# The Advent Calendar of Physics: Using Angular Momentum

Now that we’ve defined angular momentum, the next equation on our countdown to Newton’s birthday tells us what to do with it:

This is the Angular Momentum Principle, and as with energy and momentum before it, this relates the time derivative of the angular momentum (that is, how quickly it’s changing its value) to a quantity related to the interactions with other objects, in this case the torque.

So, why is this important?

As with energy, with the proper choice of system, we can often ensure that there is no net external torque on the system, in which case the right-hand side of this equation is zero. In such a system, the angular momentum is conserved, and does not change in time, so we know that the total angular momentum at the end of the problem must be the same as the total angular momentum at the start of the problem, though it can be redistributed among the components of the system.

This is a very powerful principle, which comes into all sorts of exotic scenarios. On the very small scale, it’s what lets us measure the angular momentum of a single photon— since the total angular momentum of the photon plus the optical elements must be a constant, we know that any optical element that changes the angular momentum of the photon must itself acquire some angular momentum, and start rotating. On the very large scale, it’s why we can detect black holes– as gas spirals into a black hole, the total angular momentum of the system must remain constant, which means the speed of the gas has to increase. As the gas moves faster, it heats up, eventually becoming so hot that it emits x-rays that we can detect here on Earth.

The black hole example also demonstrates a neat feature of angular momentum, which is that it depends not just on how much stuff you have and how fast it’s moving, but also on how that stuff is arranged. If you change the radius at which the mass of a moving object is located, that would tend to change its angular momentum, so in order to keep the angular momentum constant, the speed will change as the object moves. This is why ice skaters, gymnasts, and high-divers spin very rapidly when their limbs are drawn in tight, but slow down when they stretch out. And it’s why the merry-go-round spins faster when SteelyKid moves from the outer edge to the center, and slows down when she returns to the outside.

As with energy and momentum, the fact that angular momentum is conserved reflects a deep symmetry in nature. In this case, it’s a symmetry under rotation: the laws of physics work exactly the same way whether your facing north or east. Because of this, there must be some quantity that remains unchanged as you rotate from one orientation to another, and that quantity is the angular momentum.

Angular momentum is thus a very fundamental property in physics, and comes into play even for things that are not literally spinning. Fundamental particles– quarks, leptons, photons, all have intrinsic angular momentum called “spin,” which exists even though there is no sensible way to associate it with the rotation of a physical object. This spin angular momentum combines with angular momentum associated with the physical movement of objects to produce the total angular momentum for a composite system, which is then a conserved quantity. Angular momentum conservation also works in quantum physics, where an electron orbiting an atom does not have a definite position or radius, but exists as a diffuse ball of probability (unless you’re a Bohmian). That spread-out wavefunction still has angular momentum associated with it, which is conserved in cases where there are no external interactions to change it.

So, take a moment today to appreciate angular momentum and its conservation, and come back tomorrow for the next equation of the season.

1. #1 John Armstrong
December 9, 2011

So what about the Noether symmetries for momentum and angular momentum? You mentioned that conservation of energy means that the laws of physics are the same today and tomorrow, but conservation of momentum means that the laws of physics are the same here and next door, and conservation of angular momentum means that they’re the same before and after I turn my head.

2. #2 Evan Berkowitz
December 9, 2011

Your local playground still has a merry-go-round? I thought those were outlawed when Fun became Dangerous…

December 9, 2011

So what about the Noether symmetries for momentum and angular momentum?

Third paragraph from the bottom. I forgot to name-check Emmy Noether, but I did mention the symmetry.

Your local playground still has a merry-go-round? I thought those were outlawed when Fun became Dangerous…

There’s one on the playground outside SteelyKid’s day care. It has a remarkably good bearing, and gets going really fast if I start it and move to the center. I’m sort of amazed they’ve been able to keep it.

One of the local elementary schools has some one-person spinning things that are ok, but small. The merry-go-round at the JCC is the only real merry-go-round I’ve seen locally.

4. #4 Matt McIrvin
December 9, 2011

Here’s a trivia puzzle for advanced readers only: What’s the Noether conservation law associated with symmetry under changes of linear velocity?

(Or am I getting ahead of things?)

5. #5 lordaxil
December 9, 2011

Centre of energy (or momentum for non-relativistic speeds) at time zero?

