Spinning Particles of 1/2 Integer Spin: Explained

[Freddy86]

Hi, please could someone explain the notion that particles of 1/2 integer spin do not look the same when turned through 360 degrees. This notion seems to crop up when I read around QM but nobody seems to explain how this came about. So my question is this - what experiment shows/confirms that electrons do not look the same when turned through 360 degrees. I get QM is weird and non-intuitive but there must surely be some experiment or reason confirming this is the case? Thanks

 

[edguy99]

This was Dirac's example of something needing to be rotated by 720 degrees before it returns to its original orientation. http://vimeo.com/62228139

The Feynman plate trick is pretty neat too. http://www.youtube.com/watch?v=GgahTJybT5U

 

[Bill_K]

Hi, please could someone explain the notion that particles of 1/2 integer spin do not look the same when turned through 360 degrees. This notion seems to crop up when I read around QM but nobody seems to explain how this came about. So my question is this - what experiment shows/confirms that electrons do not look the same when turned through 360 degrees. I get QM is weird and non-intuitive but there must surely be some experiment or reason confirming this is the case? Thanks

This is a common misconception. Fermions, along with the rest of the world, most certainly DO "look the same" when rotated through 360 degrees.

The wavefunction of a fermion is a spinor, and spinors are double-valued functions. That is, they carry an implicit ± sign in front. Under a 360-degree rotation, ψ → - ψ, but this is the same wavefunction.

 

[Freddy86]

This is a common misconception. Fermions, along with the rest of the world, most certainly DO "look the same" when rotated through 360 degrees.

The wavefunction of a fermion is a spinor, and spinors are double-valued functions. That is, they carry an implicit ± sign in front. Under a 360-degree rotation, ψ → - ψ, but this is the same wavefunction.

Thanks. So is it not correct to say that fermions do not look the same after one revolution? I'm a bit confused now as it states in one of Hawking's books (brief history of time) that fermions do not look the same after being rotated through 360 degrees. Or is just that the notion of "look the same" means different things to different people and should not be taken literally (classically).

 

[edguy99]

You sometimes see it referred to as being "upside down" after a 360 degree rotation. I don't know how this is confirmed experimentally, I would guess through some type of stern-gerlach experimental setup? It would be fun to read a good reference on this.

 

[scoobmx]

It's a bit misleading to think of the electron as a 3-D particle in this context. If you did indeed have some kind of sphere in real space (R3) then it must look the same after a 2-pi rotation because that's a property of euclidean space. Instead the so-called spin-1/2 'spinors' are in the complex 2-D (C2) space. The question is then how does one do a '2-pi' rotation on something in C2? It turns out there's a fairly general way to derive the rotation operators so that they look like exponentiated angular momentum operators. If you go through this derivation for C2 space you find there's an extra 1/2 factor so that it looks something like e^(1/2)theta. Thus when you put in 2-pi you get a -1 factor and instead the periodicity is for 4-pi.

 

[edguy99]

It's a bit misleading to think of the electron as a 3-D particle in this context. If you did indeed have some kind of sphere in real space (R3) then it must look the same after a 2-pi rotation because that's a property of euclidean space. Instead the so-called spin-1/2 'spinors' are in the complex 2-D (C2) space. The question is then how does one do a '2-pi' rotation on something in C2? It turns out there's a fairly general way to derive the rotation operators so that they look like exponentiated angular momentum operators. If you go through this derivation for C2 space you find there's an extra 1/2 factor so that it looks something like e^(1/2)theta. Thus when you put in 2-pi you get a -1 factor and instead the periodicity is for 4-pi.

You seem know the subject very well. Along the lines of the belt or plate trick, I would appreciate your opinion on 2 additional ways that an object in euclidean 3d space could require rotation through 720 degrees.

1/ A rotating sphere where it rotates twice as fast in the horizontal direction then in the vertical direction. Ie. After rotating 2-pi in the x-y plane, it has only rotated 1-pi in the x-z direction. This sphere would require a rotation of 4-pi to return to its original orientation.

