Question about electron spin
Dave Griffiths
of the EPR experiment. In this he claims that it is possible to measure the
spin of an electron in three different directions at 120 degrees to each
other and obtain the result YES, the spin of the electron is aligned in
that direction for each measurement. Surely this is incorrect? He has already
shown that if you measure the spin in one direction and get the answer YES,
then there is a probability of (1 + cos 120)/2 = 1/4 that a measurement at
120 degrees to this will also give the answer YES. Fine. But surely, with
the wave function having collapsed, it's not possible for three measurements
at 120 degrees to each other _all_ to give YES?
If it is possible, has such a result ever been experimentally confirmed?
Please say no.
Dave Griffiths
Jim Carr
>I've just been reading Roger Penrose's explanation in the Emperor's New Mind
>of the EPR experiment. In this he claims that it is possible to measure the
>spin of an electron in three different directions at 120 degrees to each
>other and obtain the result YES, the spin of the electron is aligned in
>that direction for each measurement. Surely this is incorrect? He has already
Nope. It is correct.
>shown that if you measure the spin in one direction and get the answer YES,
>then there is a probability of (1 + cos 120)/2 = 1/4 that a measurement at
>120 degrees to this will also give the answer YES. Fine. But surely, with
>the wave function having collapsed, it's not possible for three measurements
>at 120 degrees to each other _all_ to give YES?
Why not? The wavefunction was "collapsed" in the original direction
before you ran it through the first 120-degree analyzer. The chance of
getting three YES answers from the system described is just 1/4 * 1/4 *
1/4 = 1/64 = 1.56%.
The answer is identical to the answer you get for three Polaroid sheets
laid at 120-degree angles on top of an sheet that defines the original
polarization. Surely you have seen (or done, find some friends with
Polaroid sunglasses) the experiment where you have crossed (90 degree)
polarizers and then insert one between them at 45 degrees and see it
go from black to (dim) light. Same thing with electrons.
>If it is possible, has such a result ever been experimentally confirmed?
>
>Please say no.
Sorry, the answer is yes. I know people who do this sort of thing all
the time when doing experiments with polarized beams.
Is your brain numb yet?
--
J. A. Carr | "The New Frontier of which I
j...@gw.scri.fsu.edu | speak is not a set of promises
Florida State University B-186 | -- it is a set of challenges."
Supercomputer Computations Research Institute | John F. Kennedy (15 July 60)
Dave Griffiths
>In article <1992Sep27.185539.594@prim> prim!da...@germany.eu.net (Dave Griffiths) writes:
>>I've just been reading Roger Penrose's explanation in the Emperor's New Mind
>>of the EPR experiment. In this he claims that it is possible to measure the
>>spin of an electron in three different directions at 120 degrees to each
>>other and obtain the result YES, the spin of the electron is aligned in
>>that direction for each measurement. Surely this is incorrect? He has already
>
>Nope. It is correct.
>
>>shown that if you measure the spin in one direction and get the answer YES,
>>then there is a probability of (1 + cos 120)/2 = 1/4 that a measurement at
>>120 degrees to this will also give the answer YES. Fine. But surely, with
>>the wave function having collapsed, it's not possible for three measurements
>>at 120 degrees to each other _all_ to give YES?
>
>Why not? The wavefunction was "collapsed" in the original direction
>before you ran it through the first 120-degree analyzer. The chance of
>getting three YES answers from the system described is just 1/4 * 1/4 *
>1/4 = 1/64 = 1.56%.
>
No wonder they call it spin. :-) Seems the notion of spin in a particular
direction is a bit meaningless if you can detect it in all three
directions. Your comment about polarizing sheets has almost persuaded me,
however we're dealing there with _aggregates_. I was wondering whether it
was possible to measure the spin of one single electron three times
consecutively and get "spin up" in three directions. If you did such a
measurement at 0, 120 and 240 degrees and got YES each time, and then
measured it at 180 degrees, would there be a chance (1 + cos 60)/2 (ie the
angle between the last measurement at 240 and 180) of now getting a YES,
or does the past history of a measurement at 0 preclude this? Could you
possibly get a YES at 60 degrees, exactly opposite your YES at 240 degrees?
>Is your brain numb yet?
Yes.
Dave Griffiths
Jim Carr
>
>No wonder they call it spin. :-) Seems the notion of spin in a particular
>direction is a bit meaningless if you can detect it in all three
No, it has a well-defined meaning. However, when the spin is in a
particular direction, you can still measure the *projection* of the spin
on an axis at some angle to that direction. If the angle is zero, you
get YES 100% of the time -- which is what gives the "direction" a well-
defined meaning. If the angle is 90 degrees, you get NO 100% of the
time -- ditto for well-defined meaning. It is, say, 45 degrees, then
you get YES 1/2 of the time since there is a 50-50 projection onto
these YES and NO axes. (You cannot get half the spin, since the
spin is quantized, so instead you get half the particles.)
>directions. Your comment about polarizing sheets has almost persuaded me,
>however we're dealing there with _aggregates_. I was wondering whether it
>was possible to measure the spin of one single electron three times
>consecutively and get "spin up" in three directions. If you did such a
>measurement at 0, 120 and 240 degrees and got YES each time, and then
>measured it at 180 degrees, would there be a chance (1 + cos 60)/2 (ie the
>angle between the last measurement at 240 and 180) of now getting a YES,
>or does the past history of a measurement at 0 preclude this? Could you
Past history has nothing to do with it, only the last direction of
polarization matters. It does not matter how you got there. The
reason, of course, is that each measurement "collapses" the wavefunction
(or, if you prefer, prepares the wavefunction in a new intial state with
some new set of amplitudes) and you start anew.
