it is stated as follows in about the middle of the page:
"Spin angular momentum clearly has many properties in common with orbital angular momentum. However, there is one vitally important difference. Spin angular momentum operators cannot be expressed in terms of position and momentum operators, like in Eqs. (297)-(299), since this identification depends on an analogy with classical mechanics, and the concept of spin is purely quantum mechanical: i.e., it has no analogy in classical physics. Consequently, the restriction that the quantum number of the overall angular momentum must take integer values is lifted for spin angular momentum, since this restriction (found in Sects. 5.3 and 5.4) depends on Eqs. (297)-(299). In other words, the quantum number is allowed to take half-integer values."
Digging a little deeper, when it is stated that "the concept of spin is purely quantum mechanical: i.e., it has no analogy in classical physics," is this because the electron (or whatever fermion is under consideration) is regarded to be a "point" particle with no "radius" for the spin? In other words, does this restriction apply because of the "point particle" notion or would this apply even if the electron had a finite (albeit exceedingly tiny, e.g. Planck scale) spatial expanse which permitted a "rotation"?
Related to this, how do we "know" that intrinsic spin does not in fact involve rotation on an exceptionally small scale beyond the direct reach of our experimentation? Is this just supposition?
Related to this, is the orbital angular momentum based on the l and m quantum numbers for the electron considered to entail a small-scale rotation of a sort that does not apply for the spin, or is this as equally "quantum mechanical" as the intrinsic spin?
Finally, because each of orbital and spin angular momentum, together with their conserved sum j = l + s, only have definite eigenvalues with respect to a single chosen (z) axis of quantization, and the only other good quantum number comes from the Casimir operator j(j+1), the angular momenta (orbital and spin) around other than the z axis are uncertain. Does this not place l and s on the same footing quantum mechanically?
> it is stated as follows in about the middle of the page:
> "Spin angular momentum clearly has many properties in common with > orbital angular momentum. However, there is one vitally important > difference. Spin angular momentum operators cannot be expressed in terms > of position and momentum operators, like in Eqs. (297)-(299), since this > identification depends on an analogy with classical mechanics, and the > concept of spin is purely quantum mechanical: i.e., it has no analogy in > classical physics.
At this point, I would say that this is clearly wrong. Spin is a quantity that exists very happily in classical field theory, and quantum mechanics and QED and QFT inherit spin from classical field theory.
Still, it's a common statement, and it appears to be an overgeneralisation from "spin doesn't exist in classical _mechanics_". But classical mechanics is far from all of classical physics.
> Digging a little deeper, when it is stated that "the concept of spin is > purely quantum mechanical: i.e., it has no analogy in classical > physics," is this because the electron (or whatever fermion is under > consideration) is regarded to be a "point" particle with no "radius" for > the spin?
AFAIK, the situation for electrons is analogous to photons. For photons, the spin is a property of the classical EM field, and the quantisation of spin is a result of the quantisation of the EM field. However, one could argue that an electron is more "particle-like" than a photon, the QED version of which has little in common with the classical idea of "particle".
> Finally, because each of orbital and spin angular momentum, together > with their conserved sum j = l + s, only have definite eigenvalues with > respect to a single chosen (z) axis of quantization, and the only other > good quantum number comes from the Casimir operator j(j+1), the angular > momenta (orbital and spin) around other than the z axis are uncertain. > Does this not place l and s on the same footing quantum mechanically?
I'd say so. I can suggest some references on photon spin/spin in classical EM if you're interested, but I might not be able to come up with much on electron spin, other than an old paper in Am J Phys by Romer (iirc).
> > it is stated as follows in about the middle of the page:
> > "Spin angular momentum clearly has many properties in common with > > orbital angular momentum. However, there is one vitally important > > difference. Spin angular momentum operators cannot be expressed in terms > > of position and momentum operators, like in Eqs. (297)-(299), since this > > identification depends on an analogy with classical mechanics, and the > > concept of spin is purely quantum mechanical: i.e., it has no analogy in > > classical physics.
