© Matt Strassler [September 1, 2012]
This is article 5 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
Reminder: The Quantum Ball on a Spring
Back in the first article in this series we studied the ball (of mass M) on a spring (of strength K), and recalled that its oscillations have
- a motion formula: z(t) = z0 + A cos [ 2 π ν t ]
- energy: E = 2 π2 ν2 A2 M
- an equation of motion: d2z/dt2 = – K/M (z – z0)
where the equation of motion forces ν = √ K/M / 2π, but allows the amplitude A to be of any positive size. Then in the second article, we saw that what quantum mechanics does to the oscillations (among many more subtle things) is effectively restrict the amplitude — it can’t be just anything. Instead the amplitude is quantized; it has to take one of an infinite number of discrete values.
- A = (1/2 π) √ 2 n h / ν M
where n=0, or 1, or 2, or 3, or 44, or any integer greater than or equal to zero. In particular, A can be as small as (1/2 π) √ 2 h / ν M , but it can’t be anything smaller, except zero. We say that n is the number of quanta of oscillation in the ball’s motion. The energy of the ball is also now quantized:
- E = (n+1/2) h ν
The most important fact, for us, is that the energy required to add one quantum of oscillation to the ball’s motion is h ν; we may say that each quantum carries energy hν.
The Quantum Wave
For waves, it’s basically the same; we have, for a wave of frequency ν and wavelength λ oscillating with amplitude A around an equilibrium position Z0,
- a motion formula: Z(x,t) = Z0 + A cos (2π [ν t – x/λ])
- the energy per wavelength: 2 π2 ν2 A2 Jλ
(where Jλ is a constant that depends only on, say, the rope, if these are waves on a rope), and several possible equations of motion, of which we chose two to study
- Class 0: d2Z/dt2 – cw2 d2Z/dx2 = 0
- Class 1: d2Z/dt2 – cw2 d2Z/dx2 = – (2 π μ)2 (Z-Z0)
Again quantum mechanics restricts the amplitude A, which we might have thought could be of any size we liked, to discrete values. Just as for an oscillation of a spring,
- a (simple) wave of a given frequency and wavelength is made from n quanta
- the allowed values of the amplitude A are proportional to √ n ;
- the allowed values of the energy E are proportional to (n+1/2)
More precisely, just as for a ball on a spring
- the allowed values of the energy are E = (n+1/2) h ν
- each quantum of a wave carries energy h ν
The formula for A is a bit more complicated, because here we have to know how long the wave is, and an exact formula would be messy, so let me just write one that gets the right idea. We got most of our formulas studying waves that are infinite, but any real wave in nature has a finite length. If the wave is roughly of length L, and therefore has L/λ crests, then the amplitude is approximately
- A = (1/2 π) √ 2 n h λ / ν L Jλ
which is proportional to √ n h / ν just as for the spring, but depends on L; a longer wave has a smaller amplitude, arranged just such that each quantum of the wave always has energy h ν.
And that’s it, as illustrated in the figure below (which you may wish to compare with Figures 1 and 3 in the article on the Quantum Ball on a Spring.)
One Implication
What does this mean for our Class 0 and Class 1 waves?
Since waves that satisfy an equation of Class 0 can have any frequency, they can correspondingly have any energy. Even with a tiny amount of energy ε, you can always make a single quantum of a Class 0 wave with frequency ν = ε/h. For such small energy, that quantum wave will have very low frequency and very long wavelength, but it can exist.
Waves that satisfy an equation of Class 1 are different. Since there is a minimum frequency νmin = μ that such waves can have, there is a quantum of lowest energy
- Emin = h νmin = h μ
If your tiny amount of energy ε is less than this, you cannot make a quantum of this sort of wave. Quanta of Class 1 waves with finite wavelength and larger frequency all have E ≥ h μ .
To Sum Up
Before we account for quantum mechanics, the amplitude of a wave, just like the amplitude of a ball on a spring, can vary continuously; you can make it as large or as small as you want. But quantum mechanics implies there’s a smallest possible non-zero amplitude for a wave, just as for an oscillation of a ball on a spring; and in general only discrete values of the amplitude are allowed. The allowed amplitudes are such that for both an oscillating ball on a spring and a wave of any class with a definite frequency ν
- to add a single quantum of oscillation requires energy h ν
- with n quanta of oscillation, the energy of oscillation is (n+1/2) h ν
Now it’s time to apply this knowledge to fields, and observe when and how the quanta of the waves in these fields could be interpreted as what we call the “particles” of nature.