2/ A rotating torus (rotating in a vertical direction) around and encasing a rotating ring (rotating in a horizontal direction) where the surface of the torus rotates through twice the distance that the ring does. Hence a 2-pi rotation of the ring would result in an upside down torus, and the ring would have to rotate through 4-pi to return this complex object to its original orientation. (this one is hard to explain properly in words...)

Unfortunately, I do not know enough about spinors to know if they encompass these types of objects. If these are not spinors (or coupled spinors) then what describes these objects?

 

[scoobmx]

You seem know the subject very well. Along the lines of the belt or plate trick, I would appreciate your opinion on 2 additional ways that an object in euclidean 3d space could require rotation through 720 degrees.

1/ A rotating sphere where it rotates twice as fast in the horizontal direction then in the vertical direction. Ie. After rotating 2-pi in the x-y plane, it has only rotated 1-pi in the x-z direction. This sphere would require a rotation of 4-pi to return to its original orientation.

2/ A rotating torus (rotating in a vertical direction) around and encasing a rotating ring (rotating in a horizontal direction) where the surface of the torus rotates through twice the distance that the ring does. Hence a 2-pi rotation of the ring would result in an upside down torus, and the ring would have to rotate through 4-pi to return this complex object to its original orientation. (this one is hard to explain properly in words...)

Unfortunately, I do not know enough about spinors to know if they encompass these types of objects. If these are not spinors (or coupled spinors) then what describes these objects?

None of those examples are spinors. They are both a pair of objects in real space where you rotate the two objects different angles. The electron spinor example is a single object rotated as a whole by 2-pi. A different way to think of the rotation is this (it's usually called active vs passive perspective): instead of rotating the object, rotate space itself (the axes) the opposite amount. The result is physically the same, but now there's no confusion about single vs multiple objects. By rotating the axes both examples you posted will work normally for 2-pi. The spin 1/2 particle will still fail.

 

[edguy99]

None of those examples are spinors. They are both a pair of objects in real space where you rotate the two objects different angles. The electron spinor example is a single object rotated as a whole by 2-pi. A different way to think of the rotation is this (it's usually called active vs passive perspective): instead of rotating the object, rotate space itself (the axes) the opposite amount. The result is physically the same, but now there's no confusion about single vs multiple objects. By rotating the axes both examples you posted will work normally for 2-pi. The spin 1/2 particle will still fail.

Thanks for the reply, I agree with you if the spin in the x-y direction is fixed relative to the spin in the x-z direction. Its difficult to describe the object correctly in words. A link to this video gives a better example although it contains an orientation axis and 3 shells (rather then 2) with independent spin. In this case, reorientation of the object (or space around it) cannot be done to find an axis that will result in a 2-pi complete rotation.

 

[Bill_K]

When I woke up this morning, all the electrons in the world were upside down. Did you notice it? Turns out without realizing it, I had turned over 360 degrees in my sleep. So I got back in bed and turned 360 degrees the other way, and now things are back to normal.

 

[PhilDSP]

But you sleep in a simply connected environment with respect to your surroundings, don't you?

I think the point Dirac was trying to make with his ingenious 4pi rotation examples is that the object being rotated has multiple connections to something in its environment. The modeling of fermions is possible in SU(2) because SU(2) expresses a more sophisticated topology (not merely being simply connected).

http://mathworld.wolfram.com/SimplyConnected.html
http://en.wikipedia.org/wiki/Simply_connected_space

 

[mpv_plate]

The wavefunction of a fermion is a spinor, and spinors are double-valued functions. That is, they carry an implicit ± sign in front. Under a 360-degree rotation, ψ → - ψ, but this is the same wavefunction.

What makes a double-valued function carry an implicit ± sign? Single valued function does not carry a ± sign?

 

[vanhees71]

This is a common misconception avoidable by a more careful mathematical description of what's called a "quantum state". BTW for many examples of sloppyness I never understood the hype about Hawking's famous "Brief History of Time", but that's another issue.

Usually students are instroduced to the concept of quantum states using the "wave-mechanics apparoach", which is indeed a good way to heuristically introduce modern quantum mechanics, but it's not the best. A better way is to use "canonical quantization", which to a certain extent is also pretty hand-waving and can be misleading, but it's a better heuristic way to introduce quantum theory, because it is not referring to a certain representation (for wave mechanics that's the position representation) but already in the very beginning start from the abstract Hilbert-space formalism, which of course also has its disadvantages from the didactics point of view. I think a "healthy mixture" of both approaches serves this delicate issue on the didactics of introductory QM best.

