# Thread: Quantum Wave Function sketching

1. ## Quantum Wave Function sketching

I'm trying to better understand how to draw the wavefunction given the potential without making any real calculations (French and Taylor call it "qualitative plots"). Let's say I had this potential:

I think I understand the basics of how to draw the wave function for each eigenstate, but I'm having trouble actually doing it, i.e., I think that the amplitude is larger for regions where E-V is smaller (corresponding to a lower kinetic energy), but some of the other things are just hard to see. Anyone know much about this?

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## Re: Quantum Wave Function sketching

Originally Posted by Normandy6644
I'm trying to better understand how to draw the wavefunction given the potential without making any real calculations (French and Taylor call it "qualitative plots"). Let's say I had this potential:

I think I understand the basics of how to draw the wave function for each eigenstate, but I'm having trouble actually doing it, i.e., I think that the amplitude is larger for regions where E-V is smaller (corresponding to a lower kinetic energy), but some of the other things are just hard to see. Anyone know much about this?
I just started a intro to QM, so you'll have to give me a few months. 8) Although I'm sure there is someone around here that can help you out a lot faster.

3. Didn't you tell me you're taking a class too? What book are you using?

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Originally Posted by Normandy6644
Didn't you tell me you're taking a class too?
Yep.

Originally Posted by Normandy6644
What book are you using?
Shankar's Principles of Quantum Mechanics.

5. Originally Posted by Tensor
Originally Posted by Normandy6644
Didn't you tell me you're taking a class too?
Yep.

Originally Posted by Normandy6644
What book are you using?
Shankar's Principles of Quantum Mechanics.
Oh man, getting right to the good stuff! We have a two semester sequence here, and I'm in the first one now, using French and Taylor. Next year I'll take the second one and use Griffths I believe. I also just ordered Gasiorowicz (sp?), which is supposed to be good.

6. Should this maybe get moved to GA or something? I know it isn't exactly astronomy, but I have a feeling it might get more responses in a science forum. :wink:

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Originally Posted by Normandy6644
Next year I'll take the second one and use Griffths I believe.
Hmmm. I don't see any reason to not just start with Griffiths (even if you aren't supposed to just yet); his texts are very clear and easy to understand (the only complaint about them is that they could be a titch more rigorous). Feynmann's chapters on quantum are pretty good too, especially for developing intuition after you've learned the concepts (and, well, everyone should own Feynmann anyhow).

8. Originally Posted by jnik
Originally Posted by Normandy6644
Next year I'll take the second one and use Griffths I believe.
Hmmm. I don't see any reason to not just start with Griffiths (even if you aren't supposed to just yet); his texts are very clear and easy to understand (the only complaint about them is that they could be a titch more rigorous). Feynmann's chapters on quantum are pretty good too, especially for developing intuition after you've learned the concepts (and, well, everyone should own Feynmann anyhow).
The reason we don't start with Griffiths in the physics department (engineering physics does though) is that we have a two semester sequence, and they have only one. Our first class is just an intro to QM, whereas next spring I'll take the advanced course that goes more in depth. I have the Feynman lectures, and they are really good, especially at explain concepts.

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Originally Posted by Tensor
Yep.
Have you read Feynman’s “QED The Strange Theory of Light and Matter”?

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Originally Posted by Sam5
Originally Posted by Tensor
Yep.
Have you read Feynman’s “QED The Strange Theory of Light and Matter”?
Of course. It's a good non-technical explanation for anyone interested in QED, even those without a science backround.

11. That one looked really good, but I still haven't gotten around to it. It's a good book to read on a travel day, so maybe this spring.

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Originally Posted by Tensor
Of course. It's a good non-technical explanation for anyone interested in QED, even those without a science backround.
Why do you suppose that in the book he said that Maxwell’s wave theory had to be changed, and why did Feynman refer to light waves as “particles”? Maxwell’s theory and diagram of 1873 clearly shows them as EM wave pairs that travel as “quanta” of specific energies. They are double field waves, not “particles”, and this is still the most common model of light as shown on many university websites, such as this one:

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Here’s another one. This is also Maxwell theory, and these are electric and magnetic waves, issued as individual units, wave pairs, and they are waves, not particles.