>possibly get a YES at 60 degrees, exactly opposite your YES at 240 degrees?
That would depend whether you were measuring spin "up" or just the
direction (as with polaroids). No for the former, of course, and
yes for the latter.
>>Is your brain numb yet?
>
>Yes.
Then you are off to a good start. I recommend that you think about
the problem with polaroids, and do a few experiments with them. It
is a very good way to learn about the ideas of state vectors. (I
am trying to recall if any text uses this approach, my QM lecturer
did this but not sure if it has its basis in a book you might read.)
SCOTT I CHASE
>
>The answer is identical to the answer you get for three Polaroid sheets
>laid at 120-degree angles on top of an sheet that defines the original
>polarization. Surely you have seen (or done, find some friends with
>Polaroid sunglasses) the experiment where you have crossed (90 degree)
>polarizers and then insert one between them at 45 degrees and see it
>go from black to (dim) light. Same thing with electrons.
This experiment had a very profound influence on me, when I first saw it
as an undergrad. I'm not entirely sure why. I understood, mathematically,
why it *should* work, but was still struck with wonder that it *really did*
work. I guess that I am easily amused.
-Scott
--------------------
Scott I. Chase "The question seems to be of such a character
SIC...@CSA2.LBL.GOV that if I should come to life after my death
and some mathematician were to tell me that it
had been definitely settled, I think I would
immediately drop dead again." - Vandiver
John C. Baez
"Wave function having collapsed," did I hear you say? I'm not sure
how much that picture is to blame for your puzzlement. In any event,
Penrose' account as you have described it seems correct. If you
call the 3 axes A, B, and C, one can measure angular momentum first
along A, then B, then C, and get "yes" each time. It's worth noting
though that if you then measure angular momentum along axis A again,
you need not get the answer "yes" again. In fact you will get "yes"
only with a 25% chance.
Penrose is extremely idiosyncratic in this book, but here he seems
right.
I don't know if the experiment has been done but it seems easy
enough.
Benjamin Weiner
|>Polaroid sunglasses) the experiment where you have crossed (90 degree)
|>polarizers and then insert one between them at 45 degrees and see it
|>go from black to (dim) light. Same thing with electrons.
|
| I guess that I am easily amused.
Me too. I keep several 1-inch squares of Polaroid in my desk at home
and try this out on my non-scientist friends every now and then,
because it's such a cool thing. It usually gets people very
interested.
lut...@ohstpy.mps.ohio-state.edu
> In article <1992Sep28.174343.714@prim> prim!da...@germany.eu.net (Dave Griffiths) writes:
>>
>>No wonder they call it spin. :-) Seems the notion of spin in a particular
>>direction is a bit meaningless if you can detect it in all three
>
>>
>>Yes.
>
> Then you are off to a good start. I recommend that you think about
> the problem with polaroids, and do a few experiments with them. It
> is a very good way to learn about the ideas of state vectors. (I
> am trying to recall if any text uses this approach, my QM lecturer
> did this but not sure if it has its basis in a book you might read.)
>
There are at least two texts that use polarizers (or the like) to explain
this QM concept. One is by A. P. French (called Intro Quantum Mechanics,
I think) and the other is Dirac's beautiful exposition _Principles of
Qunatum Mechanics_. Of the two French's book is probably a bit easier to
tackle but Dirac's book is worth the effort.
____________________________________________________________________________
B. Luther |
Internet: Lut...@mps.ohio-state.edu | "Seek simplicity and distrust it"
Ohio State University Physics Dept. | -Whitehead
____________________________________________________________________________
Joshua W. Burton
remember that the two independent `polarizations' of a spin-1/2 particle are
180 degrees apart, not 90. If you want to see electrons that were spin-up in
the Z direction become spin-down, here's what you do:
1.) Measure their Z-spin (Stern-Gerlach apparatus) to ensure you have a beam
of pure spin-up electrons;
2.) Put this beam through a perpendicular apparatus, measuring their X-spin.
Half of the beam will be spin-left, and the other half spin-right, of
course.
3.) Now run either beam through a second Z-spin apparatus, and you will find
that half of the electrons are now spin-down, not spin-up. Of course, a
*close* examination of the angular momentum of the apparatus will tell
you where the Z-spin disappeared to.
Leave out step 2, of course, and you will never see a spin-down electron at
step 3. This is indeed just like what happens to photons, except that there
(since a photon has spin 1) the independent states are vertical and horizontal
polarization---only 90 degrees apart---and the intermediate step that allows
light to pass through crossed polarizers is at 45 degrees.
The rule that `rotational angle between orthogonal spin states is 90 degrees
divided by the spin of the particle' applies to gravitons as well, by the way.
If you aim a small graser (Spectra-Physics catalog, August 2291; I'll look
up the part number for you, but it will take a while) at a + polarizer, and
then an x polarizer, you will find that no gravitons get through. If you
interpolate a `halfway' polarizer, twisted 22.5 degrees to the right, then
a quarter of the gravitons will sail right on through, even though the + and
x states, 45 degrees apart, are orthogonal.
What about gravitinos (hypothetical spin-3/2 particles)? Well, my dissertation
is (loosely) about them, so I should be the right person to ask, but damfino.
Gravitinos are usually dealt with in quantum field theory by thinking of them
as `vectors of spinors', a rather adhoc way of applying what we think we
understand about spin-1 and spin-1/2 to a particle that has attributes of both.
It works just fine for calculating amplitudes of fundamental processes, and
supergravity even helps us get a handle on the divergences...but my intuition
about their Lorentz-group behavior is weak. Hannu?
--
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imum Saeptum Paludosum | Joshua W. Burton bur...@het.brown.edu (401)351-5908
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