6. #6 Matt McIrvin
December 9, 2011

lordaxil wins! (The confusing thing about it is that this sounds like it’s just conservation of momentum. But the reason it sounds like it’s conservation of momentum is actually that conservation law.)

7. #7 joemac53
December 9, 2011

Do they still have those spinners that you push around, jump on (or off) changing the moment of inertia? Around here they have fallen victim to fear of broken wrists and ankles (parent fear, not kid fear).

8. #8 John Armstrong
December 9, 2011

Oops, you’re right; missed that.

9. #9 Neil Bates
December 11, 2011

Angular momentum of a single photon … Yes, in principle, although Beth’s setup couldn’t individually measure the spin of one photon by itself. The amount is too small. He measured the collected torque from many millions of them. However, what if we could use mirrors etc. to keep sending the same single photon through a half-wave plate over and over again? (Making sure to re-flip it elsewhere before each entry, so if it came in RH it returns RH, etc.) According to indistinguishability, the effect of one photon passing the same plate a million times should be just like a million photons passing through once each, etc. This could make literally one single photon detectable by its spin.

Here’s something even more interesting: a HWP flips the photon’s circular state completely and cleanly, so it even reverses the sense but not the shape of elliptical rotation. (That also doubles the spin transfer.) This seems odd since an interaction appears to be a “measurement” that should collapse the elliptical photon onto either RH or LH pure state. The chance would be based on squared amplitudes of corresponding states – but we know that isn’t what happens. Perhaps its because the amount of spin is too small to detect in any one pass, and so the expectation value itself is just summed into the uncertainty spread of the HWP.

Therefore, elliptical light (a superposition of RH and LH circular in various relative proportions) will show intermediate effect in a Beth-type experiment. This (using “S” specifically for spin angular momentum, A for the modulus of the amplitude of the RH state and B for of the LH state) is given as:

ΔS = 2n(A² – B²) ℏ

In this case, “n” is the number of photons. So let’s say we send one million elliptical-state photons of state 0.8RH, 0.6LH through the Beth detector. The BD should show a change in angular momentum of 560,000 ℏ. (Some margin of error, but with enough we could show the point.)

So far, this is uncontroversial. But let’s say we send one photon of that same elliptical state photon, one million times through the SPSED (single-photon spin expectation detector) as described above. Straightforward acceptance of the effect of each pass of one photon, says this should have the same effect as one million such photons in the Beth experiment. The change in angular momentum of the SPSED should be 560,000 ℏ. Hence the meaning of “n”changed from how many photons passed, to how many times a single like photon passed. (This presumes the photon doesn’t evolve during all this – and we already know, states are cleanly reversed by an HWP within measureability and per optical theory.)

Is this something strange? Yes, because current theory says we can’t measure the quantum state of a single particle. Supposedly we can only get probabilistic answers to queries, for A² chance of finding the sample photon as RH and B² chance of finding it as LH. But the SPSED would show the actual ellipticity of a single photon, by treating its many passes as if the passage of many identical photons. It would in effect break the no-cloning rule. (Compare the less radical idea of “weak measurements” per Aharanov et al.)

Sure, I can appreciate skepticism that the SPSED would be able to do this, it’s a long shot – but given the straightforward argument that it does work, can anyone explain, show, what would directly keep it from working? As is, it’s a paradox. (Google for “quantum measurement paradox”, see what you get. More at name link.)

‡ a|R〉 + be b|L〉

10. #10 Neil Bates
December 11, 2011

Correction: sorry, that last relation should be:
a|R〉 + be|L〉

11. #11 Frank Wappler
December 12, 2011

Neil Bates wrote (December 11, 2011 10:15 AM):
> […] the SPSED (single-photon spin expectation detector) as described above
> […] would in effect break the no-cloning rule.

Apparently the proposed apparatus can distinguish (for the purposes of detection as well as production) between the “photon states” “RH” and “LH”.

But I doubt that it could distinguish between
“(RH + LH) / Sqrt[ 2 ]” and “(RH – LH) / Sqrt[ 2 ]”,

or indeed between

“Sqrt[ 1 – (a~a) ] RH + a LH” and “Sqrt[ 1 – (a~a) ] RH – a LH”

for any complex number “a”.