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Originally Posted by Sam5
Originally Posted by Tensor
Of course. It's a good non-technical explanation for anyone interested in QED, even those without a science backround.
Why do you suppose that in the book he said that Maxwell’s wave theory had to be changed, and why did Feynman refer to light waves as “particles”? Maxwell’s theory and diagram of 1873 clearly shows them as EM wave pairs that travel as “quanta” of specific energies. They are double field waves, not “particles”, and this is still the most common model of light as shown on many university websites, such as this one:

Shorthand. You're right that 'quantum' is the most correct word for what we're talking about, but due to force of habit, most people tend to think of particles as things that exhibit a certain set of characteristics and waves as things with a different set of characteristics. Light, like everything else on quantum scales, exhibits both particle and wave properties. As such, people tend to refer to photons, particles of light, when dealing with a situation where the particle-like properties dominate. Similarly, when dealing with a situation where the wave-like properties dominate, people tend to refer to light waves. It's just easier to simplify the model in those cases.

But, the problem is that neither is a completely accurate description. Fresnel and Young proved that light, like a wave, exhibits diffraction and interference and Maxwell's equations prove that light has the properties of an electromagnetic wave (incidentally, Maxwell didn't propose that light travelled as a quantum). You can't explain any of that by modelling light as a particle. On the other hand, the wave-theory of light can explain neither the photoelectric effect nor blackbody radiation but, as Planck and Einstein showed, you can explain them once you bring in particle properties.

That's why Feynman said Maxwell's theory needed to be changed. As good as his ideas were, they were incomplete. If light is simply an electromagnetic wave, his equations should tell you all you need to know about its behavior. The theory simply can not explain the photoelectric effect. That is, if light is a classical wave, like the ones shown in the diagrams you linked to, its energy would be related to its amplitude, basically its intensity. Now, shine your light on a piece of metal and the light will dislodge electrons. For a classical wave, the electrons would travel faster (that is, with a greater kinetic energy) if you increased the amplitude (energy) of the light. Turns out it doesn't happen. Therefore, light can not be a classical wave as Maxwell argued. That's where Planck and Einstein came in. Planck showed that light's energy was proportional to its frequency - which, again, is not a typical property for a wave. Einstein took that a step further and proposed that those corresponded to the energy of particles.

It sounds strange and, in all honesty, it IS strange. Still, the only way around this is for light to be both a wave and a particle. It's the only explanation that accounts for all of its properties. Now, I'm not sure what a photon looks like - I doubt anyone can fully visualize them - but the applets you linked to aren't actually photons. They're illustrations of Maxwell's theory of light, that is, light as a classical electromagnetic wave. They make a good approximation of a photon if they're finite. If it helps, think of those waves coming in little packets. Maybe think of a laser as sort of a machinegun firing little light-wave bullets.

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For me it was Eisberg and Resnick during my first semester of quantum and Griffiths during my second. With the latter, my prof. tried to convince us that the cat on the back cover was only sleeping.

Normandy: Anyway, my QM is more than a little rusty so take this with a grain of salt, but it sounds like you're on the right track. Unfortunately, the only other advice I can give you is to think of the potential barriers as solid walls and your waves as strings. Pluck the string as hard as you want, it can't vibrate through the wall. I realise that's a hopelessly classical analogy, but hopefully it will help some.

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Originally Posted by Sam5
Originally Posted by Tensor
Of course. It's a good non-technical explanation for anyone interested in QED, even those without a science backround.
Why do you suppose that in the book he said that Maxwell’s wave theory had to be changed, and why did Feynman refer to light waves as “particles”? Maxwell’s theory and diagram of 1873 clearly shows them as EM wave pairs that travel as “quanta” of specific energies. They are double field waves, not “particles”, and this is still the most common model of light as shown on many university websites, such as this one:

Note the bolding. It's a good non-technical explanation for anyone interested in QED, even those without a science backround

Maxwell died in 1879, before the discovery of the specific particle properties of EM radiation. All the way up to the development of QED, physicists had argued over whether light was particles or waves. During Maxwell's lifetime, most physicists argued for waves. Currently, EM radiation is though of as having the properties of both a wave and a particle. For instance, the photons spin of 1 has no exact classical counterpart.

Note to Normandy or anyone else whose interested: Some lecture notes on relativistic wave equations at this link.

