Spin - why related to angular momentum?
Danny Ross Lunsford
news:50206757.03121...@posting.google.com...
> But: How can you motivate that the electrons' two states are related
> to an angular momentum? This information is not given by the
> Stern-Gerlach experiment.
The Pauli spin theory is a low-order limit of the relativistic Dirac theory,
in which the spin emerges automatically, and is not "put in by hand". So it
is not really possible to understand 2-spin in a fundamental way. The
situation is similar to electrodynamics in slowly varying fields, where the
magnetic field is defined purely phenomenologically, and only finds a real
"explanation" in terms of special relativity.
Thus, if you want to understand this issue, first understand the issue with
the relation of magnetic to electric fields under special relativity, then
learn something about the Dirac theory, then follow the reasoning that shows
how a Dirac 4-spinor has "strong" and "weak" components in the low-energy
limit.
-drl
Lubos Motl
> The Stern-Gerlach experiment tells us, that electrons have two (inner)
> states, which are called 'spin-up' and 'spin-down'...
the identification of the degeneracy with the internal angular momentum -
i.e. the spin - can be justified in many ways.
First of all, the theory that describes the spin of the electron - i.e.
Pauli's equation or its relativistic refinement by Dirac - does imply that
the two components of the electron are associated with angular momentum.
The two-component wave function of the electron, that enters Pauli's
equation, transforms as a *spinor*. It means, for example, that if you
rotate the whole system around the z-axis by angle "alpha", the "up"
component of the wave function gets multiplied by exp(i.alpha/2) and the
"down" component is multiplied by exp(-i.alpha/2).
Because the angular momentum in quantum mechanics is exactly measured by
the rate by which the wavefunction's phase is rotated when you rotate the
system around an axis, it follows that the electron carries the angular
momentum hbar/2.
The Stern-Gerlach experiment does not tell you about the angular momentum
directly, but once you know all possibilities how a wavefunction can be
transformed under the rotations of the coordinate system (i.e. all
representations of SU(2) = spin(3)), the splitting into two beams implies
that the spin of the electron must be 1/2.
The angular momentum is the only "internal label" related to space that a
particle can have. It is true classically, and it is also true quantum
mechanically - although the quantum mechanical spin has many special
features.
The Stern-Gerlach experiment tells you what is the magnetic moment of the
electron. The magnetic moment can be represented as a current that rotates
around the axis - it is excited by a circular wire, for example. It is
therefore guaranteed that the angular momentum of a particle must be
proportional to its magnetic moment.
Be sure that the internal angular momentum of the elementary particles can
be measured directly. If you absorb a large number of particles whose j_z
is positive, be sure that you will eventually rotate around "z" in the
correct direction. This fact has been measured in many cases, too, and
there is no doubt that it is true.
The angular momentum is the conserved quantity that is associated - via
Noether's theorem - with the rotational invariance of the physical laws.
In quantum mechanics it is always given by the transformation properties
of the wavefunction under the rotations around an axis (various axes are
connected with various components), but it can have many components; the
internal angular momentum of a particle - its spin - is one of them, but
only the total is conserved in general.
Best
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
Cl.Massé
50206757.03121...@posting.google.com...
> Hello everybody!
>
> The Stern-Gerlach experiment tells us, that electrons have two (inner)
> states, which are called 'spin-up' and 'spin-down'.
>
> For modelling this, one uses the formalism which was developed for
> dealing with angular momentum in quantum mechanics. Everything works
> fine, and there are no contradictions to existing experiments.
>
> But: How can you motivate that the electrons' two states are related
> to an angular momentum? This information is not given by the
> Stern-Gerlach experiment.
> I searched a solution in several books, but without results.
> Cohen-Tannoudji talks in the corresponding chapter about angular
> momentum in classical mechanics.
> I have problems in accepting a motivation of Quantum Mechanics, which
> refers to classical models, because mixing up the two is not exact in
> the sense of mathematic deduction.
>
> My question: How can you motivate the application of the formalism of
> angular momentum in quantum mechanics to the spin without referring to
> classical mechanics?
>
> Thanks!
The answer is in the Dirac equation that describe a fermion, that is a
spin 1/2 particle. The angular momentum is the quantity associated with
space rotation symmetry. A wave function carrying a given angular
momentum has a given transformation property under a rotation. For
example, and angular momentum of 1 hbar transform like a vector. That
can be seen with the Schoedinger equation for the electron of a hydrogen
atom.