12. #12 Neil Bates
December 12, 2011

Frank, thanks but you’re missing the key point of my argument: explaining exactly why, in gory detail, the SPSED would be able to distinguish among different magnitudes of circularity (ie, intermediate between full RH and LH) including elliptical. That means, it could tell if a photon was “linear” (equal magnitude of |R〉 and |L〉, as in your first example) rather than RH or LH. That already would exceed theoretical expectations of what we can do in principle. You move from mere RH v. LH distinction, which is already doable, to phase angle issues, leaving out the key novelty at issue. Please address that.

Sure, I doubt it could distinguish different phase relations in addition to various relative and amplitudes per se, which correspond to various angles for the states (orientation of the line or ellipse.) Hence, the SPSED only measures one degree of freedom, not two (the full characterization can be represented on the Bloch sphere.) Well, “so what”, it is already pushing the envelope and that’s worthwhile enough.

Tech tip: learn to use alt+[number] to get more of HTML character set. For example the “ket” ending ” 〉 ” is made by alt+9002.

13. #13 Frank Wappler
December 12, 2011

Neil Bates wrote (December 12, 2011 4:28 PM):
> You move from mere RH v. LH distinction […] to phase angle issues

Yes I did; in order to directly address your ultimate question (December 11, 2011 10:15 AM) why the apparatus under consideration could not be used for cloning in general. AFAIU, the “no-cloning-theorem” involves similar consideration of “phase angle issues“.

> but you’re missing the key point of my argument: explaining exactly why, in gory detail, the SPSED would be able to distinguish among different magnitudes of circularity (ie, intermediate between full RH and LH) including elliptical.

On second thoughts I’m not quite convinced that the described SPSED (based on the Beth experiment) would even accomplish that. For instance:

How reliable could the necessary frequently repeated “re-flip upon reflection” be implemented for a single photon? (Sorry, that’s way out of my department. &)

On the other hand: yes — if “magnitude of circularity” of a single photon could actually be distinguished this might really be more spectacular than I first gave it credit for …

p.s.
> Tech tip: learn to use alt+[number] to get more of HTML character set. For example the “ket” ending ” ? ” is made by alt+9002.

Thanks; I rather go with typing “&#9002;”.
But unfortunately this doesn’t even render on some browsers I get to use, which is why I tend to do without “bra-ket”s entirely.

14. #14 Frank Wappler
December 13, 2011

Having thought a little more about “the re-flip issue” and why it gave me pause in the first place I come to the following closely related but perhaps more principal question about the workings of a “SPSED (single-photon spin expectation detector)” proposed by Neil Bates (December 11, 2011 10:15 AM).

(For the sake of the argument let’s assume that it is “completely and cleanly” made “sure to re-flip [the photon] before each entry, so if it came in RH it returns RH, etc“, as prescribed.)

It was also noted that
> a HWP [halfwave plate] flips the photon’s circular state completely and cleanly, so it even reverses the sense but not the shape of elliptical rotation.

I wonder whether this applies reliably in each single passage of the HWP even if the photon were described not as purely “RH” or “LH” but as “partially linear“.

After all: even (and especially) a perfect HWP is not expected to gain any angular momentum from passages of (a) fully “linear” photon(s).

If the “sense of circularity” is not reliably flipped with each passage of the HWP (but only on average over many passages, and surely occuring in proportion to the “magnitude of circularity“), and if “re-flip upon reflection” occurs reliably before each entry, then …

15. #15 Neil Baytes
December 13, 2011

Frank, read up more on the optics of half-wave plates and get back to me. They are *known* to reliably invert sense of rotation of any photon state, and linear stays linear. This can be proven by experiments designed to detect the specific inverted ellipse of polarization that is produced by the HWP. It is consistent, it is reliable and for single photons, it is not statistical (ie, it is like the consistent detection of “polarized light rotated twenty degrees” by a rotary medium, etc.)

And sure, a good HWP does not gain angular momentum from passage of linear photons, that is how the repeated passage of such a photon would distinguish it from either a RH or a LH photon, since the latter produce distinct angular momentum of one sign or another. Such distinguishing of three states overturns previous assumptions about what is possible.

Tütüne son: (next time please use clear quote marks or distinction): OK, you have a point: this process does not *totally* clone the photon. It clones only it’s circularity, not the phase angle, if it works. However, it allows for more similarity than is currently thought possible. Let’s call it “partial cloning.”

December 13, 2011

“Tutune son” was Turkish comment spam that I hadn’t gotten around to deleting yet. It’s gone now.

