"Orbital angular momentum" is a label characterizing a representation of the group of spatial rotations on a space of orbital wave functions describing a certain finite number, N, of particles in non-relativistic quantum mechanics. Now, it is assumed (for good reasons) that orbital wave functions are single-valued functions on the configuration space of those N particles, which is an N-fold Cartesian product of physical space, \mathbb{E}^{3}. Then the group that is represented on any space of orbital wave functions is SO(3), rather than its universal covering group SU(2). One now observes that only integral angular momenta correspond to representations of SO(3). That's it! - However, if one studies particles with "intrinsic angular momentum", i.e., spin, one finds that spin can also be half-integral, because, on the state space of particles with spin, the group SU(2) is represented, rather than SO(3). It is usually claimed that particles with half-integral spin (electrons, protons, neutrons,...) are fermions, while particles with integral spin (the nucleus of Helium_{4}, ...) are bosons. This "connection between spin and statistics" can only be understood or "explained" in relativistic local quantum (field) theory.

To Andrew Messing: The first proof of the connection between spin and statistics in relativistic, local quantum theories of freely moving particles is due to Markus Fierz and was published by him in 1939. At the time, Fierz was a twenty-seven years old postdoctoral researcher with Pauli. In 1940, Pauli published a somewhat simpler proof. The general connection between the spin and the quantum statistics of fields and particles in relativistic local quantum field theories on space-times of dimension at least 4 was derived by Lüders and Zumino: Half-integral spin corresponds to Fermi statistics, integral spin corresponds to Bose statistics. (In space-times of dimension 2 or 3, other forms of quantum statistics, called braid-group statistics are possible. There remains a connection between spin and braid-group statistics; but it is more subtle.)

The boundary condition: the value of the wavefunction at angle=0 equals it value after on cycle discards all half integral values

The differential operators defined on the sphere and obeying the commutation rules of angular momentum operators involve only two angular variables. One of  these angular  variables is restricted on half a circle, in order to avoid double-valuedness of the position on the sphere. While it is possible and legal (that is, in accordance with the interpretation of the wave function in QM) for the wave function to change its sign after one full circle, we simply do not have full circle with respect to the other angular variable, so the possible change of the sign is meaningless, as far as this variable is concerned. However, it should be possible to construct differential operators, with three, or more, angular variables, whose eigen-values would take integer and half-integer values. Such angular variables obviously would not correspond to the sphere in three spatial dimensions.

Short answer: For OAM, we are working in a coordinate space, and in particular one of position, we require continuous solutions, which forces us (if we want physically reasonable solutions) to impose conditions on the possible solutions. Namely, we throw out all half integer values. If we don't, we can't ensure that the wave function be single valued.

Long answer: the long answer is that this was a long answer, both in the time it took to develop and the breadth of topics. Pauli's 1940 work on the subject was more particle physics than QM, and his theory held that the only particles with integer spins were bosons, and the only with half integer were fermions. As particle physics developed, naturally this changed. If memory serves integer only particles obey Bose-Einstein statistics, while half integer spin particles obey Fermi-Dirac statistics. I recall a textbook, though I can't remember the name, stating explicitly that orbital angular momentum were not allowed (and included a derivation) but referred to the "long history" only to go on to say that they wouldn't go into it. I remember thinking "then why mention it?" At any rate the reasons are due to the mathematical structure/coordinate space, in particular the eigenvalue schema, of angular momentum that the equations for orbital angular momentum are derived from (or can be). It is in fact not necessary to rely on the wave-function criterion of single values in order to construct a proof by contradiction (namely, assume there exists a Hilbert space with dimensions satisfying the theory of angular momentum and that is valid of OAM and show that the Hilbert space can't exist). The important thing is for the mathematical structure to be consistent, but as shown in early treatments of OAM the assignment of different quantum numbers yielded different yet equally consistent results. In fact, modern results do admit fractional OAM, as in e.g., the linked paper below.

The literature on this is vast. In a nearly 800 page book Angular Momentum in Quantum Physics: Theory and Applications, apart from a bit more comprehensive treatment than one would find in your average QM textbook, there is only a chapter endnote emphasizing the extent of the literature and the nature of the problems & proposed solutions from Eddington's attempt to use multivalued wave functions Pandres' "nonzero function".

http://arxiv.org/pdf/quant-ph/0611169.pdf

Thank you all and special thanks to Andrew messing for the reference. I think we invent reasons for the non-observation of fractional OAM because we have not yet observed them. The day we observe them we will be only too ready to claim that the general AM algebra demanded that !

Regards,

Rajat

Because algebra is not physics.  There must be a SU(2) symmetry, the orbital angular momentum has only a SO(3) symmetry.

Dear Claude,

The question is why it is so?