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Originally Posted by Taibak
Shorthand. You're right that 'quantum' is the most correct word for what we're talking about, but due to force of habit, most people tend to think of particles as things that exhibit a certain set of characteristics and waves as things with a different set of characteristics.
Ok, very good. That’s what I suspected. I’m going to go over your post line by line today and comment on it. I’d like your opinions about what I say. I’ll give you mine and you give me yours. Let me know what you think about mine. I want to run some ideas past you and see what you think about them.

I’ve been studying this stuff for some time, working with light rays and photons since the 1950s. Photons break up silver halide molecules into free silver and halide gas. Both frequency and amplitude are involved in this process. There are two types of “energy” involved over a square surface area of film, one is related to the frequency and another is relate to the amplitude per second (the number of photons arriving per second per square cm). The silver halides in film require certain frequencies to break them up, and they require certain light amplitudes to break more of them up more rapidly over a large area of film.

Maxwell did talk about the energy level of a single wave of light. He recognized that light carried a certain amount of energy in each em wave pair. He spoke of amplitude in terms of fewer or greater wave pairs per area per second, as if one wave pair always had only one single standard amplitude. More amplitude required more waves to travel side by side, or back to back, or both, in the form of long “rays” of wave pairs. The wave pairs are what we call “photons” today. I’ve got his book right here and I’m looking at it. So tell me what you think of this interpretation of what he said.

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Originally Posted by Sam5
Originally Posted by Taibak
Shorthand. You're right that 'quantum' is the most correct word for what we're talking about, but due to force of habit, most people tend to think of particles as things that exhibit a certain set of characteristics and waves as things with a different set of characteristics.
Ok, very good. That’s what I suspected. I’m going to go over your post line by line today and comment on it. I’d like your opinions about what I say. I’ll give you mine and you give me yours. Let me know what you think about mine. I want to run some ideas past you and see what you think about them.
Sure. Why not? Like I told Normandy though, my understanding of quantum mechanics isn't all that great so there may be things I'll have to defer on and I'd much rather admit my ignorance rather than try to lead you down a wrong track.

Originally Posted by Sam5
I’ve been studying this stuff for some time, working with light rays and photons since the 1950s. Photons break up silver halide molecules into free silver and halide gas. Both frequency and amplitude are involved in this process. There are two types of “energy” involved over a square surface area of film, one is related to the frequency and another is relate to the amplitude per second (the number of photons arriving per second per square cm). The silver halides in film require certain frequencies to break them up, and they require certain light amplitudes to break more of them up more rapidly over a large area of film.
A better way of thinking about it is that each photon carries a specific amount of energy and that there are two ways you can control the total amount of energy the silver halide receives: changing the number of photons (the amplitude of the light wave) and changing the energy of each photon (changing the frequency of the light wave). Your example sounds very similar to the photoelectric effect, actually. Each photon can only react with one atom, so the more photons you use (the higher the amplitude of the light wave) the more reactions you cause. But, like you say, each photon needs a certain frequency, a certain energy, to cause the reaction to happen and if the photon doesn't have enough energy, it's not going to react.

Originally Posted by Sam5
Maxwell did talk about the energy level of a single wave of light. He recognized that light carried a certain amount of energy in each em wave pair.
Understandable. All waves do that.

Originally Posted by Sam5
He spoke of amplitude in terms of fewer or greater wave pairs per area per second, as if one wave pair always had only one single standard amplitude.
Huh? That's not amplitude. That's frequency. Different concept. Do you have a citation for this?

Originally Posted by Sam5
More amplitude required more waves to travel side by side, or back to back, or both, in the form of long “rays” of wave pairs. The wave pairs are what we call “photons” today. I’ve got his book right here and I’m looking at it. So tell me what you think of this interpretation of what he said.
To be honest, it'd help if you could provide a citation here because what you're describing is most definitely not amplitude. Amplitude is the height of each wavecrest; the number of crests that go past in a second is frequency. Related, but ultimately different concepts. For light, like any other wave, you can change one of them without changing the other (it's pretty easy to get a feel for this if you have a Slinky handy).

Also, I'm not sure what you mean by 'wave pairs.' Do you mean the electric and magnetic components of the wave? If so, let's refer to that as a single wave to avoid confusion (besides, you can't have one without the other so it really is one wave with two components). Either way, that's not necessarily a photon because that model doesn't capture the photon's particle-like properties. Alternatively, do you mean one crest followed by one trough? If so, then that's not a photon. Limiting the number of crests in a wave limits the wave's frequency (again, this is easy to show if you have a guitar or, preferably, the aforementioned Slinky) and experiments have shown that photons have no constraints on their frequencies.

