How To Rotate Fermions? [Archive]

Yi-Zen Chu; Yiren Qu

Oct15-04, 01:54 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nHello everyone\n\nI\'ve been reading the following paper:\n\nObservability of the Sign Change of Spinors under 2Pi Rotations,\nY. Aharanov and L. Susskind,\nPhysical Review vol 158 (pp1237-1238)\n\nIt dicusses a thought experiment where there\'s an electron in a box that\nhas a partition in the middle. There is a constant magnetic field of the\nsame magnitude and direction in each compartment of the box. The\npartition has a hole with a shutter in it so one can choose to close the\nshutter, seperate the two halves of the box, turn one half of the box by\n2Pi odd number of times and then put them together again before opening\nthe shutter to allow for interference due to the possible minus sign\naquired.\n\nMy question is how do we know turning the box also "turns" the electron?\nThere is a magnetic field present - that does help to "turn" the\nelectron when we\'re turning the box, in addition to the spin precession\nthat an electron would experience even when the box is not being turned?\nEven for electrons in a conductor say - the paper above goes on to\ndiscuss a situation involving tunneling current (which I don\'t\nunderstand fully) - does rotating a given conductor "rotate" the\nelectrons in it?\n\nElectrons as I know it are point particles and I really do not know how\nto think about whether one is rotating it. On a related note in\nSakurai\'s Modern Quantum Mechanics text he uses the spin precession of\nneutrons to illustrate that rotations of 2Pi give rise to a minus sign,\nbut I really don\'t understand how one can equate spin precession in a\nmagnetic field and the actual rotation of the fermions themselves,\nunless we postulate there\'s some sort of "equivalence principle" between\nthe two.\n\nThanks,\nYi-Zen\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello everyone

I've been reading the following paper:

Observability of the Sign Change of Spinors under 2Pi Rotations,
Y. Aharanov and L. Susskind,
Physical Review vol 158 (pp1237-1238)

It dicusses a thought experiment where there's an electron in a box that
has a partition in the middle. There is a constant magnetic field of the
same magnitude and direction in each compartment of the box. The
partition has a hole with a shutter in it so one can choose to close the
shutter, seperate the two halves of the box, turn one half of the box by
2Pi odd number of times and then put them together again before opening
the shutter to allow for interference due to the possible minus sign
aquired.

My question is how do we know turning the box also "turns" the electron?
There is a magnetic field present - that does help to "turn" the
electron when we're turning the box, in addition to the spin precession
that an electron would experience even when the box is not being turned?
Even for electrons in a conductor say - the paper above goes on to
discuss a situation involving tunneling current (which I don't
understand fully) - does rotating a given conductor "rotate" the
electrons in it?

Electrons as I know it are point particles and I really do not know how
to think about whether one is rotating it. On a related note in
Sakurai's Modern Quantum Mechanics text he uses the spin precession of
neutrons to illustrate that rotations of 2Pi give rise to a minus sign,
but I really don't understand how one can equate spin precession in a
magnetic field and the actual rotation of the fermions themselves,
unless we postulate there's some sort of "equivalence principle" between
the two.