The Dirac equation describes the evolution of a Dirac spinor, a four
complex components column vector. It contains the four first order
partial derivatives with respect to space and time. Under a rotation,
these derivatives transform like a 4-vector, so the Dirac matrices must
also transform, and consequently the Dirac spinor, in order that the
equation remains invariant. It happens the Dirac spinor transform
through the multiplication by a matrix that is similar to the rotation
matrix for a space vector. Under a rotation of 2pi, the Dirac spinor
isn't equal to the initial one, but is it's opposite. It takes a
rotation of 4pi to find the initial spinor again. That's why the spin
is 1/2 hbar.
That is verified by experiment, where a spin flip yield an angular
momentum of 1 hbar.
--
~~~~ clmasse at free dot fr
Liberty, Equality, Profitability.
alejandro.rivero
> But: How can you motivate that the electrons' two states are related
> to an angular momentum? This information is not given by the
> Stern-Gerlach experiment.
Well, first, I'd vote by not using the infamous h=1 notation. While it
is nice for expert calculations, it does not help to motivate anything except
perhaps the wigner classification theorem and rest of the stuff on the
Poincare group. For all the other, it is better to say that electron
has spin h/2 and photon has spin h, and of course to notice that
the units of h are units of angular momentum.
Secondly, I am thinking now on a motivation from the end... ie from
the interaction of electrons. We know that to interact one electron
must transfer momenta, and angular momentum, to the another, so
the total is preserved. This is to be done in two separate operations:
trasference from the electron to the background electromagnetic field,
and transference from the background EM field to the other electron. It
just happens that everything fits if the electron has internal angular
momenta h/2 and the background EM carries internal angular momenta
nh. We call it photon.
Alejandro
Maurice Barnhill
Stephan Camerer wrote:
> Hello everybody!
>
> The Stern-Gerlach experiment tells us, that electrons have two (inner)
> states, which are called 'spin-up' and 'spin-down'.
>
> For modelling this, one uses the formalism which was developed for
> dealing with angular momentum in quantum mechanics. Everything works
> fine, and there are no contradictions to existing experiments.
>
> But: How can you motivate that the electrons' two states are related
> to an angular momentum? This information is not given by the
> Stern-Gerlach experiment.
> I searched a solution in several books, but without results.
> Cohen-Tannoudji talks in the corresponding chapter about angular
> momentum in classical mechanics.
> I have problems in accepting a motivation of Quantum Mechanics, which
> refers to classical models, because mixing up the two is not exact in
> the sense of mathematic deduction.
>
> My question: How can you motivate the application of the formalism of
> angular momentum in quantum mechanics to the spin without referring to
> classical mechanics?
>
> Thanks!
>
> Stephan
>
To put it briefly, spin is an angular momentum because
in a wide variety of experiments the sum of orbital
angular momentum and spin does not change with time,
while orbital angular momentum or spin alone does change.
It is therefore most natural to treat the two as aspects of the
same quantity. The Dirac equation produces this result
automatically, in my opinion part of Wigner's "unreasonable
effectiveness of mathematics in physics."
--
Maurice Barnhill
m...@udel.edu [Use ReplyTo, not From]
[bellatlantic.net is reserved for spam only]
Department of Physics and Astronomy
University of Delaware
Newark, DE 19716
J. J. Lodder
ws:
> 50206757.03121...@posting.google.com...
>
> > Hello everybody!
> >
> > The Stern-Gerlach experiment tells us, that electrons have two (inner=
)
> > states, which are called 'spin-up' and 'spin-down'.
> >
> > For modelling this, one uses the formalism which was developed for
> > dealing with angular momentum in quantum mechanics. Everything works
> > fine, and there are no contradictions to existing experiments.
> >
> > But: How can you motivate that the electrons' two states are related
> > to an angular momentum? This information is not given by the
> > Stern-Gerlach experiment.
> > I searched a solution in several books, but without results.
> > Cohen-Tannoudji talks in the corresponding chapter about angular
> > momentum in classical mechanics.
> > I have problems in accepting a motivation of Quantum Mechanics, which
> > refers to classical models, because mixing up the two is not exact in
> > the sense of mathematic deduction.
> >
> > My question: How can you motivate the application of the formalism of
o
> > classical mechanics?
Ultimately the answer is that spin can couple to 'ordinary' angular
momentum. Only the sum of the two will be a conserved quantity.