35 Responses
So if this class 1 wave equation is similar in nature to the Higgs field, and the class 1 wave equation implies this cutoff frequency, does that mean that massless particles are somehow above that frequency? Like, are you saying that the particles that do not interact with this Higgs field are somehow above that cutoff frequency? And if so, how are you coming up with a specific frequency for each particle? Are you thinking of something like a Compton wavelength? Or is that way off?
I’m not understanding your questions, so I think there’s something mixed up in your thinking at the moment. Can you try to ask one of your questions more precisely? Just to throw out some facts that may help: A particle with mass can have any frequency above its minimum, which is its mass times c^2/h; the higher the frequency, the higher the particle’s momentum and energy. The minimum frequency occurs when the particle isn’t moving, so it has no momentum and has its minimal possible energy; this stationary particle is then a standing wave, vaguely like a vibrating guitar string, so we identify this minimum frequency as a resonance frequency. Meanwhile, a massless particle can have any frequency; the minimum frequency is zero. So massless particles can have frequencies that extend *below* the frequencies of ones with mass — which is why I’m not following your question.
Thanks for the reply. Yes. I’m trying to understand the relation between this class 1 wave equation, its solutions, and the properties ascribed to this higgs field.
If indeed you are saying that this equation describes something much like this higgs field, then I don’t quite see how it would have the same behavior as the the higgs field is said to have.
When I try to solve this equation, I get periodic solutions (exp(iwt)) above a certain frequency, and I get decaying solutions (exp(-wt)) below that frequency. So I am guessing that you are saying that that is a characteristic of this higgs field- it would be characterized by a cutoff frequency, above which waves would propagate in space, and below which, they would decay in space.
So my first thought was that you were saying that the massive particle is like the solution that decays in space, and the massless particle is like the solution that propagates in space.
But that did not make sense to me, for the reason you state- massless particles have no minimum frequency, but massive particles do.
So I think you must be relating this class 1 wave equation to the Higgs field in some other way, that I do not understand correctly, or else you are not relating it to this higgs field, at all, and I misunderstood, or possibly, I am not even solving the equation correctly. Which is to?
I also did not quite understand the idea of the coupling coefficients. I had assumed that if the prediction of the Higgs field was correct, that it would have to also predict a single coupling coefficient that applied to everything the same way. If the coupling coefficients have to be imposed, in order to match experiment, and the coefficients have to be different for each type of particle, doesn’t that discredit the theory, to a great extent? It would be like introducing a theory of gravity that required a different coefficient for every person that stepped on the scale, would it not?
It seems you are mixing up the Higgs field’s own equations with the effect of the Higgs field on other fields’ equations. (Which is why you can’t follow the pointt about the coupling coefficients.)
All the field’s equations are more complicated than the ones I have written down, because in addition to the simple forms that I have written in which each term contains a field once, there can be terms that include more than one field. For understanding simple ripples, those extra terms can mostly be dropped. But for understanding why the Higgs field has a non-zero value in spacetime and how it gives mass to other fields, you need the complications described in the mathy discussion in https://profmattstrassler.com/articles-and-posts/particle-physics-basics/how-the-higgs-field-works-with-math/1-the-basic-idea/ and https://profmattstrassler.com/articles-and-posts/particle-physics-basics/how-the-higgs-field-works-with-math/2-why-the-higgs-field-is-non-zero-on-average/, both of which involve more complicated equations than the simple Class 0 and Class 1 of the article here.
Thanks. This is good.
I think I was imagining that this minimum frequency was related to massive particles in a different way than what you meant. I was imagining that the massive particle was the one that did not radiate, in a field described by the class 1 equation. And the massless particle was the one that did radiate. I thought that because massless particles, by nature, radiate away, at speed c, and massive particles are different in the sense that are are localized.
But it sounds like you are saying something different- that the minimum frequency imposes a minimum energy that such a wave can hold, and that that minimum energy is equivalent to its rest mass. And I think you are even saying that the restoring force in the equation is related in a simple way to the rest mass and the compton wavelength.
Is that right?