This is of the style 'eat or perish' and is a sequence of postulates rather than arguments - poor students
> At this point, I would say that this is clearly wrong. Spin is a quantity > that exists very happily in classical field theory, and quantum mechanics > and QED and QFT inherit spin from classical field theory.
> Still, it's a common statement, and it appears to be an overgeneralisation > from "spin doesn't exist in classical _mechanics_". But classical > mechanics is far from all of classical physics.
This is not true in this generality
> > Digging a little deeper, when it is stated that "the concept of spin is > > purely quantum mechanical: i.e., it has no analogy in classical > > physics," is this because the electron (or whatever fermion is under > > consideration) is regarded to be a "point" particle with no "radius" for > > the spin? > AFAIK, the situation for electrons is analogous to photons. For > photons, the spin is a property of the classical EM field, and the > quantisation of spin is a result of the quantisation of the EM field.
Indeed
> However, one could argue that an electron is more "particle-like" than a > photon, the QED version of which has little in common with the classical > idea of "particle". > > Finally, because each of orbital and spin angular momentum, together > > with their conserved sum j = l + s, only have definite eigenvalues with > > respect to a single chosen (z) axis of quantization, and the only other > > good quantum number comes from the Casimir operator j(j+1), the angular > > momenta (orbital and spin) around other than the z axis are uncertain. > > Does this not place l and s on the same footing quantum mechanically? > I'd say so.
Me too, since all three, L, S and J=L+S, obey the same Lie algebra (iirc)
> I can suggest some references on photon spin/spin in classical > EM if you're interested, but I might not be able to come up with much on > electron spin, other than an old paper in Am J Phys by Romer (iirc).
> it is stated as follows in about the middle of the page:
> "Spin angular momentum clearly has many properties in common with > orbital angular momentum. However, there is one vitally important > difference. Spin angular momentum operators cannot be expressed in terms > of position and momentum operators, like in Eqs. (297)-(299), since this > identification depends on an analogy with classical mechanics, and the > concept of spin is purely quantum mechanical: i.e., it has no analogy in > classical physics. Consequently, the restriction that the quantum number > of the overall angular momentum must take integer values is lifted for > spin angular momentum, since this restriction (found in Sects. 5.3 and > 5.4) depends on Eqs. (297)-(299). In other words, the quantum number is > allowed to take half-integer values."
> Digging a little deeper, when it is stated that "the concept of spin is > purely quantum mechanical: i.e., it has no analogy in classical > physics," is this because the electron (or whatever fermion is under > consideration) is regarded to be a "point" particle with no "radius" for > the spin? In other words, does this restriction apply because of the > "point particle" notion or would this apply even if the electron had a > finite (albeit exceedingly tiny, e.g. Planck scale) spatial expanse > which permitted a "rotation"?
> Related to this, how do we "know" that intrinsic spin does not in fact > involve rotation on an exceptionally small scale beyond the direct reach > of our experimentation? Is this just supposition?
> Related to this, is the orbital angular momentum based on the l and m > quantum numbers for the electron considered to entail a small-scale > rotation of a sort that does not apply for the spin, or is this as > equally "quantum mechanical" as the intrinsic spin?
> Finally, because each of orbital and spin angular momentum, together > with their conserved sum j = l + s, only have definite eigenvalues with > respect to a single chosen (z) axis of quantization, and the only other > good quantum number comes from the Casimir operator j(j+1), the angular > momenta (orbital and spin) around other than the z axis are uncertain. > Does this not place l and s on the same footing quantum mechanically?
Let me start by using an airplane, orientated from tail to nose by using Y axis, with wings being wings being on the X axis , and Z upward. The Pitch is around X, the Roll around Y and the Yaw around Z.
Denote Pitch = A23, Roll=A31, Yaw=A12,
where for legalese,
Auv = x_u dx^v/dt - x_v dx^u/dt.
Presume our airplane has equal moments of inertial in all 3 axes, to unitized the momentum. I can maintain a Pitch=0, and Roll at twice the rate of Yaw, meaning I can Roll 720 degs for a Yaw of 360 degs. It appears to me that Yaw/Roll =1/2 and I suggest that is an intrinsic scalar.