Thanks,
Yi-Zen

BW

Oct17-04, 01:46 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nYi-Zen Chu; Yiren Qu wrote:\n> Electrons as I know it are point particles and I really do not know\nhow\n> to think about whether one is rotating it. On a related note in\n> Sakurai\'s Modern Quantum Mechanics text he uses the spin precession\nof\n> neutrons to illustrate that rotations of 2Pi give rise to a minus\nsign,\n> but I really don\'t understand how one can equate spin precession in a\n\n> magnetic field and the actual rotation of the fermions themselves,\n> unless we postulate there\'s some sort of "equivalence principle"\nbetween\n> the two.\n\nAs far as I understand it, there is, although the "postulate" is really\ndependent on experiments measuring the intrinsic angular momentum\n(spin). This subtle point has bothered me for some time as well\nactually. Maybe it should be completely obvious sometimes but in QM\nbooks it is seldom explicitely stated what facts are empirically\narrived at and which are "predicted", and if there are other classes of\nsolutions equally useable although experimentally falsified.\n\nIn this case, for example in Shankar, Principles of Quantum Mechanics\nch 14 page 374, he writes that a rotation of a wave function will do\ntwo things - rotating the field values in space, but also transform the\nfield components into each other (with the suitable matrix called S).\nThe underlying assumptions on why the latter is necessary are absent\nhowever - my guess is them being 1) that the wave function components\nhave a relation at all and are not independent scalar fields and 2)\nthat the wave function relations are related to normal 3D space so that\nrotations in the latter will need rotations in the former.\n\nIn the electrons case, the wave function components are related in that\nthey consist of representations of the SU(2) group and that rotations\nin this group correspond to rotations of space. The SU(2) components\nmight as well have been in some sort of completely internal space\nthough (for example the electron/neutrino SU(2) symmetry does not\nrotate with spatial rotations..), it is the experiments which\nultimately decide what degree of spin a field has and that it is\nconnected to angular momentum.\n\nI\'d be happy if someone could correct or comment on this understanding\n(or maybe rather non-understanding!).. like I said it\'s a subtle point\nthat has been bugging me.\n\n/Bjorn\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Yi-Zen Chu; Yiren Qu wrote:
> Electrons as I know it are point particles and I really do not know
how
> to think about whether one is rotating it. On a related note in
> Sakurai's Modern Quantum Mechanics text he uses the spin precession
of
> neutrons to illustrate that rotations of 2Pi give rise to a minus
sign,
> but I really don't understand how one can equate spin precession in a

> magnetic field and the actual rotation of the fermions themselves,
> unless we postulate there's some sort of "equivalence principle"
between
> the two.

As far as I understand it, there is, although the "postulate" is really
dependent on experiments measuring the intrinsic angular momentum
(spin). This subtle point has bothered me for some time as well
actually. Maybe it should be completely obvious sometimes but in QM
books it is seldom explicitely stated what facts are empirically
arrived at and which are "predicted", and if there are other classes of
solutions equally useable although experimentally falsified.

In this case, for example in Shankar, Principles of Quantum Mechanics
ch 14 page 374, he writes that a rotation of a wave function will do
two things - rotating the field values in space, but also transform the
field components into each other (with the suitable matrix called S).
The underlying assumptions on why the latter is necessary are absent
however - my guess is them being 1) that the wave function components
have a relation at all and are not independent scalar fields and 2)
that the wave function relations are related to normal 3D space so that
rotations in the latter will need rotations in the former.

In the electrons case, the wave function components are related in that
they consist of representations of the SU(2) group and that rotations
in this group correspond to rotations of space. The SU(2) components
might as well have been in some sort of completely internal space
though (for example the electron/neutrino SU(2) symmetry does not
rotate with spatial rotations..), it is the experiments which
ultimately decide what degree of spin a field has and that it is
connected to angular momentum.

I'd be happy if someone could correct or comment on this understanding
(or maybe rather non-understanding!).. like I said it's a subtle point
that has been bugging me.