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Taibak,

Ok, later I’ll make some photocopies or text copies of Maxwell’s book and post them to show you what I’m talking about. I’m getting the Maxwell stuff out of Volume 2 of “A Treatise on Electricity and Magnetism”. He also published a diagram of electric and magnetic wave pairs traveling together.

I think one reason for some of the confusion is that there are two ways of thinking and two sets of terms: classical, and QM. I think in classical terms and you tend to think in QM terms. My own experiments and observational experience over the past 45 years make sense to me in classical terms.

For example, light amplitude in photography involves more photons hitting the film per square cm all at once (ie a brighter light producing more side-by-side photons) OR more exposure time (a dim light allowed to hit a square cm of film for a longer amount of time, thus producing more back-to-back photons). I’ve never heard of any way to add amplitude to a single photon. Can you refer me to any text about how to cause a single photons to be brighter or have more amplitude?

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Originally Posted by Sam5
Taibak,

Ok, later I’ll make some photocopies or text copies of Maxwell’s book and post them to show you what I’m talking about. I’m getting the Maxwell stuff out of Volume 2 of “A Treatise on Electricity and Magnetism”. He also published a diagram of electric and magnetic wave pairs traveling together.
Or you could just quote the relevant lines. Whatever's easier. Either way, as Tensor pointed out, Maxwell's book isn't the whole story. If you want to understand what a photon is, you have to put Maxwell aside and start looking at quantum mechanics.

Originally Posted by Sam5
I think one reason for some of the confusion is that there are two ways of thinking and two sets of terms: classical, and QM. I think in classical terms and you tend to think in QM terms. My own experiments and observational experience over the past 45 years make sense to me in classical terms.
Well, I appreciate the vote of confidence but, to be honest, I usually think in classical terms. In fact, so doesn't just about everyone else. Most of the things we'll observe throughout our lifetime can be explained perfectly well using classical physics. You only need to bring in quantum mechanics when you start dealing with ridiculously small objects, such as subatomic particles.

But just because you can usually think about things in classical terms doesn't mean that you can think about everything that way. To use a fairly mundane example, you can't explain all the properties of solder without using quantum mechanics. From a classical point of view, the two wires you soldered together are conductive but the solder itself is not. To a classical electron, the solder is an inpenetrable barrier. Quantum mechanics, on the other hand, provides a mechanism (quantum tunneling) that explains how the current manages to get through there. Granted, if you're just interested in the practical side of this (that is, if all you care about is that you can solder two wires together to make a circuit) there's no reason to bother with the quantum mechanical side of it, but to understand what's actually happening at the atomic level there's no way around the theory.

Originally Posted by Sam5
For example, light amplitude in photography involves more photons hitting the film per square cm all at once (ie a brighter light producing more side-by-side photons) OR more exposure time (a dim light allowed to hit a square cm of film for a longer amount of time, thus producing more back-to-back photons). I’ve never heard of any way to add amplitude to a single photon. Can you refer me to any text about how to cause a single photons to be brighter or have more amplitude?
Not off the top of my head, no. Really, this is something that's implicit in most any quantum mechanics textbook since the authors assume that anyone reading them already knows what a wave is. The key is that each individual photon is both a particle AND a wave. As such, each individual photon, despite acting exactly like a particle in some situations, has a wavelength, frequency, and amplitude. Also, keep in mind that photons add like waves to produce one big wave. That is, where two crests meet you get a bigger crest and where a crest meets a trough you get nothing. For the photon's particle-like properties, the peaks in that larger wave are where you find the most photons. Conversely, you won't find any photons where the wave is flat.

For better or worse, this is just one of those places where common sense and intuition break down. There is no analog for the wave-particle duality in everyday life. It's not something our brains can visualize particularly easily and we're stuck describing it with terms that, in some cases, were developed centuries before this stuff was even thought of. Sooner or later, if you want to understand this, you're going to have to learn the math. There really is no other way to fully explain quantum mechanics.

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Another really important effect that requires QM to be quantised is blackbody radiation. You can derive the long wavelength part of the function without too much difficulty on classical grounds, but to defuse the UV catastrophe you have to start quantising.

