# Quantum spin, wavelength and frequency

• I
• Peter J Carroll

#### Peter J Carroll

TL;DR Summary
Does the quantum spin of a particle (its intrinsic angular momentum) have anything to do with its wavelength and frequency?
Greetings, I'm new here, I have an interest in the nature of reality, and a question.

Does the quantum spin of a particle (its intrinsic angular momentum) have anything to do with its wavelength and frequency?

One of the experts on Quora said no, and I cannot find anything about it on the web, however a simple relationship has occurred to me which I attach below, does it make sense? Regards, Pete.

##h/4\pi=\lambda mc=vmc/f##

does it make sense?
No, it does not.

Peter J Carroll
Just because you can find Planck's constant in some expression and isolate it doesn't mean the variables on the other side of the equals sign relate to spin. If you study quantum mechanics, you will find that Planck's constant appears... well... everywhere.

Peter J Carroll
Does the quantum spin of a particle (its intrinsic angular momentum) have anything to do with its wavelength and frequency?
Obviously not, since you can have different particles with the same spin but different wavelength (momentum) and frequency (energy).

Peter J Carroll, PeroK, Vanadium 50 and 1 other person
For a massive particle, in both non-relativistic and special relativistic QT the spin (0, 1/2, 1,...) tells you how the single-particle states in the rest frame of the particle transform under rotations. So spin is defined independently of the momentum of the particle and thus the wavelength of the corresponding one-particle mode.

The details are pretty subtle representation theory of the Galilei or Poincare groups and can be found nicely in

L. E. Ballentine, Quantum Mechanics, World Scientific,
Singapore, New Jersey, London, Hong Kong (1998).

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles, Springer, Wien (2001) (for relativistic local QFT)

or for the most general case of arbitrary spin (for relativistic QT):

S. Weinberg, The Quantum Theory of Fields, vol. 1, Cambridge University Press (1995).

Peter J Carroll
Thank you for your replies and the links from vanhees71 which I will seek out.

As Peter Donis notes “you can have different particles with the same spin but different wavelength (momentum) and frequency (energy).”

Indeed, and this simple formula

##h/4\pi = \lambda mc = vmc/f##

applies only to spin one half fermions such as electrons. Yet it shows the dependency of the reduced Compton wavelength on mass, and the dependency of frequency on momentum, and relates them all quantitatively to the accepted value of their quantised angular momentum.

Quantum physics seems full of accurate mathematical descriptions, but such descriptions often do not provide explanations.

So, can we dismiss this as a purely mathematical coincidence, or does it suggest an intriguing connection between certain properties of quanta that could reveal mechanisms and explanations?

Regards, Pete.

weirdoguy and PeroK
Quantum physics seems full of accurate mathematical descriptions, but such descriptions often do not provide explanations.
Like the rest of physics from Newton's laws to the present day. Where's the explanation for ##F = ma## or ##F = \frac{GMm}{r^2}##?

The mathematics is the explanation.

Peter J Carroll and vanhees71
this simple formula
(1) Does not even contain the spin;

(2) Is not the formula for "the wavelength" of the particle, it's the formula for the Compton wavelength of the particle, which does not depend on its actual wavelength (momentum) or frequency (energy).

the dependency of frequency on momentum
Is given by the simple equation ##E^2 - p^2 = m^2## (in natural units where ##c =1##), plus the QM definitions of ##E## and ##p## in terms of frequency and wavelength and Planck's constant. And that works for particles of any spin, so it has nothing to do with spin.

Peter J Carroll and vanhees71
PeroK, your example of Newton’s laws nicely illustrates the difference between a description and an explanation. F = ma acts as both. It works as a tautology that fits, (for moderate velocities) by defining each term in terms of the others.