/Bjorn

Igor

Oct17-04, 01:46 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n"Yi-Zen Chu; Yiren Qu" <y_i_-_z_e_n_._c_h_u_@_y_a_l_e_._e_d_u> wrote in message news:<cko3ak\\$lj3\\$1@news.wss.yale.edu>...\n> Hello everyone\n>\n> I\'ve been reading the following paper:\n>\n> Observability of the Sign Change of Spinors under 2Pi Rotations,\n> Y. Aharanov and L. Susskind,\n> Physical Review vol 158 (pp1237-1238)\n>\n> It dicusses a thought experiment where there\'s an electron in a box that\n> has a partition in the middle. There is a constant magnetic field of the\n> same magnitude and direction in each compartment of the box. The\n> partition has a hole with a shutter in it so one can choose to close the\n> shutter, seperate the two halves of the box, turn one half of the box by\n> 2Pi odd number of times and then put them together again before opening\n> the shutter to allow for interference due to the possible minus sign\n> aquired.\n>\n> My question is how do we know turning the box also "turns" the electron?\n> There is a magnetic field present - that does help to "turn" the\n> electron when we\'re turning the box, in addition to the spin precession\n> that an electron would experience even when the box is not being turned?\n> Even for electrons in a conductor say - the paper above goes on to\n> discuss a situation involving tunneling current (which I don\'t\n> understand fully) - does rotating a given conductor "rotate" the\n> electrons in it?\n>\n> Electrons as I know it are point particles and I really do not know how\n> to think about whether one is rotating it. On a related note in\n> Sakurai\'s Modern Quantum Mechanics text he uses the spin precession of\n> neutrons to illustrate that rotations of 2Pi give rise to a minus sign,\n> but I really don\'t understand how one can equate spin precession in a\n> magnetic field and the actual rotation of the fermions themselves,\n> unless we postulate there\'s some sort of "equivalence principle" between\n> the two.\n>\n> Thanks,\n> Yi-Zen\n\n\nIt\'s not the particle that is rotating, since it is impossible for a\npoint to rotate. Instead, it is the wave function that is rotated by\na phase factor exp(ia), where a is the angle of rotation. For bosons,\nthis phase rotation resembles ordinary planar rotation in that the\nwave function returns to its orginal state after going through 2pi\nradians. For fermions, however, the angles are essentially halved\n(i.e a becomes a/2), which is a consequence of spinor analysis. Thus,\nfor a = 2pi, we only have an effective rotation of pi and exp(i*pi) =\n-1, so a rotation of 2pi corresponds to an inversion of the wave\nfunction. That\'s also why you need to go around twice (a = 4pi) to\nget back to where you started at a = 0.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Yi-Zen Chu; Yiren Qu" <y_{i_}-_z_e_n_._c_h_u_@_y_a_l_e_._e_d_u> wrote in message news:<cko3ak$lj3$1@news.wss.yale.edu>...
> Hello everyone
>
> I've been reading the following paper:
>
> Observability of the Sign Change of Spinors under 2Pi Rotations,
> Y. Aharanov and L. Susskind,
> Physical Review vol 158 (pp1237-1238)
>
> It dicusses a thought experiment where there's an electron in a box that
> has a partition in the middle. There is a constant magnetic field of the
> same magnitude and direction in each compartment of the box. The
> partition has a hole with a shutter in it so one can choose to close the
> shutter, seperate the two halves of the box, turn one half of the box by
> 2Pi odd number of times and then put them together again before opening
> the shutter to allow for interference due to the possible minus sign
> aquired.
>
> My question is how do we know turning the box also "turns" the electron?
> There is a magnetic field present - that does help to "turn" the
> electron when we're turning the box, in addition to the spin precession
> that an electron would experience even when the box is not being turned?
> Even for electrons in a conductor say - the paper above goes on to
> discuss a situation involving tunneling current (which I don't
> understand fully) - does rotating a given conductor "rotate" the
> electrons in it?
>
> Electrons as I know it are point particles and I really do not know how
> to think about whether one is rotating it. On a related note in
> Sakurai's Modern Quantum Mechanics text he uses the spin precession of
> neutrons to illustrate that rotations of 2Pi give rise to a minus sign,
> but I really don't understand how one can equate spin precession in a
> magnetic field and the actual rotation of the fermions themselves,
> unless we postulate there's some sort of "equivalence principle" between
> the two.
>
> Thanks,
> Yi-Zen

It's not the particle that is rotating, since it is impossible for a
point to rotate. Instead, it is the wave function that is rotated by
a phase factor \exp(ia), where a is the angle of rotation. For bosons,
this phase rotation resembles ordinary planar rotation in that the
wave function returns to its orginal state after going through 2pi
radians. For fermions, however, the angles are essentially halved
(i.e a becomes a/2), which is a consequence of spinor analysis. Thus,
for a = 2pi, we only have an effective rotation of \pi and \exp(i*\pi) =
-1, so a rotation of 2pi corresponds to an inversion of the wave
function. That's also why you need to go around twice (a = 4pi) to
get back to where you started at a = .