Momentum and Spin
Andrew - Palfreyman
Young's Slits system for electrons, whereby knowledge of either or both
quantities may affect the interference pattern.
We also know that both quantities are quantised; however, momentum may
take on a continuous range of values as a function of energy, whereas
spin has a binary nature.
Lastly, we know that the complementarity principle applies to momentum
with respect to spatial position. I know of no such corresponding relation
for spin. My question: Is there a deep reason for this "asymmetry", and
is this related to the discrete/ continuous properties of these quantities?
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| lord snooty @the giant | Would You Like Fries With That? |
| poisoned electric head | andrew palfr...@cup.portal.com |
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John C. Baez
>We know that momentum and spin both contribute state information to a
>Young's Slits system for electrons, whereby knowledge of either or both
>quantities may affect the interference pattern.
>
>We also know that both quantities are quantised; however, momentum may
>take on a continuous range of values as a function of energy, whereas
>spin has a binary nature.
Spin doesn't have a binary nature. For example, the spin of a spin one
particle can take on 3 values. It is quantized, however. That's
because it arises from a representation of a compact Lie group (the
rotation group, or more precisely its double cover, SU(2)). Position
and momentum arise from a noncompact symmetry group, the Heisenberg
group, which makes it instantly plausible to a mathematician that they
have a continuous spectrum (although one must work a bit to see this;
there's no theorem saying that any observable arising from a noncompact
group of symmetries has continuous spectrum).
>Lastly, we know that the complementarity principle applies to momentum
>with respect to spatial position. I know of no such corresponding relation
>for spin. My question: Is there a deep reason for this "asymmetry", and
>is this related to the discrete/ continuous properties of these quantities?
The analog of complementarity for spin is that it's not possible to know
all three components (x, y, and z) of the angular momentum exactly.
Complementarity arises whenever you have some observables which arise
from a NONCOMMUTATIVE group of symmetries.
This may seem cryptic to some, but it will, hopefully, intrigue them
in the idea, basic to modern physics, that many of the basic properties of
observables in quantum theory are consequences of group theory: whether
or not the group is compact, or commutative, makes a gigantic difference
in the physics.
Mcirvin
>Lastly, we know that the complementarity principle applies to momentum
>with respect to spatial position. I know of no such corresponding relation
>for spin.
It's there. Look at the relations between the components of spin in
different directions! There are uncertainty principles much like the
ones relating position and momentum. The idea is the same: that of
noncommuting operators acting on the same Hilbert space. If you
measure the z component, the x component becomes as uncertain as
possible (the uncertainty is finite in this case), and so on.
I consider electron spin a less confusing system than position and
momentum with which to examine the fundamentals of how QM works, because
of its finite and discrete nature. Everything gets much simpler.
Look at the Feynman lectures, vol. III, for much detail on this.
--
Matt McIrvin, grad student, Dept. of Physics, Harvard University
mci...@husc.harvard.edu mumble mumble mumble mumble mumble
Stephen Collyer
>We know that momentum and spin both contribute state information to a
>Young's Slits system for electrons, whereby knowledge of either or both
>quantities may affect the interference pattern.
>
>We also know that both quantities are quantised; however, momentum may
>take on a continuous range of values as a function of energy, whereas
>spin has a binary nature.
I would disagree with your terminology here - if momentum takes on a
continuous range of values, it can't be said to be quantised, can it ?
That aside, it is not true that spin has a "binary nature" in general.
This is true for spin-1/2 particles, like electrons (a Stern-Gerlach expt
will split them into 2 beams) but spin-1 particles can assume 3 spin states.
Also if you take the term spin to cover orbital AM, such as possessed by
an electron in an H atom, spin can take on 0, 1, 2, 3, .. possible values.
>
>Lastly, we know that the complementarity principle applies to momentum
>with respect to spatial position. I know of no such corresponding relation
>for spin. My question: Is there a deep reason for this "asymmetry", and
>is this related to the discrete/ continuous properties of these quantities?
I'm not sure you mean complementarity principle here, but assuming you're
talking about the uncertainty principle, there is in fact a spin uncertainty
principle - for example, you can simultaneously measure total AM of an
orbital electron, and the z component, but not all three components
simultaneously.
