20 Spin, Zeno, and the stability of matter

In 1922, Otto Stern and Walther Ger­lach dis­cov­ered some­thing impor­tant: when a narrow beam of silver atoms passes through an inho­mo­ge­neous mag­netic field, it splits into two beams. An inho­mo­ge­neous field has a gra­dient, which points in the direc­tion in which the field strength increases fastest. If the gra­dient of the mag­netic field is ori­ented upward while the beam is hor­i­zontal, then some atoms are deflected upward by a cer­tain angle and the rest are deflected down­ward by the same angle. The mea­sure­ment of the angle of deflec­tion, more­over, is repeat­able: if the atoms in the upper beam are made to pass through a second, iden­tical appa­ratus, all of them are again deflected upward, and the same goes (mutatis mutandis) for the atoms in the lower beam.

Let us rep­re­sent the pos­sible out­comes of this mea­sure­ment by the basis vec­tors z↑ and z↓. Although the phys­ical prop­erty whose value is indi­cated by this mea­sure­ment lacks a clas­sical ana­logue, it is, like angular momentum, related to the isotropy of space — the phys­ical equiv­a­lence of all ori­en­ta­tions in space. For this reason we refer to it as the (z-​​component of the) atom’s spin.

The gra­dient of the mag­netic field can of course point in any direc­tion. If it is par­allel to the x-​​axis, we are set to mea­sure the x-​​component of the atom’s spin. Its pos­sible values are rep­re­sented by the vec­tors x↑ and x↓, which form another basis in the same 2-​​dimensional vector space. By Born’s rule, the prob­a­bility of obtaining the out­come z↑ after having obtained the out­come x↑ is given by <z↑|x↑>.

The World According to Quantum MechanicsAccording to a common phrase­ology, the first mea­sure­ment pre­pares the atom (in such a way that sub­se­quently it is) in the state x↑. Expres­sions of this sort are seri­ously mis­leading. (How can a phys­ical system be in a prob­a­bility algorithm?)

By con­sid­ering mea­sure­ments of spin com­po­nents with respect to var­ious axes, one readily finds how spin states trans­form under rota­tions of the axis of mea­sure­ment. For example, if both appa­ra­tuses mea­sure the z-​​component but the second appa­ratus is rotated rel­a­tive to the first by an angle α about the z-​​axis, then, in terms of the states asso­ci­ated with the first appa­ratus, those asso­ci­ated with the second are

z’↑ = [1:−α/2] z↑,   z’↓ = [1:α/2] z↓.

Because of the factor 1/​2, par­ti­cles (or atoms) with two spin states (per com­po­nent) are said to have a spin equal to 12. (To con­vert this to con­ven­tional units, mul­tiply by ℏ.)

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The quantum Zeno effect

The fol­lowing result is also readily obtained. Sup­pose that the spin of a spin-​​1/​2 par­ticle is mea­sured twice, at the time t=0 with respect to any axis (the z-​​axis, say), and at the time t=T with respect to an axis that is rotated rel­a­tive to the z-​​axis by an angle α (about, say, the x-​​axis). If the first out­come is “up”, then the prob­a­bility that the second out­come is “up” as well equals cos2(α/​2).

Let us inter­pose a spin mea­sure­ment at the time T/​2, with respect to an axis that is rotated by α/​2 about the x-​​axis. If the first out­come is “up”, the prob­a­bility that both the second and the third out­come are “up” now equals [cos2(α/​4)]2 = cos4(α/​4).

If instead we inter­pose 2 mea­sure­ments at the times (1/3)T and (2/3)T, with respect to axes rotated by (1/​3)α and (2/​3)α, and if the first out­come is “up”, the prob­a­bility that the three other out­comes are all “up” equals [cos2(α/​6)]3 = cos6(α/​6).

Let us con­tinue inter­posing mea­sure­ments in the same manner. If the number of inter­posed mea­sure­ments is N−1, and if the first out­come is “up”, the prob­a­bility that the remaining out­comes are all “up” equals cos2N(α/​2N). In the limit N→∞, this tends to 1.

In this example, a quantum state that would nor­mally be inde­pen­dent of time changes as deter­mined by a “con­tin­uous” sequence of mea­sure­ments. The reverse sce­nario is that of a quantum state that would nor­mally change with time but is pre­vented from doing so by a “con­tin­uous” sequence of measurements.