After the dust of the initial disturbance of the students' (and usually also the teachers') minds by quantum theory has settled, the state description boils down to the following.

A quantum state is uniquely defined by a self-adjoint positive semidefinite operator $\hat{\rho}$ (the Statistical Operator) with trace 1 on an appropriate Hilbert space $\mathcal{H}$.

Special Statistical operators are projection operators, $\hat{\rho}_{\psi}=|\psi \rangle \langle \psi |$, called "pure states" as opposed to "mixed states" that are no projection operators. Here $|\psi \rangle$ is some Hilbert-space vector with norm 1.

Defining states in this way, no trouble like the here discussed thing occur with half-integer spins. The reason is that obviously not the normalized Hilbert-space vector $|\psi \rangle$ characterizes a certain pure state but a whole equivalence class of such vectors, because obviously with $|\psi' \rangle=\exp(\mathrm{i} \varphi) |\psi \rangle$ with $\varphi \in \mathbb{R}$ we have
$$\rho_{\psi'}=|\psi ' \rangle \langle \psi'|=|\psi \rangle \langle{\psi}|=\rho_{\psi},$$
and thus $|\psi' \rangle$ represents the same state as $|\psi \rangle$. Thus a phase factor doesn't matter to the state representation, i.e., a state is not characterized by the state ket itself but by the whole equivalence class of state vectors, which just deviate from each other by some phase factor. This is called a (unit) ray in Hilbert space.

This is a quite important issue, if it comes to the very foundations of the notion of a half-integer spin from the first place. A much better "correspondence principle" than the pretty handwaving "canonical quantization procedure" (in brief substituting Poisson brackets in classical mechanics by commutators of self-adjoint operators (times a purely imaginary factor), representing observables in the Hilbert space of quantum mechanics): You use the symmetry properties of the system. The most general symmetry is that of the underlying space-time model (e.g., as discussed here the Galileo symmetry of Newtonian space-time) to define the observable algebra of quantum mechanics (e.g., generators of spatial translations are called components of momentum, of temporal translations (time evolution) energy or Hamiltonian, and so on).

If it comes to the analysis of the rotation group and thus, attacking the somewhat simpler representation theory of the corresponding Lie algebra first, one automatically finds not only the integer-spin representations but also the half-integer-spin representations (if you don't stick to orbital angular momentum but first consider the abstract Lie algebra of the rotation group, which is just the angular-momentum algebra of quantum theory because angular momentum is the generator of rotations).

If you would now assume that the Hilbert-space vectors themselves (and not more correctly the rays in Hilbert space), you'd conclude that you must exclude half-integer-spin right away, because otherwise you'd really have the paradox stated in the original posting. But this contradicts the observation that there are particles with half-integer spin like the most common around us, nucleons and electrons! This shows that indeed we must represent pure states by the rays (or equivalently by the pure-state Statistical Operators) rather than the Hilbert-space vectors themselves als discussed above.

There is another interesting twist to these considerations, namely the appearance of super-selection rules! The phase factors of state kets become observable as soon as one considers superpositions of vectors and the various vectors in the superposition change their relative phase (rather than an overall phase for all of them, which doesn't change the state as discussed above), because the relative phase is in principle observable when measuring observables that are sensitive to interference effetcts of the superposition.

This, however implies, that there must never by superpositions of states with half-integer spin and integer spin, because then you really have a paradox, even in the refined definition of states discussed above! Suppose we have a vector $|\psi_i \rangle$ with integer and one with half-integer spin $|\psi_h \rangle$. Now consider the superposition
$$|\psi \rangle=|\psi_i \rangle + |\psi_{h} \rangle$$
and assume that we have properly normalized this vector to have norm 1 as to serve as an representant of a ray to describe a state.