I think the important question is: why are momentum/position evalues
continuous, but spin evalues are discrete. I dont know myself, so someone
else will have to answer this. (By the way, could someone please tell me
if complementarity principle/HUP are synonymous - I really cant remember
what complementarity principle is supposed to mean !)
> --------------------------------------------------------------------------
>| lord snooty @the giant | Would You Like Fries With That? |
>| poisoned electric head | andrew palfr...@cup.portal.com |
> --------------------------------------------------------------------------
Steve Collyer.
Matt Austern
> I consider electron spin a less confusing system than position and
> momentum with which to examine the fundamentals of how QM works, because
> of its finite and discrete nature. Everything gets much simpler.
> Look at the Feynman lectures, vol. III, for much detail on this.
I agree. Ultimately, a finite-dimensional vector space is much easier
to work with than an infinite-dimensional space. This is one reason
why I like Sakurai's (black) book on quantum mechanics: it, also,
begins with a discussion of spin. And, unlike Feynman's book, it has
equations in it.
--
Matthew Austern I dreamt I was being followed by a roving band of
(510) 644-2618 of young Republicans, all wearing the same suit,
ma...@physics.berkeley.edu taunting me and shouting, "Politically correct
aus...@theorm.lbl.gov multiculturist scum!"... They were going to make
aus...@lbl.bitnet me kiss Jesse Helms's picture when I woke up.
Mcirvin
>In article <61...@cup.portal.com> lordS...@cup.portal.com (Andrew - Palfreyman) writes:
>>We also know that both quantities are quantised; however, momentum may
>>take on a continuous range of values as a function of energy, whereas
>>spin has a binary nature.
>I would disagree with your terminology here - if momentum takes on a
>continuous range of values, it can't be said to be quantised, can it ?
It's quantized in the canonical sense: the variable has been replaced
with an operator that has certain commutation relations unlike those
of numbers. "To quantize" usually means "to turn into a quantum theory."
>I think the important question is: why are momentum/position evalues
>continuous, but spin evalues are discrete. I dont know myself, so someone
>else will have to answer this.
A partial answer comes from the fact that eigenstates of momentum and
angular momentum can both be classified by their behavior under a family
of transformations (the symmetry associated with the conserved
quantity). In our world a rotation of 4pi (two whole rotations) brings
a physical system back to its original state. Eigenstates of angular
momentum, the conserved quantity associated with rotations, have to be
unchanged by two whole rotations, so the change in phase associated with
a rotation has to come to some multiple of 2pi when the rotation is a
multiple of 4pi. The change in phase has to scale as some half-integer
or integer multiple of the change in angle, so the possible states are
discrete.
Momentum, on the other hand, is associated with linear translations.
Assuming infinite space, you can translate as far as you want without
getting back to where you started, so there's no maximum translation
like the rotation of 4pi, and momentum eigenstates have a continuous
spectrum. If you study a situation in which the available space is
*not* infinite, such as a particle in a box or in a finite universe,
the momenta become discrete.
You could invert both of these arguments: since both angular and linear
momentum can get arbitrarily large, spatial and angular position are
both continuous.
As to why it's *two* whole rotations that leave the world unchanged,
rather than one or twelve or one-half, I have to confess ignorance,
though I think some people have greater insight into the matter.
(By the way, could someone please tell me
>if complementarity principle/HUP are synonymous - I really cant remember
>what complementarity principle is supposed to mean !)
Good question. I don't know if it ever had a precise definition.
I think that the word "complementarity" was originally applied to
particle and wave descriptions of matter; the idea, using the old
vocabulary, was that these descriptions were not contradictory but
in some sense "complementary." I seem to remember hearing x and
p referred to as "complementary observables" as well. I think it
is a broader term than "uncertainty principle." Any historians
out there?
Mcirvin
(such as in solid-state problems where we treat the possible sites for
an electron as positions in a lattice of atoms) momentum can't get
arbitrarily large. Of course, we don't really know that space is
infinite and continuous, just that it's a good approximation of
it over an enormous range of scales.
John C. Baez
>ste...@inmos.co.uk (Stephen Collyer) writes:
>
>>In article <61...@cup.portal.com> lordS...@cup.portal.com (Andrew - Palfreyman) writes:
>>I think the important question is: why are momentum/position evalues
>>continuous, but spin evalues are discrete. I don't know myself, so someone
>>else will have to answer this.