To prove that, let's employ (Maxwell's 2nd set), (a property of asymmetric tensors),
A23,1 + A13,2 + A21,3 = 0
(Pitch + Roll + Yaw , rates of change).
Let's use integers "n" for "rates of change",
n1 * A23 + n2 * A13 + n3 * A21 = 0,
so in the above example, n1=0, n2=2, n3=1, with "intrinsic spin" being n3/n2 = 1/2.
A question arises as to why the assumption of using "n" as an integer value is reasonable. Suppose "n" is in units of action which can be equated to units of angular momentum, and spin, and the equation evolves to,
n1*h * A23 + n2*h * A13 + n3*h * A21 = 0,
in a clearer presentation, to permit,
h * (n1 + n2 + n3) = 0
by using the above assumption of equal moments of inertial to allow unitizing A23 , A13 and A21.
Let n1=0 , n2 =2 , n3 = -1 then
n2/n2 + n3/n2 = 0 => 1 - 1/2 = 1/2
which is a scalar invariant. Regards Ken S. Tucker
>> it is stated as follows in about the middle of the page:
>> "Spin angular momentum clearly has many properties in common with >> orbital angular momentum. However, there is one vitally important >> difference. Spin angular momentum operators cannot be expressed in >> terms >> of position and momentum operators, like in Eqs. (297)-(299), since >> this >> identification depends on an analogy with classical mechanics, and >> the >> concept of spin is purely quantum mechanical: i.e., it has no analogy >> in >> classical physics.
> At this point, I would say that this is clearly wrong. Spin is a > quantity > that exists very happily in classical field theory, and quantum > mechanics > and QED and QFT inherit spin from classical field theory.
> Still, it's a common statement, and it appears to be an > overgeneralisation > from "spin doesn't exist in classical _mechanics_"
That is exactly my thought about this -- we are extrapolating physics results from what are nothing more than sloppy linguistic statements about classical mechanics, which is a great way to run into unintended mischief. Do others perceive this as well?
> >> it is stated as follows in about the middle of the page:
> >> "Spin angular momentum clearly has many properties in common with > >> orbital angular momentum. However, there is one vitally important > >> difference. Spin angular momentum operators cannot be expressed in > >> terms > >> of position and momentum operators, like in Eqs. (297)-(299), since > >> this > >> identification depends on an analogy with classical mechanics, and > >> the > >> concept of spin is purely quantum mechanical: i.e., it has no analogy > >> in > >> classical physics.
> > At this point, I would say that this is clearly wrong. Spin is a > > quantity > > that exists very happily in classical field theory, and quantum > > mechanics > > and QED and QFT inherit spin from classical field theory.
> > Still, it's a common statement, and it appears to be an > > overgeneralisation > > from "spin doesn't exist in classical _mechanics_"
> That is exactly my thought about this -- we are extrapolating physics > results from what are nothing more than sloppy linguistic statements > about classical mechanics, which is a great way to run into unintended > mischief. Do others perceive this as well? > Thanks, > Jay.
My 1st post to this thread was intended to support Timo's conjecture. The *tricky* part (IMHO) is to produce an "intrinsic" spin compatible with classical mechanics. In my understanding, we need to prove a classical spin (aka angular momentum), is relativistically invariant, IOW's there is no FoR (Frame of Ref) even in GR to eliminate that so-called "intrinsic" spin.
Suppose you have 2 spinning disc's side by side. You may attach a CS to either one, but, the relational spin cannot be transformed away by any choice of CS.
It's the *difference* of the spins that's intrinisic, that I posted on.
((An old canuck joke, "What's the difference between a moose?")).
> That is exactly my thought about this -- we are extrapolating physics > results from what are nothing more than sloppy linguistic statements > about classical mechanics, which is a great way to run into unintended > mischief. Do others perceive this as well?