However, Newton’s formula for gravity works only as a description that implies an incomprehensible action at a distance. This troubled Newton deeply, and it troubled many that followed him. The description worked well but it didn’t explain how. Eventually Einstein realized that matter curves spacetime and spacetime makes matter move. This provides a high level tautology that fits even better, and which acts as both a description and an explanation.

So, I don’t always accept that ‘The mathematics is the explanation’, particularly in the quantum realm.

Thank you all for your feedback on this topic.

Regards, Pete.

vanhees71
Well, in the quantum realm the only way to express the results is mathematics. There's no other way to talk in a concise enough way about how to describe matter on the most fundamental level.

Peter J Carroll
Indeed, mathematics remains invaluable, but if science depends on a dialogue between mathematics and concepts, then quantum physics becomes particularly tricky when we have difficulties finding concepts to fit the maths.

Thanks for your responses, no further questions at present. Regards, Pete.

weirdoguy
Indeed, mathematics remains invaluable, but if science depends on a dialogue between mathematics and concepts, then quantum physics becomes particularly tricky when we have difficulties finding concepts to fit the maths.

Thanks for your responses, no further questions at present. Regards, Pete.
Who's "we" in this context? I don't find mathematics unduly tricky. I like using mathematics. That's the way I think.

DrChinese, vanhees71 and weirdoguy
Indeed, mathematics remains invaluable, but if science depends on a dialogue between mathematics and concepts, then quantum physics becomes particularly tricky when we have difficulties finding concepts to fit the maths.

As PeroK says, "tricky" is in the mind of the beholder. Your concept of what physics should explain reflects a Pollyanna viewpoint of science. The first job of science is to describe. "Why" questions, especially in quantum physics, tend to go in circles. Presumably, there is no answer to why the laws of physics are as they are. Why is c about equal to 300,000 km/sec, rather than 350,000 km/sec? Science is about finding useful descriptions of patterns and pattern exceptions. If an answer appears to a "why" question, and provides something useful to our knowledge in the process, then that is a true advance.

Lord Jestocost and vanhees71
I don't find mathematics unduly tricky.
Work some more combinatorics problems. You'll come around.

vanhees71
Work some more combinatorics problems. You'll come around.
They are my speciality!

Well as you ask PeroK, I speak for myself but perhaps also for a lot of people who wonder what the maths implies in terms of concepts. For example, does the intrinsic angular momentum ‘spin’ of a fermion involve the concept of anything moving or storing energy in circular motion, or is it just an arbitrarily chosen word for something that submits to a mathematical description only? Is anything expressed in words or images about quantum physics basically meaningless?

Regards, Pete.

Eventually Einstein realized that matter curves spacetime and spacetime makes matter move. This provides a high level tautology that fits even better, and which acts as both a description and an explanation.
That’s just another (better, which is what makes the exercise worthwhile) description still without explanation. Why does matter curve spacetime? Why are the paths of massive objects not subject to external forces spacetime geodesics? The theory says nothing about these questions; like Newton’s theory it “worked well but it didn’t explain how”.
Is anything expressed in words or images about quantum physics basically meaningless?
Not necessarily meaningless, but likely to be misleading if taken seriously. Natural language evolved to describe the classical phenomena that surround us so works pretty well for describing these. It’s not as well-suited for describing quantum behavior: for example “the particle goes through both slits” and “the particle goes through neither slit” are both defensible natural language descriptions of the double-slit experiment.

vanhees71 and Peter J Carroll
Hi Nugatory, for me, GR does explain ‘Why does matter curve spacetime? Why are the paths of massive objects not subject to external forces spacetime geodesics?

If it means that Matter Is Curved Spacetime, and that Geodesics Are the true shape of Spacetime in which matter moves.

I can understand the double slit experiment in non-contradictory natural language if the wave passes through both slits but the particle passes through only one of them, or more precisely in terms of TIQM, if retarded waves pass through both slits but an advanced wave returns through only one of them, completing the particle exchange, if the particle Is the result of a combined retarded and advanced wave.

Regards, Pete.