22. Originally Posted by Fortis
Another really important effect that requires QM to be quantised is blackbody radiation. You can derive the long wavelength part of the function without too much difficulty on classical grounds, but to defuse the UV catastrophe you have to start quantising.
I always thought the name "Ultraviolet catastophe" was pretty cool.

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Originally Posted by Taibak
Or you could just quote the relevant lines. Whatever's easier. Either way, as Tensor pointed out, Maxwell's book isn't the whole story. If you want to understand what a photon is, you have to put Maxwell aside and start looking at quantum mechanics.
Well, as long as university websites and books still show drawings and animations of the 1873 Maxwell electric and magnetic wave, I’ll continue to consider it as a good model. You might want to read a couple of Maxwell books yourself. In Volume 1, 1873, he was the first guy to suggest that the earth go on a fundamental atomic clock time standard.

And when Feynman finally has to show a drawing of a photon in his book, and he uses a little wavy line similar to the ones in the Maxwell model, rather than using a little round marble type “particle” drawing, I’ll not be too quick to give up the Maxwell model.

Originally Posted by Taibak
But just because you can usually think about things in classical terms doesn't mean that you can think about everything that way. To use a fairly mundane example, you can't explain all the properties of solder without using quantum mechanics. From a classical point of view, the two wires you soldered together are conductive but the solder itself is not. To a classical electron, the solder is an inpenetrable barrier. Quantum mechanics, on the other hand, provides a mechanism (quantum tunneling) that explains how the current manages to get through there. Granted, if you're just interested in the practical side of this (that is, if all you care about is that you can solder two wires together to make a circuit) there's no reason to bother with the quantum mechanical side of it, but to understand what's actually happening at the atomic level there's no way around the theory.
I have a roll of Radio Shack solder right here, and I find that a few-inch length of it conducts electricity just as well as a copper wire. I’ve never heard of any scientist say it blocks electricity. Is that some new idea or what?

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Originally Posted by Taibak
For better or worse, this is just one of those places where common sense and intuition break down. There is no analog for the wave-particle duality in everyday life. It's not something our brains can visualize particularly easily and we're stuck describing it with terms that, in some cases, were developed centuries before this stuff was even thought of.
Sure there is. You can knock over a magnet without touching it, by moving another magnet near it. It’s the moving field in between the two magnets that carries the force. You don’t have to knock the one over with a “particle” when a field wave can act like a particle and knock it over. Anyway, you hear by means of field waves. No “particle” ever hits your ear drum or your cilia, only fields. That’s the way you see too.

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Originally Posted by Normandy6644
I always thought the name "Ultraviolet catastophe" was pretty cool.
I thought that was the name of a Seattle rock band.

26. Originally Posted by Sam5
Originally Posted by Normandy6644
I always thought the name "Ultraviolet catastophe" was pretty cool.
I thought that was the name of a Seattle rock band.
That would be really cool! I wish I'd thought of that for my band.

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Originally Posted by Sam5
Originally Posted by Taibak
For better or worse, this is just one of those places where common sense and intuition break down. There is no analog for the wave-particle duality in everyday life. It's not something our brains can visualize particularly easily and we're stuck describing it with terms that, in some cases, were developed centuries before this stuff was even thought of.
Sure there is. You can knock over a magnet without touching it, by moving another magnet near it. It’s the moving field in between the two magnets that carries the force. You don’t have to knock the one over with a “particle” when a field wave can act like a particle and knock it over. Anyway, you hear by means of field waves. No “particle” ever hits your ear drum or your cilia, only fields. That’s the way you see too.
How is the classical wave quantised? Quantisation of the photon energy in units of h.nu doesn't appear in the classical Maxwell's equations.

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Originally Posted by Sam5
Originally Posted by Taibak
Or you could just quote the relevant lines. Whatever's easier. Either way, as Tensor pointed out, Maxwell's book isn't the whole story. If you want to understand what a photon is, you have to put Maxwell aside and start looking at quantum mechanics.
Well, as long as university websites and books still show drawings and animations of the 1873 Maxwell electric and magnetic wave, I’ll continue to consider it as a good model. You might want to read a couple of Maxwell books yourself. In Volume 1, 1873, he was the first guy to suggest that the earth go on a fundamental atomic clock time standard.
Maxwell's model is still taught because it does an excellent job at explaining light in the classical realm. If, for instance, you wanted to design a radio transmitter you would use Maxwell - the theory works fine for that. It's a nice, convenient, simple model when you're trying to work on, well, rather a lot. But - and this is a BIG but - Maxwell won't give you a complete understanding of what a photon is. Simply put: his theory is completely incapable of explaining the particle-like properties of light. It can't explain blackbody radiation. It can't explain the photoelectric effect. It can't explain Compton scattering.