>
>A partial answer comes from the fact that eigenstates of momentum and
>angular momentum can both be classified by their behavior under a family
>of transformations (the symmetry associated with the conserved
>quantity). In our world a rotation of 4pi (two whole rotations) brings
>a physical system back to its original state. Eigenstates of angular
>momentum, the conserved quantity associated with rotations, have to be
>unchanged by two whole rotations, so the change in phase associated with
>a rotation has to come to some multiple of 2pi when the rotation is a
>multiple of 4pi. The change in phase has to scale as some half-integer
>or integer multiple of the change in angle, so the possible states are
>discrete.
>
>Momentum, on the other hand, is associated with linear translations.
>Assuming infinite space, you can translate as far as you want without
>getting back to where you started, so there's no maximum translation
>like the rotation of 4pi, and momentum eigenstates have a continuous
>spectrum. If you study a situation in which the available space is
>*not* infinite, such as a particle in a box or in a finite universe,
>the momenta become discrete.
Ah, what a beautifully down-to-earth way of explaining what I only
alluded to in my cryptic remarks about observables having discrete
spectra if the associated symmetry group is compact. Roughly, if your
symmetry group is compact you have to wiggle around n times before you
wind up back where you started, so the spectrum has to be discrete. In
fact, if you look at Bohr's early argument for why the energy levels of
hydrogen were discrete, you'll see a beautiful handwaving argument along
these lines. Something wiggles as it goes around the electrons orbit;
for it not to cancel out it must wiggle an integer number of times --
mind you, this was long before Schrodinger's equation or a detailed
understanding of what the @#$#* was "wiggling"! -- and by this reasoning
that genius was able to CALCULATE the energy levels.
>As to why it's *two* whole rotations that leave the world unchanged,
>rather than one or twelve or one-half, I have to confess ignorance,
>though I think some people have greater insight into the matter.
Well, I do this periodically on this newsgroup, with varying levels of
mathematical erudition. Since I'm feeling down-to-earth at the moment I
will just say: grab a coffee cup (empty please until you have
practiced!). You can turn it around by one whole rotation but your arm
will be rather twisted. Now you can turn it around ANOTHER whole
rotation in the SAME direction and your arm will no longer be twisted.
(Please -- do this at your own risk -- I am not liable for dislocated
elbows!) (Actually it's easy once you get the hang of it.) Figuring out
the mathematics of WHY this works the way it does is nontrivial and
explains why two whole rotations always leave things unchanged.
>I think that the word "complementarity" was originally applied to
>particle and wave descriptions of matter; the idea, using the old
>vocabulary, was that these descriptions were not contradictory but
>in some sense "complementary." I seem to remember hearing x and
>p referred to as "complementary observables" as well. I think it
>is a broader term than "uncertainty principle." Any historians
>out there?
No, but some old fogeys anyway! Bohr was the one who introduced the
notion of complementarity as a general principle: measuring X may make
it hard to measure Y accurately and vice versa. He was so enamored with
this principle that his coat of arms was the yin/yang symbol. (Honest!)
These days, less romantic mathematical physicists myself model
complementarity with noncommuting observables and note that one can
derive the uncertainty principle easily from the Cauchy-Schwarz inequality.
Allen Knutson
>In article <mc.711071761@husc8> mci...@husc8.harvard.edu (Mcirvin) mumbles:
>>quantity). In our world a rotation of 4pi (two whole rotations) brings
>>a physical system back to its original state.
>>As to why it's *two* whole rotations that leave the world unchanged,
>>rather than one or twelve or one-half, I have to confess ignorance,
>>though I think some people have greater insight into the matter.
[John's parlor trick deleted]
>Figuring out
>the mathematics of WHY this works the way it does is nontrivial and
>explains why two whole rotations always leave things unchanged.
I'll bite. What it comes down to is the following: since one can't determine
a phase of a wave function, one doesn't really care about requiring the
product of two group elements acting exactly the same way as the would come
from applying one and then the other - the result is allowed to be off by
a phase. (Mathematically, one would say we don't have unitary representations
but projective unitary.)