Now for spin to make sense at all the whole formalism must admit rotations as a unitary transformations (this is also a pretty interesting statement, related to the Bargmann-Wigner theorem, which is worth to be studied in this context too). Now you can consider a rotation with rotation angle $2 \pi$ around an arbitrary axis. Then the above superposition becomes
$$\hat{R}(2 \pi) |\psi \rangle = |\psi_i \rangle - |\psi_{h} \rangle.$$
But this is not just the original state multiplied by an overall phase factor, but the relative phase between the two vectors in the superposition changes, i.e., through a rotation around $2 \pi$ you get really another ray, and thus another state of the system. On the other hand, such a rotation must be as good as doing nothing to the system, and this means that such superpositions must be forbidden to keep the whole edifice of the theory consistent! This is an example for a superselection rule, and indeed, until today one has never observed any states that are such "forbidden superpositions" of an integer- and a half-integer-spin state.

 

[edguy99]

But you sleep in a simply connected environment with respect to your surroundings, don't you?

I think the point Dirac was trying to make with his ingenious 4pi rotation examples is that the object being rotated has multiple connections to something in its environment. The modeling of fermions is possible in SU(2) because SU(2) expresses a more sophisticated topology (not merely being simply connected).

http://mathworld.wolfram.com/SimplyConnected.html
http://en.wikipedia.org/wiki/Simply_connected_space

At 12:20 of this video, Dirac discusses in detail the difficulty of representing an object that has 2 states and how a matrix can represent this type of object. He goes on in some detail to describe how this matrix represents an object, ie. the electron.

I feel somewhat different from you, in the sense that he does not discuss the up or down state of the electron relative to the outside world, but that it is an intrinsic property of the electron to be in either an up or down state. The easiest way to do this is allow the particle to have 2 independent components of spin. The first component of spin in a right-handed world defines the axis of the particle. The second independent component of spin defines the rotation of the second axis of spin around the first axis of spin, if it rotates right, its up, if it rotates left, its down. This way, he can define a two state particle independent of the space around it. (in the video, I think he actually talks of Heisenberg doing it like this first)

Using the "turning in bed" example, I guess if you look at the person and the bed together, you can sleep on your stomach or your back. The matrix Dirac used, represented both the person and the bed together as a particle, independent of the rest of the world. Turning both the bed and you upside down does not change the state of you sleeping on your stomach or your back. Unfortunately you may fall out of bed and wreck the whole experiment! :)

 

[scoobmx]

For a very detailed discussion of this go to: http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf
It was taken from here, in case you need the preceding lectures as well: http://bohr.physics.berkeley.edu/classes/221/1011/221a.html

 

[Bill_K]

Sakurai's book on Quantum Mechanics mentions on p166 a "Neutron Interferometry Experiment to Study 2π Rotations". In which a monoenergetic beam of neutrons is split into two paths A and B, path B goes through a region with a static magnetic field, which induces a phase change. The interference pattern exhibits a sinusoidal variation as the magnetic field is increased. The ΔB needed to produce successive maxima shows a 4π periodicity.

A more detailed discussion of the experiment is given in "Neutron Interferometry" by Helmut Rauch and Samuel A Werner, p167.

 

[PhilDSP]

The most thorough study of rotation and spin from the mathematical side I know of is the book by Simon Altmann "Rotations, Quaternions and Double Groups". In a small section called "Rotations by 2 pi" on p. 22 Altmann declares that from a purely geometric consideration involving symmetry there isn't a difference between 2 pi and 4 pi rotations and that the negative in the sign of the wave function (phase factor) is not an observable because energy in QM is a quadratic expression (as you said Bill_K). But that applies only in the case of an isolated system.

However the dynamic process of spin does account for phase factors between an object and another frame. Altmann also references the Rauch and Werner experiments as well as the following papers:

Werner S. A., et al 1975 Observation of the phase shift of a neutron due to precession in a magnetic field. Phys. Rev. Lett. 35, 1053-5
Rauch H., et al 1975 Verification of coherent spinor rotation of fermions. Phys. Lett. 54 A, 427-427
Bacry H. 1977 La rotation des fermions. La Recherche 8, 1010-1010
Anandan J. 1980 On the hypothesis underlying physical geometry. Foundations of Physics 10, 601-29
Page D. N., Wooters W. K. 1983 Evolution without evolution: dynamics described by stationary observables. Phys. Rev. D 27, 2885-92