# B Is angular momentum of a quantum object, metaphorical?

1. Oct 21, 2016

### bland

OK, I understand that 'spin' is just an unfortunate name some someone gave to a particular quantum property and that particles do not actually spin. I also accept that I may never be able to understand what spin is without also understanding the mathematics involved.

However, I am also told that 'spin' is an adequate analogy for the lay person because it gives rise to another quantised property of quantum objects called 'angular momentum'. This is what I would like to clear up or understand if possible. Is the 'angular momentum' of a particle also a metaphor for a particular mathematical property, or is a particle with angular momentum, able to exhibit a force that really is angular momentum.

I have read thread no. 886279 which discussed the divorcing of angular momentum from spin, but I'm hoping to get a clearer picture.

2. Oct 21, 2016

### Staff: Mentor

What do you mean by "a force that really is angular momentum"?

3. Oct 22, 2016

### bland

What I meant was, and correct me if I'm wrong here but I am given to understand that Newton's f=ma equation can be applied to electrons as if they were little billiard balls if they move along a wire faster i.e. mass times acceleration then they have more force, that is they hit other atoms in the wire harder causing them to vibrate faster. So if an electron had angular momentum that momentum could manifest as a force if the electron was a billiard ball and the angular momentum was harnessed, stop me if I'm talking rubbish. That is what I meant about trying to figure out if the 'angular momentum' of a quantum object is real as opposed to the metaphorical spin of an electron. That is can the angular momentum of an electron be harnessed to do work, in the same way that a classical object with angular momentum could do work. And if so how would this angular momentum actually be used to do work. Or is angular momentum just a metaphorical term and it's not an intrinsic property that is in any way analogous to classical angular momentum, discounting of course that it is quantised.

4. Oct 22, 2016

### bhobba

As just about any QM book explains if you apply the quantum rules to the equations of a classical spinning particle you get quantum spin - see for example page 142 of Dirac

Its not metaphysics - its in fact deeply related to rotational symmetry but that is an advanced area - however its vital for understanding. See two books - Landau - Mechanics and Chapter 3 Ballentine.

After you will have a deep appreciation of one of, if not the greatest discovery of modern physics, it's way way up there. Its deep, profound and well worth your while coming to grips with it.

After it will be all put together by Noether Beautiful theorem (it shocked Einstein - its that deep and profound):
https://www.amazon.com/Noethers-Wonderful-Theorem-Dwight-Neuenschwander/dp/0801896940

Philosophers rarely if ever talk about it, but physicists/mathematicians when exposed to it often (and correctly) sit in shocked awe.

Thanks
Bill

Last edited by a moderator: May 8, 2017
5. Oct 22, 2016

### vanhees71

Yes, if you want to really understand physics, you cannot avoid group theory. Ironically Pauli called it "die Gruppenpest", when more and more mathematicians and physicists used group-theoretical methods in quantum theory (Weyl, van der Waerden, Wigner). Later he himself became a master in using these methods with profound findings (spin-statistiscs and CPT theorem).

In general you can say, that physics becomes the more difficult the less you are willing to learn the adequate math. I don't understand why, but often math is taken as some "annoying evil" in teaching physics. However, the contrary is true! It's great fun to learn math together with its application in physics!

In fact in highschool we learnt about the Schrödinger equation, and also about the momentum operator being realized as $\hat{p}_x=-\mathrm{i} \partial_x$, and I asked my physics teacher why this is the right operator describing momentum (it was derived a bit handwavy using the usual arguments a la Einstein and de Broglie). Since I've self-learnt a lot of math beyond the highschool stuff, she gave me a theoretical-physics book (Weizel, "Theoretische Physik"; a somewhat older but great German textbook in two big volumes). There I learnt about "symmetry". It took me a great effort to understand it, but then it really made "click" :-)).

6. Oct 22, 2016

### bhobba

Last edited by a moderator: May 8, 2017
7. Oct 22, 2016

### Staff: Mentor

Only as an approximation that works under certain very restricted circumstances. (I'm not even sure electrons flowing in a wire is one of them--do you have any references?)

I'm not aware of any scenarios where this is useful for single electrons. But the angular momentum of electrons certainly contributes to the angular momentum of larger objects that contain electrons.

For any classical object with angular momentum, part of that angular momentum is coming from the angular momentum of the electrons inside it. In that sense, certainly the angular momentum of electrons can be harnessed to do work.

For single electrons, their intrinsic angular momentum can certainly be harnessed to do work on them--that's how the direction of their angular momentum is measured, by using the magnetic moment associated with their intrinsic angular momentum to deflect them (for example, look up the Stern-Gerlach experiment). But trying to view electrons as little billard balls has, as above, very limited usefulness.

8. Oct 22, 2016

### Staff: Mentor

There is experimental evidence that the intrinsic angular momentum ("spin") of electrons contributes to the total macroscopic angular momentum of a magnetized object. I first read about this in the Feynman Lectures on Physics many years ago:

http://www.feynmanlectures.info/docroot/II_37.html#Ch37-F3

See Fig. 37-3 and the paragraph preceding it, at the link above. This is known as the Einstein-de Haas effect. It's analogous to a common introductory physics lecture demonstration in which someone sits on an initially stationary but rotatable stool holding a spinning bicycle wheel with its axis aligned vertically. If the wheel spins clockwise as seen from above, and the person flips the wheel over so it is now spinning counterclockwise, the person and stool start to rotate clockwise so as to conserve the total angular momentum of the wheel + person + stool.

When Einstein and de Haas studied this in the mid 1910s, electron spin was unknown (along with the rest of quantum mechanics as we know it!), so they interpreted this phenomenon in terms of "bound currents" associated with magnetization (polarization of magnetic dipoles) in classical electrodynamics. Now we interpret it in terms of electron spin.