Highly idealized thought experiment: stand on a turntable holding an
electron spinning up. Turn it through 180 degrees, so that it is
spinning down. You will have acquired an angular momentum equal to hbar.
Somewhat more practical: Einstein-De Haas effect.
Changing the magnetization (carried mainly by the spins)
of a ferromagnet will cause it to rotate.
And very practical:
Angular momentum conservation at the elementary particle level
is verified routinely in all high energy physics experiments.
Spins of newly discovered particles are fixed
by applying the conservation law.
Best,
Jan
Doug Sweetser
Both integral and half-integral angular momentum have classical
mechanical analogs. The easiest one is integral spin. Just spin.
In a mere 2 pi, whatever was spinning gets back to where it started.
Now spin your palm, face up. You will notice that it takes two
rotations to get everything back in place. Your hand is a classical
system, and is a mechanical analog of a half-integral angular momentum
system: it turns, and takes 2 full rotations to get back to the
starting point.
There is a dominant school of thought that spin is just too an abstract
for us to ever visualize. I've just watched my hand move in circles
and take two full rotations at least a dozen times this morning, so
disagree with that position.
What is the difference between me or my hand spinning? First, I
always, even for classical problems, think of myself in spacetime. If
I spin freely in spacetime, there are no constraints on me, and the
result is integral spin. My hand, on the other hand :-) is constrained
by my shoulder. The constraint is what makes the system an example of
half integral angular momentum.
An electron has both integral and half-integral spin. When an electron
is wandering around the room, that may have integral spin, just like me
walking down the street can be twirling as I walk. At this point, I do
not have a clear idea how the constraint on an electron works, the
proverbial shoulder constraint needed for the half-integral angular
momentum of an electron.
doug
quaternions.com
Rob Woodside
the original state, while a 2pi rotation does not. He correctly sees
the reason as the attachment of the hand to the shoulder "at infinity"
as the culprit. With a charge it is the attachment of the field lines
at infinity that give the result. Misner, Thorne and Wheeler's big
black "Gravitation" has a bit on attaching rubber bands to the poles
of a sphere and taping the loose ends of the bands to a table top.
Rotating the sphere 2pi twists the bands, just as the hand and arm are
contorted on a 2pi rotation. A further rotation of 2pi allows the
bands to be slipped over the sphere and completely undoes the
twisting, restoring the initial state. The difference is that spinors
change sign on a rotation of 2pi which is much nicer than this twisted
contortion of arm or rubber bands.
When Uhlenbeck and Goudschmidt first proposed quantum spin, Lorentz
suggested they publish in Dutch rather than German as the idea seemed
dodgy. When Pauli saw the work he called it a "joke". The reason was
that hbar/2 is a lot of angular momentum for something as small as an
electron. In those days the classical radius of the electron was taken
seriously. If the electron was that small; then no matter how its mass
was distributed, most of the electron's surface would be moving much
faster than the speed of light to give it hbar/2 rotational angular
momentum. Everyone says that Quantum Spin is sort of like rotational
angular momentum and the electron is sort of like a charged spinning
top. The "sort of like" gives some intuition about Quantum Spin and
covers the fact that Quantum Spin is definately not the result of
rotating matter. Never the less, if you want to conserve angular
momentum as experiment says you must, you must also include Quantum
Spin.
This situation is a little reminiscent of the role of heat in energy
conservation. Heat is the energy that flows due to temperature
gradiants and is a very special form of energy that is cloaked in
entropy. Until the discovery of Quantum Spin all angular momentum was
the result of material rotation. Just as heat has an energy
equivalence that must be accounted for, Quantum Spin has a rotational
equivalence that must also be accounted. It is amusing to speculate on
other forms of angular momentum, just as there are many forms of
energy, but experiment has not taken us there yet.
John Devers
>
> When Uhlenbeck and Goudschmidt first proposed quantum spin, Lorentz
> suggested they publish in Dutch rather than German as the idea seemed
> dodgy. When Pauli saw the work he called it a "joke". The reason was
> that hbar/2 is a lot of angular momentum for something as small as an
> electron. In those days the classical radius of the electron was taken
> seriously. If the electron was that small; then no matter how its mass
> was distributed, most of the electron's surface would be moving much
> faster than the speed of light to give it hbar/2 rotational angular
> momentum. Everyone says that Quantum Spin is sort of like rotational
> angular momentum and the electron is sort of like a charged spinning
> top. The "sort of like" gives some intuition about Quantum Spin and
> covers the fact that Quantum Spin is definately not the result of
> rotating matter. Never the less, if you want to conserve angular
> momentum as experiment says you must, you must also include Quantum
> Spin.
