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↑>.
According 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 1/2. (To convert this to conventional units, multiply by ℏ.)
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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 1/2. And, indeed, all of the constituents of atoms — electrons and nucleons or electrons and quarks — do have a spin equal to 1/2. (If electrons and nuclei were bosons, the volume occupied by 2N atoms would decrease like −N7/5, 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.)
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.
This Quantum World 