# A universe of angular momentum?

[...] That most tangible way of expressing the essence of quantum mechanics is that we live in a universe of quantized angular momentum and the Planck constant is the quantum. [...]

And I'm wondering if anyone can explain this a little more. Is it saying that ALL action is made up of angular momentum which is quantised in units of Planck's constant? Or is it just saying that all matter is composed of particles which have quantised angular momentum?

If the answer is the prior, this would mean that a photon's movement through space is as a result of angular momentum, which seems strange.

Anyway, if you could help me out with a little explanation, that would be great.

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Angular momentum is the most basic observable where QM differs from classical mechanics. In QM angular momentum takes only discrete values whereas (unless the particle is trapped in a finite region of space) momentum and position take continuous values in QM just as in classical mechanics. –  user10001 Mar 5 '13 at 12:30
However real essence of QM is not in the fact that angular momentum takes discrete values but in the fact that observables generally don't commute (whereas in classical mechanics they always commute). –  user10001 Mar 5 '13 at 12:39
@user10001 Can you please explain what do you mean by "observables don't commute"? What is "commute"? Thanks in advance! –  Cheeku Mar 5 '13 at 13:48
@Cheeku Observables in QM are matrices rather than functions so they do not commute in general(ie AB=!BA). In more simple terms suppose S is a physical system (say a particle) and A and B be two (quantum) observables associated with S. Say A= position and B=momentum. Now to know the value of A and B you have two choices - i) first measure A and then measure B ii) first measure B and then measure A. Classically both the ways are equivalent and will give the same result while for quantum systems these two ways of measurement are in general not equivalent and give different values of A and B. –  user10001 Mar 5 '13 at 14:20
@user10001: surely angular momentum is not quantised for an unbound state any more than position and momentum are. –  John Rennie Mar 5 '13 at 16:26

Over the decades generations of teachers have attempted to come up with an intuitive way of introducing quantum mechanics, and all have failed since it's fundamentally non-intuitive. These days students are usually just presented with the Schrödinger equation and told to get on with it. By solving the Schrödinger equation you are deriving energy eigenfunctions, so your starting point would be that the basis of QM is that energy is quantised (well, for most bound systems anyway).

The Wikipedia article is attempting to introduce QM as quantised angular momentum. I don't know if this is technically possible, though I would have guessed not since I can't see how you'd derive the radial part of the hydrogen wavefunction without considering energy. Whether it's technically possible or not, it seems to me to be unhelpful and confusing for students new to QM. I would ignore that article and find some other better introduction.

A formal introduction would probably start from the axioms of quantum mechanics, though these are far from intuitive. The Schrödinger equation follows from the axiom that time evolution acts linearly on states.

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Thanks. I'm going to find another intro like you suggested. This is all a little confusing :| –  JoeRocc Mar 6 '13 at 13:49

This is a comment turned into an answer.

Classically one can define an angular momentum of a straight track with respect to any axis as

The real meaning though comes from rotational states about a central axis. In this case a potential exists which constrains the particle to revolve about the axis.

Quantum mechanically one defines an angular momentum in potential problems where there exists a solution as a wave function and the corresponding angular momentum operator acts on these functions:

The result of this action will display the quantization of angular momentum, the value being a multiple of h_bar as shown in the table of the link.

The not very clear argument you are quoting from the wiki article is based on the assumption that quantum mechanically there will be an angular momentum only if a potential exists binding a particle, which will then inevitably have a quantized angular momentum. In the microcosm there cannot be a rotation about a center without a potential problem so it is true, but not, in my opinion the easiest way to understand quantum mechancis !.

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