>
So how does hbar/2 relate to spin echos, spin currents and tunneling
of a quantum spin?
Is the correct word to use for this quasi-particle a spinon?
Tunneling Measurement of a Single Quantum Spin
http://link.aps.org/abstract/PRL/v90/e040401
Quantum Interference Control of Ballistic Pure Spin Currents in
Semiconductors
http://link.aps.org/abstract/PRL/v90/e136603
See link for picture.
http://focus.aps.org/story/v11/st24
Magnetoelectonic Spin Echo
http://link.aps.org/abstract/PRL/v91/e166601
Spin Current through a Quantum Dot in the Presence of an Oscillating
Magnetic Field
http://link.aps.org/abstract/PRL/v91/e196602
A side question, I once came up with this from my reading, is there a
simular thing going on with quantum spin as there is with magnetic
spin?
Another type of quasi-particle is the magnon, which is particle which
propagates the motion of magnetic spin through a magnetic medium.
Particles are connected with a property of the ground state known as
broken symmetry this leads to new particles when the broken symmetry
is continuous. Atoms can have magnetic spin, at certain temperatures
they are random at lower temperatures they spontaineously align and
their rotational symmetry breaks. This breaking allows new
disturbances in the solid, these are recognised as waves or particles.
Rotational inertia prevents spins from responding instantly, wave
propogation occurs when restoring forces encounter inertia. Spin
disturbances propagate as spin waves, or particles called magnons.
Rob Woodside
snip
> So how does hbar/2 relate to spin echos, spin currents and tunneling
> of a quantum spin?
Quantum spin comes in discrete units of hbar/2. What you mention are
effects having to do with quantum spin.
> Is the correct word to use for this quasi-particle a spinon?
Which quasi-particle?
snip
> A side question, I once came up with this from my reading, is there a
> simular thing going on with quantum spin as there is with magnetic
> spin?
What do you mean by magnetic spin? The magnetic dipole moment that is
associated with a charged particle's quantum spin?
snip
I'm sorry I haven't been that helpful.
Cl.Massé
"Doug Sweetser" <swee...@alum.mit.edu> a écrit dans le message news:
brhsdq$4p9$1...@pcls4.std.com...
> Hello Stephan:
>
> Both integral and half-integral angular momentum have classical
> mechanical analogs. The easiest one is integral spin. Just spin.
> In a mere 2 pi, whatever was spinning gets back to where it started.
>
> Now spin your palm, face up. You will notice that it takes two
> rotations to get everything back in place. Your hand is a classical
> system, and is a mechanical analog of a half-integral angular momentum
> system: it turns, and takes 2 full rotations to get back to the
> starting point.
That's a fallacy, since in the first turn, the hand is hang from above,
and in the second turn, it is hang from below. Thus, from a sleight of
hand :-) the arm once twisted is untwisted thanks to a different
position. It is possible to turn the hand further while twisting the
arm a second time, it's no more than an issue of suppleness.
Doug Sweetser
I am a bit confused by your description. The hand appears to be a=20
system that requires 2 full rotations to get back in the starting spot.=20
After one rotation, palm up, things should feel very awkward. If it=20
doesn't, join the circus.
There are three non-surgical ways to get the palm back to its starting=20
spot. The first is to reverse the direction of the initial rotation. =20
The second is to continue to rotation, palm up. The third is to rotate=20
the hand. This too is a full rotation, but around a different axis.
After these two full rotations, there is no difference between the=20
initial and final state. That is an unambiguous statement that palm=20
rotation has 4 pi rotational symmetry, not 2 pi rotational symmetry.
doug
quaternions.com
Cl.Massé
> Hello Cl.Masse:
>
> I am a bit confused by your description. The hand appears to be a
> system that requires 2 full rotations to get back in the starting
> spot. After one rotation, palm up, things should feel very awkward.
> If it doesn't, join the circus.
>
> There are three non-surgical ways to get the palm back to its starting
> spot. The first is to reverse the direction of the initial rotation.