Regards,

Rajat

The orbital angular momentum is described by a complex wave function in space, there is no SU(2) symmetry.

L = r x p.  Linear momentum p only takes any quantised values -i hbar (d/dx) which would not result in integer values. Asking for integer, whole or fraction, is irrelevant and inaaproriate.

Commutation relations for angular moment components result in integer and half-integer values (in h-bar units).

Stanislas, that's not true.  In two dimensions there is no commutation relation since there is only one operator, yet the orbital angular momentum is quantized.  The spins are the eigenvalues of the operators, which are generators of the symmetry group.  For the orbital angular momentum, the symmetry group is SO(3), or SO(2), whose eigenvalues are integer.  It can be SU(2) only for internal degrees of freedom.

IMHO, the answer to Rajat Pradhan's "why it is so?" is, "Because we humans only access the world by means of the electromagnetic interaction." The EM current is bilinear in fermionic operators. This hides the underlying SU(2) and other microscopic symmetries, and leaves us with SO(3).

"Orbital angular momentum" is a label characterizing a representation of the group of spatial rotations on a space of orbital wave functions describing a certain finite number, N, of particles in non-relativistic quantum mechanics. Now, it is assumed (for good reasons) that orbital wave functions are single-valued functions on the configuration space of those N particles, which is an N-fold Cartesian product of physical space, \mathbb{E}^{3}. Then the group that is represented on any space of orbital wave functions is SO(3), rather than its universal covering group SU(2). One now observes that only integral angular momenta correspond to representations of SO(3). That's it! - However, if one studies particles with "intrinsic angular momentum", i.e., spin, one finds that spin can also be half-integral, because, on the state space of particles with spin, the group SU(2) is represented, rather than SO(3). It is usually claimed that particles with half-integral spin (electrons, protons, neutrons,...) are fermions, while particles with integral spin (the nucleus of Helium_{4}, ...) are bosons. This "connection between spin and statistics" can only be understood or "explained" in relativistic local quantum (field) theory.

To Andrew Messing: The first proof of the connection between spin and statistics in relativistic, local quantum theories of freely moving particles is due to Markus Fierz and was published by him in 1939. At the time, Fierz was a twenty-seven years old postdoctoral researcher with Pauli. In 1940, Pauli published a somewhat simpler proof. The general connection between the spin and the quantum statistics of fields and particles in relativistic local quantum field theories on space-times of dimension at least 4 was derived by Lüders and Zumino: Half-integral spin corresponds to Fermi statistics, integral spin corresponds to Bose statistics. (In space-times of dimension 2 or 3, other forms of quantum statistics, called braid-group statistics are possible. There remains a connection between spin and braid-group statistics; but it is more subtle.)

Why is it so dear Jurg, that the covering group SU(2) is reserved for spin while OAM has to remain content with only SO(3) ? It seems we are going by experimental results and not theoretical justifications. The day we get such a situation where OHI OAM values are detected we would surely say that lo! OAM also has the same covering group!

Regards,

Rajat

Dear Plimak,

Then we should never observe even half-integral spins!

Regards,

Rajat

No, Rajat, never. All we know about electrons is inferred from electromagnetic measurements.

Often, the value of the spin of some particles is inferred somewhat indirectly. But there ARE particles whose spin can be measured quite directly; for example atoms. One verifies experimentally that a silver atom has spin =1/2. How? Well, for example with the help of Stern-Gerlach experiments! To assign spin 1/2 to an electron is based on somewhat less direct evidence, for example on atomic spectroscopy. By studying SPECTRA of alkaline earth atoms, Pauli was led to introduce a new quantum number for electrons which, fairly soon thereafter, was interpreted as meaning that electrons have spin 1/2! (A Stern-Gerlach experiment for electrons is hard to accomplish, because electrons are charged particles. It is conceivable that, in certain semi-conductor devices, Stern-Gerlach experiments for electrons may become possible.)

Jurg,

Given that there are two groups to be assigned to two distinct kinds of Ang. Mom. why don't we get it the other way around  i.e. SU(2) for OAM and SO(3) for SPM.

There are, as usual, four possibilities as you know:

(1) BOTH SO(3),

(2)BOTH SU(2)

(3) OAM-SU(2) and SAM -SO(3)

(4)OAM-SO(3) and SAM-SU(2)

Of these only the last one is realized so far. What is the reason?

Regards,

Rajat

Dear Prof. Pradhan,

The general definition of angular momentum is: any set of three operators that satisfy the commutation relation [L_i, L_j] = i \eps_ijk L_k , i.e., the SU(2) algebra. From  this one can infer that it can only have integral or odd half-integral values in units of hbar.