9. Oct 22, 2016

### secur

It seems to me the answer is "no". When a classical object with angular momentum is used to do work, it must lose some of its angular momentum, transferring it to the object upon which the work is done. But spin can't be lost, or reduced. Quantity of an electron's spin angular momentum is constant (of course the direction can change). Therefore it can't be transferred to anything else, therefore it can't do work. In the examples above (e.g. Stern-Gerlach) the work is being done on the electron by the externally generated magnetic field. The spin is the "handle" (you might say) by which the field "grabs" the electron. But the spin angular momentum itself is not doing any work.

Note - considering your original question - that doesn't mean it's "metaphorical"! It's not.

10. Oct 24, 2016

### bland

I really appreciate the thought put into the replies. I am satisfied that my question has been adequately answered. I will not pretend to myself that I can learn the math involved, I've tried but I do my best. But while I struggle with the math, I try to make up for it by understanding the historical context and human stories concerning quantum physics and science in general. It's funny how many brave people chickened out, like Dirac not outright declaring that anti matter must exist, or more famously and curiously Einstein not trusting his general relatively to predict the expansion, or even the irrepressible Pauli, being hesitant about the neutrino.

That's a great answer, thanks. I have listened many times to the Feynman Lectures where he goes into the Stern-Gerlach experiment, and he refers to devices as Stern-Gerlach 'apparatii', which is kinda cute.

I have run into Noether's work many times in various lectures on symmetry and also the history of science, and even though I won't be able to understand the math, even I am astounded at the relationship between the conservation laws and symmetry. But I will have a look at one of the biographical books on her lower down on the Amazon page.

Pauli called it "die Gruppenpest", I did actually lol at that. What about when someone maybe Born, said Pauli had to be at lectures by 10am, he said he didn't think he could stay up that late! That's a nice story you told and it makes me want to put more effort in.

I've just heard it mentioned so many times as an explanation of why a wire glows when it resists electrons flowing through it. I have found the lectures on classical mechanics of Steven Pollock useful. But I feel that I have a better understanding of the angular momentum that I sought after reading this thread.

That's a fantastic resource you have linked to and I see that I can view all the other pages too, thanks.

11. Oct 24, 2016

### strangerep

Wasn't that Schrodinger?

12. Oct 25, 2016

### Jimster41

this bent my head. I would appreciate some clarification - In collision experiments don't particles with some spin get busted up into particles with different (but constrained) collections of spin?

When you bust an particle up into smaller/other particles is spin perfectly conserved? Does some portion always have to go to entropy? Is that portion always constrained to be of a minimum size (a valid group member)?

Last edited: Oct 25, 2016
13. Oct 25, 2016

### Staff: Mentor

Total angular momentum is perfectly conserved. In the kind of experiments you are talking about, the only angular momentum present is the intrinsic spins of the particles, so yes, the sum of all the incoming spins is equal to the sum of all the outgoing spins. I believe there are other scenarios where intrinsic spin can be exchanged with orbital angular momentum, so intrinsic spin by itself is not conserved (but total angular momentum, the sum of intrinsic spin and orbital angular momentum, is).

14. Oct 25, 2016

### secur

Spin is always conserved in collisions, decays, or anything else. Yes, "particles with some spin get busted up into particles with different (but constrained) collections of spin" but the total must always be equal. No, none is lost to entropy.

Dirac says, in "Principles of Quantum Mechanics" 4th edition sec. 70 p 266, "The spin angular momentum does not give rise to any potential energy". Therefore, I figure, it can't do work. See my previous post for more explanation.

[EDIT] Didn't see above post until now. I didn't know there are scenarios where intrinsic spin is not conserved - that would make seem to make a difference - wouldn't that allow spin to do work? Maybe it depends on the (somewhat vague) definition of the word "work". Usually in QM we talk only about different forms of "energy".

Last edited: Oct 25, 2016
15. Oct 25, 2016

### Staff: Mentor

I might have been wrong about that; I thought I remembered reading about such situations a while back, but I haven't been able to find any references.

16. Oct 25, 2016

### Jimster41

Some dumb questions:
If intrinsic spin is always perfectly conserved where does entropy come in?

Assuming the spin_in = spin_out + angular_momentum_out case:
So the curving paths of the particles resulting from the collision could account for some portion of incoming purely intrinsic spin?

If we collide two particles exactly the same way twice would the results be the same? I can imagine it is effectively impossible but Is there a theoretical prohibition against being able to do this?

17. Oct 25, 2016

### secur

They're different things. Forget about spin for a moment, consider energy in general. Energy is conserved but entropy increases (or, can stay constant; and, this is only true statistically, and of closed systems). Entropy is a measure of "disorder" (there are many other ways to look at it) and as such not explicitly related to energy - or spin. It is however related to the "disorder" of the spins of a bunch of particles or atomic systems.

I think not. The rotational energy, or angular momentum, of the curve of the particle, comes from the magnetic field. AFAIK.

It's about the same chance as finding two identical snowflakes. However - there is a theoretical prohibition against being able to know for certain that the two outcomes were the same! (Uncertainty principle).

18. Oct 25, 2016

### Staff: Mentor

Not in a collision experiment. What can happen, though, is an exchange of angular momentum between the orbital angular momentum of the particles and the angular momentum stored in magnetic fields, if there are such fields present and they are deflecting the particles. So to properly account for angular momentum conservation you have to include the angular momentum stored in the fields.

19. Oct 25, 2016

### Jilang

Do magnetic fields store angular momentum then? I haven't come across this before.

20. Oct 25, 2016

### Staff: Mentor

Yes, they have to. Otherwise an electric motor or generator would violate angular momentum conservation.