17. #17 Frank Wappler
December 13, 2011

Neil B wrote (@ 15):
> They [half-wave plates] are *known* to reliably invert sense of rotation of any photon state […]
> Frank, read up more on the optics of half-wave plates […]

Fair advice, thanks; I’m certainly struggling to reconcile
(inverting) sense of rotation” with
“(transferring) angular momentum”
when thinking of a “partially linear photon”.

Though I’d also appreciate some online (crash course) pointers on this issue …

Meanwhile two remarks that are not really specific to the optics of half-wave plates:

> experiments designed to detect the specific inverted ellipse of polarization that is produced by the HWP.

Yes, such detection experiments should have a specific design in relation to the (expected) production setup;
otherwise such a detector might perhaps perform even more spectacular stunts than the one you’ve proposed above.

But why should “the effect that is produced by the HWP” as an element of one particular setup (or class of setups) be expected equally when the HWP is an element of some principally different setup?
And:

> It is consistent, it is reliable and for single photons, it is not statistical

Don’t at least these “experiments designed to detect the specific inverted ellipse” themselves require “some statistics” (i.e. multiple photons being detected) in order to quantify whether some given optical element performs as HWP (at least within the particular setup)?

18. #18 Neil Bates
December 14, 2011

Frank, I appreciate your interest and mutual forbearance (don’t I remember you from that big thread on UseNet, ca. 2000, “The problematical nature of photon spin”, about how can we conserve angular momentum in different frames when considering the spin axis of emitted circular photons? – and/or was it my early post on the same subject here, about a year later?) I’ll consider some good sites (but you know of course the easy way to find out), meanwhile: the idea you need statistical patterns to prove a point in QM is not quite right, it’s kind of an urban legend and an exaggeration from some cases. If the detector is in the same eigenstate, matches the input, then detection is essentially 100%. Hence I can set up optical elements and prove over and over that a single linear photon gets turned by an angle of 20°, that a photon representing a given ellipse keeps the same eccentricity and angle, but revolves the other way (the latter, by careful matching of a quarter-wave plate and linear detectors at just the right relative angles) etc.

19. #19 Frank Wappler
December 16, 2011

Neil Bates wrote (December 14, 2011 8:10 AM):
> I’ll consider some good sites

Thank you.(My own googling mostly turned up items like
http://scienceblogs.com/principles/2010/04/measuring_the_angular_momentum.php
— and apparently our topic has not particularly turned on our gracious host since, either.)

> (but you know of course the easy way to find out)

Could you spell that out, please, to be sure?
(Sadly, there’s not much on “Quantum Optics” in my local library; and I’m still debating whether to inter-library-loan Grynberg/Aspect, for Christmas … ;)

> meanwhile: […]

… it might perhaps be helpful, for gaining a somewhat rigorous perspective, to consider “stationary states” (“ground state”, or various “excited states”) of the two mirrors (as a “cavity”) with a half-wave plate between.

“Stationary” in the sense that the whole system is thought as “boxed in”, with constant “angular momentum” etc.,
and the identifiable parts of the system (mirrors and HWP separately) each keep at least their average “angular momentum” constant.
(And, perhaps, the HWP “having been excited the fast way” occured in some regular fashion, if at all; and correspondingly the HWP “having been excited the slow way”.)

On this basis we still would have to figure out how “when the box was first locked” was related to “when the box was again opened” …

> If the detector is in the same eigenstate, matches the input, then detection is essentially 100%. Hence I can set up optical elements and prove over and over that a single linear photon gets turned by an angle of 20°, that a photon representing a given ellipse keeps the same eccentricity and angle, but revolves the other way (the latter, by careful matching of a quarter-wave plate and linear detectors at just the right relative angles)

Well … wait a minute … isn’t it the other way around?:

If detection has been essentially 100 %
(– which may have been essentially established after … how many detection trials? –)
then you’ve proven that the input-generating system and the detector had been essentially “matched“, as far as the trials under consideration were concerned.
(Hmm — I wonder whether I would enjoy reading Grynberg/Aspect … &)

p.s.

> don’t I remember you

I don’t seem to remember specificly;
but if you remember having received a response along the lines of “How do you match in the first place?”, then it might as well have been me responding.

> the idea you need statistical patterns to prove a point in QM is not quite right, it’s kind of an urban legend and an exaggeration from some cases.

One particular point (“in QM”) that surely cannot proven by statistical patterns seems to be the proof of any individual element of such pattern; for instance the proof whether the result of an individual trial had been “detection” or “not”.