Yes. That’s correct. E=hf for any wavicle, moving or not. Meanwhile E>mc^2 in general, with E=mc^2 for a standing wavicle, also known as a stationary particle. And the minimum f is indeed related to sort of “stiffness” of the field, which involves the restoring force in the equation. Finally, the Higgs field is responsible for the restoring force for most of the known fields (other than the Higgs field itself, which is more complicated.)
To make this point without equations was the main goal of my recent popular book on this subject.
OK, thanks for the help, I fell like I’m getting somewhere now. That opens up a bunch of other questions, but I’ll have to come back to them later.
hey i have a question what happens if a trough is smaller than a crest on a any electromagnetic wave.
Professor Strassler ive been looking for a constantly updating and exeedingly current view on theoretical physics (something i pursue with pure tenacity) and ive found it here. thank you, i can further improve my understanding of these sciences.
Prof Strassler,
What is the equation that must be solved to determine the energy of a wave (ripple) in a field?
And, what are the constraints or boundary conditions on this equation such that its solution forces the possible values of the energy to be quantized?
I believe you can just add up or integrate all the squares of the amplitude, everywhere, to get the total energy.
Is “matter wave” transverse or longitudinal wave? or neither (its amplitude has no direction?)?
Be interesting to see how you merge into theoretic.:)
A simple string conversion perhaps?
Best,
Try this for printing and paste within html language at end of post.
This was a quick scan so there maybe others- http://www.printfriendly.com/button
Thanks — will try that.
Print: try the following, go to the top of the page and click on the right side of the mouse and scroll to “print” Cannot control whether this works because my printer needs to be replaced.
Very interesting. Thank you.
Thanks for the great series of articles, and thanks for not shying away from the mathematics either. Looking forward to the next one!
* the energy required to add one quantum of oscillation to the quantum ball’s motion is hv.
* to add a single quantum of oscillation to the quantum wave also requires energy hv.
My question is: do the quantum ball and the quantum wave have the same inertia?
Even two quantum balls with the same frequency v, and therefore the same energy per quantum, need not have the same mass and momentum. If I double the mass M of the ball and double the strength of the spring (the spring constant K), the energy per quantum remains the same. So the answer to your question (“do a quantum ball and a quantum wave that have the same frequency v have the same inertia”) is no.
Hello Professor Strassler. I may be humping in here a few years late, but I needed to point out that you continue to refer to the spring constant, K, as the strength. Technically it is actually the “stiffness of the spring”; ten force required to produce a unit extension
I also tried to print and it doesn’t work properly. I would love to have the wave articles on paper right in front of me. I’m just a layperson who is fascinated by particle physics but my math skills are poor. Will need to chew through those equations a few more times. Your articles, Professor Strassler, have all been very valuable and helpful for me in order to understand how nature works. Thank you very much.
good article, but not able to print the pages either to a printer or pdf file without truncating part of the page.
There are all kinds of button on this page but no print button.
Most (if not all) of other physics blogs have a “print” button or the page can be printed properly.
Hope this can be fixed.
Really not sure about the cause here or even where to look for a solution. Any readers with suggestions?
Hi Matt, could you please help me clarify some confusion that I have in my mind? How do you “picture” the quanta of a wave? When you say quanta I immediately think of photons (quantum of light), and I imagine them as something separated by space and time. I can understand the properties of one quanta of wave, i.e. it has the energy h ν and increases the amplitude (according to the formula you mentioned), but I cannot quite “picture” it.
I understand your question: let me think about whether to insert the answer into the current article, or into the later one on particles. I should have foreseen it.
I am trying to make a comparison between the quantum ball and a photon on the one side and between the quantum wave and a light wave on the other side.
That not (as stated) the right comparison to make. The right one is that
* a quantum of oscillation for the quantum ball is to the oscillation of a classical ball
as
* a photon (a quantum of oscillation) for the quantum electric/magnetic field is to the light wave of the classical electric/magnetic field.
We’ll get to this in a few days. Meanwhile, I still don’t see where inertia is hiding in your thinking.
Light seems to have inertia.
Light has energy and it has momentum, though I haven’t explained that yet. Still trying to understand the source and nature of your question.
Both the quantum ball and the quantum wave have the same inertia?
Hmmm. Where did “inertia” come in to the discussion? Could you clarify the source of your question?