So let's say we have a group G, a G-symmetric Hamiltonian H, and thus an
assignment of a unitary matrix (with the phase forgotten) that commutes
with H, to each element of G. Let's try and *pick* phases that agree with the
group multiplication. Start with an infinitesimal region near the identity of
the group. Right now the identity of the group goes to the identity matrix
"up to a phase": we'll choose it go to the identity matrix. There is a
canonical choice of what phases to pick for the infinitesimal generators,
as it turns out. (Mathematically put, the Lie algebra u(n) = su(n) x u(1),
and pu(n) = u(n)/u(1), so there's a natural section pu(n) -> su(n).)
We now start exponentiating to choose phases for the finite group elements.
Let's wander around the group, fleshing out our choice of phase. Wander
back to the identity. There's no guarantee that we'll have chosen the
same phase: but this will be true if the loop can be continuously shrunk
to a point in the group. (To see this, say we come home to a new phase c.
Shrink the loop through the region where we've already chosen phases so
as for the group multiplication to work right: this causes c to not change
while we shrink. Eventually the loop is a point, and by continuity c=1.)
So the possible phase ambiguity is tied up with the structure of loops
in the group. In our case the group is SO(3) (for special orthogonal 3x3
matrices), rotations of 3-space. What does it look like (and loops in
it look like)? Make a ball of radius pi, and correspond rotations about
a given axis and rotation through a certain angle A to the point in the
ball on that axis, at that radius A. So the identity is in the center.
But it's not really a ball, because rotating by pi is the same as
rotating by -pi. So opposite points on this ball are really the same
point. Which means we can go from the identity out one side, back in
the other, back to the identity. Which exactly corresponds to rotating
something by 2pi. But that loop isn't shrinkable to a point, so we
can expect to come back to the identity times a phase C. (To prove it
isn't shrinkable to a point, start deforming it, and keep track of the
number of times you intersect the boundary - it's always odd.)
I will leave it to those with pencil and paper to convince themselves
that if you follow such a path twice, you DO get a path that's shrinkable
to a point. So rotating by 4pi leaves things unchanged. QED.
(As John said, the groups involved determine a lot of the physics.)
Advertisement: the group of rotations of the circle looks like a circle
itself, and there is no moment of reckoning as came above when we went around
a second time. So any phase ambiguity can come with a 2pi rotation, hence
Wilczek's name "anyons" for such particles.
The group of conformal transformations was long ignored as a symmetry group
because used in unitary reps, it leads to noncausality. This problem goes
away if you allow projective unitary reps. (So it's a little sad that this
particular phase ambiguity goes by the name of "the conformal anomaly"!)
Allen K.
Mcirvin
>In fact, if you look at Bohr's early argument for why the energy levels of
>hydrogen were discrete, you'll see a beautiful handwaving argument along
>these lines. Something wiggles as it goes around the electrons orbit;
>for it not to cancel out it must wiggle an integer number of times --
>mind you, this was long before Schrodinger's equation or a detailed
>understanding of what the @#$#* was "wiggling"! -- and by this reasoning
>that genius was able to CALCULATE the energy levels.
Wasn't it de Broglie who suggested this? I thought Bohr just assumed
discrete angular momenta for otherwise classical orbits.
John C. Baez
>jb...@nevanlinna.mit.edu (John C. Baez) writes:
>
>>In fact, if you look at Bohr's early argument for why the energy levels of
>>hydrogen were discrete, you'll see a beautiful handwaving argument along
>>these lines. Something wiggles as it goes around the electrons orbit;
>>for it not to cancel out it must wiggle an integer number of times --
>>mind you, this was long before Schrodinger's equation or a detailed
>>understanding of what the @#$#* was "wiggling"! -- and by this reasoning
>>that genius was able to CALCULATE the energy levels.
>
>Wasn't it de Broglie who suggested this? I thought Bohr just assumed
>discrete angular momenta for otherwise classical orbits.
You may be right. My impression was based on a talk by Weisskopf, but I
may have gotten mixed up.
Other errata -
Yes indeed, rotational invariance of hexagonal cellular automaton models
of gases only holds in a certain limit, in which one can ignore certain
tensors of higher rank.
There could be a distinction between space and time in a flat space of
signature ++-- if one insisted that all particles move along curves
whose tangent vectors are "timelike", i.e.
v_1^2 + v_2^2 - v_3^2 - v_4^2 >= 0.
Since the lightcone x_1^2 + x_2^2 - x_3^2 - x_4^2 = 0 would have a
different topology from that of Minkowski space, the way information
propagated would be very different than in our world, and "life" if it
could exist would (I think) be quite different.