that I own assume the SW model. I am not sure why you talk about a sort of depuration of the SW model since Bohr. My understanding is that Bohr did not have a SW model, he saw the electron as a particle. It was only years after that de Broglie proposed the idea of matter (electrons) as waves, which Schrödinger soon developed into the current model of the atom, where, yes, the electron is a SW. This model already accounts for all the finer measurements that you refer to, that is why it contains the various quantum numbers...

Of course, there may be better analogies right now. If you can cite any source discussing them, I will be glad to consult them.

houlahound: I would also like to know about sources discussing such ringing bell analogy.

But in this thread I just want to understand better the SW model. Even if it were wrong, it would be a good basis to be afterwards corrected wherever required.

In particular, before re-taking my original questions, I am realizing that there is a key issue that I don't understand with this model: in it, are really the "emission/absorption lines" analogous to the tones? I mean: those lines are the frequencies at which photons are emitted when the electron transits downward between levels. So they refer to something that the electron first gains and then loses... But in the macroscopic case, if I have plucked one string and it is vibrating in the fundamental tone (f1) and I want to bring it to the next level (first overtone = f2), do I use a source vibrating at (f2 - f1), which would better fit the analogy but would sound strange, or at f2, which sounds better but may not be a good analogy with the atom case...?

If someone is familiar with the SW model of the electron and can solve this doubt, I will be grateful!

vanhees71
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I guess "SW" stands for "Schrödinger wave"? If so, then your intuition is known to be wrong for about 90 years now. The Schrödinger wave function has only one meaning that is compatible with all observations, and that's that ##|\psi(t,\vec{x})|^2## is the probability distribution for the position vector ##\vec{x}## of the particle at time ##t##. So the electron is not the wave in the same sense as light is an electromagnetic wave (in the classical picture of light). It's neither a classical particle ("billiard ball en miniature"). As far as we know it's an elementary particle. In the Standard Model it's described by a quantized Dirac field, and that's the most precise description we can give today. It's well understood why for atoms with sufficiently small ##Z## we can as well describe it as a non-relativistic particle in terms of a wave function (or many electrons as a many-body wave function).

Gold Member
I will be very brief...

SW stands for "standing wave".

And I just want to understand the model of the electron as a SW, regardless whether it is accurate or not, right or wrong...

houlahound: I would also like to know about sources discussing such ringing bell analogy.

it's just one I made up.

PeterDonis
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all the physics books that I own assume the SW model
What books are they?

I am not sure why you talk about a sort of depuration of the SW model since Bohr.
You do realize that we have a great deal more knowledge, both experimental and theoretical, since Bohr, right? Science progresses.

Of course, there may be better analogies right now
If you really want to understand QM, you should not be looking for analogies. Quantum objects are different from anything you are familiar with. You need to understand them as they are, without trying to draw analogies with anything else.

weirdoguy
vanhees71
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I will be very brief...

SW stands for "standing wave".

And I just want to understand the model of the electron as a SW, regardless whether it is accurate or not, right or wrong...
A "standing wave" is, by definition, a eigenvector of the Hamiltonian and represents a stationary (time-independent) state. The electron is NOT this wave but the modulus squared of the solution ##|u_E(\vec{x})|^2## is the probability distribution for finding the electron at the position ##\vec{x}## (Born's rule).

Gold Member
Peter, why do you insist on Bohr again? As commented, Bohr did not talk about waves. Say at least since Schrödinger model... It is just 10 years difference but a little closer to the truth of my comments.

vanhees71, this is no secret: I have no math knowledge to follow your comment. Not that I am proud of that, I wish I had it and I try hard to learn, but that is the way it is so far. But I wonder if you and PeterDonis know what an "analogy" is. I promise I could explain it in a manner of which you would not understand a word, though I could also explain in an easy manner for "laymen".

Anyhow, I have found a very nice explanation of this particular analogy we are discussing here. It is in the following link, second answer by James Buban:
https://www.quora.com/What-is-a-good-analogy-to-explain-quantum-numbers

A smart guy, he goes so far as reconciling the SW analogue not only with the principal but also with angular and magnetic numbers, although he thinks the trick breaks down when reaching spin number. I wouldn't say the analogue "breaks down" there, it is simply that such area is "out of scope" of the concept, but never mind...

What is still bugging and the only thing I want to discuss in this thread is what I commented before. Actually it is not a "quantum physics" issue, although it is an issue about a quantum (albeit classical) system after all.

Imagine that you have plucked a string and it is vibrating as a SW at its fundamental tone (frequency f1). You leave it there and later on come back with the intention of bringing it up to the second tone (frequency f2). What should you do? It seems obvious: apply a stimulous oscilating at such second frequency (f2). But what happens then with the energy of the first vibration oscilating at f1? Is it wasted? Is there no way to benefit from such energy? Can't you build on the existing vibration and apply a stimulous of frequency (f2 - f1)? Allow me a stupid improvisation for the sake of dicussion: if I arrive with my arm vibrating at (f2 - f1) and then grasp the string, will it somehow be raised to vibration level f2?

PeterDonis
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Peter, why do you insist on Bohr again?
I only mention him because you do. And I wasn't "insisting on" him, I was pointing out that we know a lot that he didn't.

I have no math knowledge to follow your comment.
Then you should not be labeling your thread as "I". That assumes undergraduate knowledge of the subject matter, which is impossible with no math knowledge.

I have found a very nice explanation of this particular analogy
Analogies are not physics. You need to learn the actual physics, not analogies.

Actually it is not a "quantum physics" issue
Then it doesn't belong in this forum.

This thread is going nowhere and is now closed.