Originally Posted by Sam5
And when Feynman finally has to show a drawing of a photon in his book, and he uses a little wavy line similar to the ones in the Maxwell model, rather than using a little round marble type “particle” drawing, I’ll not be too quick to give up the Maxwell model.
But that's the key problem: you need to incorporate BOTH the marble and the wavy line. The usual solution is to draw a football-shaped particle enclosing a wavy line. It's an imperfect drawing, but it works reasonably well for most purposes.

Originally Posted by Taibak
But just because you can usually think about things in classical terms doesn't mean that you can think about everything that way. To use a fairly mundane example, you can't explain all the properties of solder without using quantum mechanics. From a classical point of view, the two wires you soldered together are conductive but the solder itself is not. To a classical electron, the solder is an inpenetrable barrier. Quantum mechanics, on the other hand, provides a mechanism (quantum tunneling) that explains how the current manages to get through there. Granted, if you're just interested in the practical side of this (that is, if all you care about is that you can solder two wires together to make a circuit) there's no reason to bother with the quantum mechanical side of it, but to understand what's actually happening at the atomic level there's no way around the theory.
Originally Posted by Sam5
I have a roll of Radio Shack solder right here, and I find that a few-inch length of it conducts electricity just as well as a copper wire. I’ve never heard of any scientist say it blocks electricity. Is that some new idea or what?
I might be misremembering the details, actually. If my memory serves me right, both the wire and solder are conductive but the interface between them acts as a potential barrier through which the electrons tunnel. Let me do some digging here, see if I can corroborate this.

Or, to switch to a better example, you can't explain how a transistor works without quantum tunneling.

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Originally Posted by Sam5
Originally Posted by Taibak
For better or worse, this is just one of those places where common sense and intuition break down. There is no analog for the wave-particle duality in everyday life. It's not something our brains can visualize particularly easily and we're stuck describing it with terms that, in some cases, were developed centuries before this stuff was even thought of.
Sure there is. You can knock over a magnet without touching it, by moving another magnet near it. It’s the moving field in between the two magnets that carries the force. You don’t have to knock the one over with a “particle” when a field wave can act like a particle and knock it over.
Bad example. You're right, we can model magnets that way, but you've chosen an example that, in the classical realm, involves neither waves nor particles. The magnetic fields are constant, nothing's oscillating, therefore there is no wave. Similarly, in Maxwell's theory, there is no need for particles.

Originally Posted by Sam5
Anyway, you hear by means of field waves. No “particle” ever hits your ear drum or your cilia, only fields. That’s the way you see too.
Better example, but you're misunderstanding how sound works. In this case, there is no field. A sound wave, by definition, is caused by particles vibrating back and forth in the direction of the wave. As such, you don't hear anything unless those particles slam into the appropriate organs. Either way, sound is definitely a wave. There is no 'soundon' that accompanies the wave.

Regardless, I stand by what I said. There are no classical analogs of the wave-particle duality. As far as Newton and Maxwell are concerned, something is either a wave or a particle, never both. A baseball is a particle. A plucked guitar string carries a wave. There is no 'baseball wave' or 'guitarstringon' particle. Once you get into quantum mechanics, the wave-particle duality is inescapable. Classically, electrons are particles, but they also show interference patterns the same way waves do - electrons are also waves. Light exhibits interference patterns, so it's a wave, but if that's the whole story you can't explain the photoelectric effect. There's no way around this: on a quantum mechanical level things are both particles AND waves.

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Originally Posted by Taibak
You're right, we can model magnets that way, but you've chosen an example that, in the classical realm, involves neither waves nor particles. The magnetic fields are constant, nothing's oscillating, therefore there is no wave.
Yes. I’m moving the magnet and the field. That is the field “wave”. One wave. Waves don’t necessarily have to oscillate. One wave of a strong field will knock over the other magnet. We can cause other effects by oscillating the magnetic fields at various rates.

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