Yes
The extremely sharp thinking and aiming at consequent concluding using well- defined notions that we have heritaged in classical mechanics is underestimated in contemporary physics and its representations, respectively. Sloppy thinking and sloppy language are quite close one to another.
>> it is stated as follows in about the middle of the page:
>> "Spin angular momentum clearly has many properties in common with >> orbital angular momentum. However, there is one vitally important >> difference. Spin angular momentum operators cannot be expressed in terms >> of position and momentum operators, like in Eqs. (297)-(299), since this >> identification depends on an analogy with classical mechanics, and the >> concept of spin is purely quantum mechanical: i.e., it has no analogy in >> classical physics.
The spin operator isn't expressed in terms of ordinary space, but of internal space, and there is no a priori reason to metaphysically distinguish them.
> At this point, I would say that this is clearly wrong. Spin is a quantity > that exists very happily in classical field theory, and quantum mechanics > and QED and QFT inherit spin from classical field theory. > Still, it's a common statement, and it appears to be an overgeneralisation > from "spin doesn't exist in classical _mechanics_". But classical > mechanics is far from all of classical physics.
The distinction classical/quantum isn't clean cut. Is the Dirac field quantum mechanical, or rather it's second quantification? It is still harder to say with the EM field, since historically it was classical. If the Dirac field is classical, then the intrinsic spin is also classical, and the orbital angular momentum has integer values. It may be considered the quantization of a point particle, thus quantum mechanical. Either way, that distinction is useless.
>> Digging a little deeper, when it is stated that "the concept of spin is >> purely quantum mechanical: i.e., it has no analogy in classical >> physics," is this because the electron (or whatever fermion is under >> consideration) is regarded to be a "point" particle with no "radius" for >> the spin?
The "point" particle concept must be handled with care. It means "non composite", and even a composite electron couldn't have a half integer spin from the constituents motion, even if it is very small. Or it means "obtained from the quantization of a point particle", and that is the exact synonym of a field or wave.
Samely, "no classical analogy" is at best wordy, since we know Nature is wholly quantum mechanical. It is but a historical term.
> AFAIK, the situation for electrons is analogous to photons. For > photons, the spin is a property of the classical EM field, and the > quantisation of spin is a result of the quantisation of the EM field.
No, it is the consequence of the internal symmetry.
> However, one could argue that an electron is more "particle-like" than a > photon,
That would be true with the Schrödinger equation, but the Dirac equation can't be written as the quantization of a classical particle. Today, both are on the same footing, which isn't claimed too noisily for historical reasons.
> the QED version of which has little in common with the classical idea of > "particle".
Indeed.
>> Finally, because each of orbital and spin angular momentum, together >> with their conserved sum j = l + s, only have definite eigenvalues with >> respect to a single chosen (z) axis of quantization, and the only other >> good quantum number comes from the Casimir operator j(j+1), the angular >> momenta (orbital and spin) around other than the z axis are uncertain. >> Does this not place l and s on the same footing quantum mechanically?
The operators associated with l and s are both generators of a representation of space rotation. But s need a very special representation using the Pauli matrices as the basis of the representation space, on which it acts on both sides at the same time, which mathematically amount to 2x2 complex matrices acting on a complex vector called a spinor. All the details can easily be found in the literature.
-- ~~~~ clmasse on free F-country Liberty, Equality, Profitability.
> My 1st post to this thread was intended to support > Timo's conjecture. The *tricky* part (IMHO) is to > produce an "intrinsic" spin compatible with classical > mechanics.
In that respect, topology shows that a half integer spin can emerge from scalar fields only. A famous example is the skyrmion, on which there is an extended literature.
-- ~~~~ clmasse on free F-country Liberty, Equality, Profitability.
> > My 1st post to this thread was intended to support > > Timo's conjecture. The *tricky* part (IMHO) is to > > produce an "intrinsic" spin compatible with classical > > mechanics.
> In that respect, topology shows that a half integer spin can emerge from > scalar fields only. A famous example is the skyrmion, on which there is an > extended literature.
Thank for the direction Mr. Masse. A nice brief on "skyrmions" is here, (Author Dr. Wong),