In the unphys­ical limit in which an infi­nite number of mea­sure­ments is per­formed in a finite interval of time, the particle’s spin state is con­trolled entirely by the mea­sure­ments to which it is sub­jected. If a large but finite number of mea­sure­ments is inter­posed, the quantum state is largely but not entirely con­trolled by them. This gen­eral and exper­i­men­tally well-​​established result[1,2] has been named after Zeno of Elea, who in the 5th Cen­tury BCE put forth a series of apparent para­doxes designed to demon­strate that motion was impossible.

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The sta­bility of matter

Beams of par­ti­cles (or atoms) passing through an inho­mo­ge­neous mag­netic field may split into any number of beams. The spin L of the par­ti­cles employed is related to the number of beams B pro­duced according to B=2L+1. Hence beams of par­ti­cles without spin (L=0) don’t split (B=1), whereas beams of spin-​​1/​2 par­ti­cles split into two beams, as we have seen. More gen­er­ally, a beam of par­ti­cles with half-​​integral spin L = 1/​2, 3/​2, 5/​2,… splits into an even number of beams (2,4,6,…), whereas a beam of par­ti­cles with inte­gral spin L = 1,2,3,… splits into an odd number of beams (1,5,7,…).

According to a cel­e­brated the­orem by Wolf­gang Pauli[3,4] — one of those the­o­rems that are as easy to state as they are hard to prove — a fermion cannot pos­sess an inte­gral spin, while a boson cannot have a half-​​integral spin. But only fermions obey the exclu­sion prin­ciple, and its validity is essen­tial for the sta­bility of matter.

As we have seen, the sta­bility of “ordi­nary” mate­rial objects rests on the sta­bility of atoms and mol­e­cules, and this requires that the rel­a­tive posi­tions and momenta of their con­stituents be fuzzy. By “ordi­nary” mate­rial objects we mean objects that

  • have spa­tial extent (they “occupy space”),
  • are com­posed of a (large but) finite number of objects without spa­tial extent (par­ti­cles that do not “occupy space”),
  • and are stable (they nei­ther explode nor col­lapse as soon as they are created).

The sta­bility of mate­rial objects con­taining a large number N of atoms fur­ther requires that the energy asso­ci­ated with, and the volume occu­pied by, twice that many atoms be twice the energy asso­ci­ated with, and the volume occu­pied, by N atoms. If we take into account that, as a con­se­quence of Maxwell’s equa­tions, the force between elec­trons and nuclei varies as 1/​r2, the nec­es­sary validity of this linear law in turn requires the Pauli exclu­sion prin­ciple to hold.[5,6] Since only fermions obey the exclu­sion prin­ciple, the con­stituents of “ordi­nary matter” must have a half-​​integral spin equal to at least 12. And, indeed, all of the con­stituents of atoms — elec­trons and nucleons or elec­trons and quarks — do have a spin equal to 12. (If elec­trons and nuclei were bosons, the volume occu­pied by 2N atoms would decrease like −N75, and matter would col­lapse into a super­dense state in which “the assembly of any two macro­scopic objects would release energy com­pa­rable to that of an atomic bomb,” Dyson and Lenard[5] wrote.)

Next


1. [↑] Misra, B., and Sudar­shan, E.C.G. (1977). The Zeno’s paradox in quantum theory, Journal of Math­e­mat­ical Physics 18, 756–763.

2. [↑] Peres, A. (1980). Zeno paradox in quantum theory, Amer­ican Journal of Physics 48, 931–932.

3. [↑] Pauli, W. (1940). The con­nec­tion between spin and sta­tis­tics, Phys­ical Review 58, 716–722.

4. [↑] Duck, I., and Sudar­shan, E.C.G. (1998). Toward an under­standing of the spin-​​statistics the­orem, Amer­ican Journal of Physics 66 (4), 284–303.

5. [↑] Dyson, F.J., and Lenard, A. (1967/​1968). Sta­bility of matter I and II, Journal of Math­e­mat­ical Physics 8, 423–434, and 9, 698–711.

6. [↑] Lieb, E.H., and Thirring, W.E. (1975). Bound for the kinetic energy of fermions which proves the sta­bility of matter, Phys­ical Review Let­ters 35, 687–689, 1116.