But the orbital angular momentum is something more .. it has a simple co-odinate space representation ... it is .. L = r X p ... Now, with the help of this we may proceed to evaluate the eigenvalues of the operator L^2 :

L^2 f = l(l+1) f  ... l(l+1) is the eigenvalue of L^2 ... This will be a differential equation .. in the spherical-polar co-ordinate it will be the Associated Legendre eqn ... Now for "well-behavedness" of the solution you can see that  "l" must take integral values in units of h-bar.

This is a standard textbook stuff ... See the solution of Schrodinger Eqn for central potential in 3d spherical polar co-ords .. and after separation of variables ... the angular part of the eqn is just the eigenvalue eqn for L^2 with the spherical harmonics as the solution.

Hope this helps.

Dipankar 

Dear Dipankar,

Thanks for joining. The question is beyond standard text-book level. Assumption about Azimuthal quantum no. m is utilized in the derivation of integral values for l. But who tells us that microscopic systems also get to the original configuration when rotated by every 2pi and not every 4pi, since all systems contain spin half particles as constituents.? We've not bothered about the details when fixing the rotational symmetry properties of the wave function exp(+/- im \phi).

Regards,

Rajat

Dear Prof. Pradhan,

I think it depends on how you define orbital angular momentum. Usually, the term "Orbital Angular Momentum" is reserved for r X p in usual co-ordinate space ... In that case there's no way that "l" can have odd half-integral values. I think you are proposing a new kind of angular momentum which is not the SPIN and which satisfies the SU(2) generator algebra (but not r X p ... so it's not orbital angular momentum either). 

Regards,

Dipankar

Dear Dipankar,

Please go deeply into the derivation of orbital angular momentum quantization in solving Schrodinger equation in 3-dimensions.

We assume during the derivation that rotation by 2\pi brings us back to the same state, which is not valid for spin. For spin we require rotation by 4\pi to get back to the original state.

Now, this means that the assumption about 2\pi rotation symmetry which we have used basing on our experience with classical systems may not be working in the quantum world.

The idea is to have only one kind of angular momentum and not two. We want unification and not diversification. Why should there be two kinds of angular momenta? Somewhere we are missing something.

There was a lot of discussion on a related thread on spin. We know spin is problematic to understand. We say it is internal quantum number and get away. But the uneasiness refuses to go. Researchers are again and again trying to have some mechanical model for spin so that it will be another orbital ang. mom.

The algebra does not distinguish between the two. It is only experimentally observed fact so far that orb. ang. mom. is quantized in integral units. But, there have been attempts to find odd half integral orbital angular momentum for the photons.

see e.g.arXiv.org/quant-ph/0611169

Regards,

Rajat

Dear All,

Interestingly enough,

The Orbital ang. mom. in Bohr theory for H-atom ground state was 1 unit  while after about ten years Schrodinger showed that it was zero !

So something that we accept today as a given truth and a fact of science we should be ready to abandon when more accurate data or more compelling theoretical arguments are provided.

Moreover, we have continued calling spin as a an internal DOF. It is an intrinsic ang. mom. having no classical analog.

Can we say the same thing about linear momentum in certain situations? That is,  can we say that there can be "INTRINSIC LINEAR MOMENTUM" of a particle in some situations that we may come across  in future?

I just want to point out that not all is well with the concept of spin as an internal DOF and with the bifurcated concept of Angular momentum.

Regards,

Rajat

All is well with spin and orbital angular momentum! It is wiser to worry about problems that are still open and hold promise for big discoveries!

If that be the case Dear Jurg, then please take up the issue of the role of conscious observer in science in general and physics in particular and modern physics in more particular. (Refer to the thread started by me on relativity). This is the hardest nut to crack and surely worthwhile for a senior professional like you.

By the way in case of Ang. Mom. you may be satisfied with the text book description but there are people (please make a search on arXiv or google search for mechanical model of spin) who think that something more needs to be done to clarify the nature of spin. And, I hope that a mechanical model will be found in near future, once we probe beyond the length scales at which we consider the electron to be  a point particle. Spin is not a fundamental intrinsic DOF for composites like Proton and Neutron as you know since they are made up of quarks. Again for quarks of course we consider them now to be elementary and thus spin becomes a tenable fundamental intrinsic DOF.

Regards

Rajat

There is an excellent answer in Ballentine: "Quantum Mechanics: A Modern Development", page 169-171.

Because the EM current (a) is quadratic in the field operator, and (b) is the only quantity that may be observed directly.

Thank you very much Kaare Borchsenius for the reference to Ballentine. I think it is a sound explanation and provides a most appropriate and elegant mathematical solution to the problem posed here in this thread. 

I would request all others on this thread to kindly go through the wonderful proof and find loopholes in in it, if there are any!

Best Regards.