> The second is to continue to rotation, palm up. The third is to
> rotate the hand. This too is a full rotation, but around a different
> axis. After these two full rotations, there is no difference between
> the initial and final state. That is an unambiguous statement that
> palm rotation has 4 pi rotational symmetry, not 2 pi rotational
> symmetry.
Take a band and twist it a full rotation. If you want to get back to
the initial configuration, twist it a second time a full rotation. You
get back to the initial configuration when you change the direction of
the rotational axis. That's just what happen with the hand, save that
the change of direction isn't conspicuous. Indeed, you make the first
rotation with the hand down, and the second one with the hand up,
changing the direction of the rotation axis. The orientation of the
palm is irrelevant, save for suppleness issue. The only thing you do is
twisting your arm 2pi, then untwisting it 2pi, illusioned by the
orientation of your hand, this angle being the average one allowed by
the arm articulations.
Kevin A. Scaldeferri
In article <3fef1cc7$0$24025$626a...@news.free.fr>,
Cl.Massé <jjmm...@hotmail.com> wrote:
>
>"Doug Sweetser" <swee...@alum.mit.edu> a écrit dans le message news:
>brhsdq$4p9$1...@pcls4.std.com...
>>
>> rotations to get everything back in place. Your hand is a classical
>> system, and is a mechanical analog of a half-integral angular momentum
>> system: it turns, and takes 2 full rotations to get back to the
>> starting point.
>
>That's a fallacy, since in the first turn, the hand is hang from above,
>and in the second turn, it is hang from below. Thus, from a sleight of
>hand :-) the arm once twisted is untwisted thanks to a different
>position. It is possible to turn the hand further while twisting the
>arm a second time, it's no more than an issue of suppleness.
No, this isn't correct. If your arm were sufficiently supple, as you
say, you could make the double turn below. But then, you would also
notice that your supple arm could loop around your hand, without your
hand or shoulder moving, and after that it would be completely
untwisted.
This is much easier to do with a belt than with your arm, by the way.
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Doug Sweetser
So we agree that a 4 pi rotation is required. You have further refined
the description to note that the palm orientation/direction of axis of
rotation is not relevant. If that is your point, it is a good one, I
had overlooked it.
doug
Cl.MassX
news: bt2m35$id4$1...@clyde.its.caltech.edu...
> No, this isn't correct. If your arm were sufficiently supple, as you
> say, you could make the double turn below. But then, you would also
> notice that your supple arm could loop around your hand, without your
> hand or shoulder moving, and after that it would be completely
> untwisted.
>
> This is much easier to do with a belt than with your arm, by the way.
I got no belt, can I take a tie?
I'm afraid there is another fallacy. Twist the belt by 4pi and take an
extremity in each hand. When you loop the belt around a hand, you get
an arm twisted with the belt. If you pass the belt behind and then
bellow, you disentangle them but the belt remain twisted.
sol
"Cl.MassX" <jjmm...@hotmail.com> wrote in message news:<3fff0f2a$0$28716$626a...@news.free.fr>...
Dirac's Belt Trick
http://gregegan.customer.netspace.net.au/APPLETS/21/21.html
Mathematics of Twisting
http://gregegan.customer.netspace.net.au/APPLETS/21/21.html
A complete rotation in the Calabi Yau, would be equal too, 720 degree
rotation? And no tearing?
Sol
Ken S. Tucker
"Cl.Massé" <jjmm...@hotmail.com> wrote in message news:<3ff86704$0$22301$626a...@news.free.fr>...
In this thread, it has been suggested the 1/2 spin is
unwise to imagine or analogize, however some of us
(me) do try to use mechanical pictures.
For fun I imagine being in a tumbling aircraft.
I was flying due north and stalled out. My plane
yawed about from pointing North to pointing
South and back again to North. At this point the
aircraft was upside down! I spun around again
from North through South and back to North and
my airplane was right side up after two rotations,
about the altitude axis, whew...close call.
In math, when my airplane rotated about the altitude
axis (say z) once, it did a half roll around y (north).
Another rotation around z let's another 1/2 rotation
around y to right the aircraft ie. right side up.
So one can *imagine* an internal difference in
the spins around the z (altitude) and y (direction)
in the ratio of spin around y = 1/2 spin around z
in an airplane model. I posted this because anyone
with a model aircraft might see the analogy to
1/2 spin, and in particular how 2 rotations are
required to reform the aircrafts orientation, with
this 1/2 spin ratio.
Regards
Ken S. Tucker