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Measurement of the z component of spin  in a Stern-Gerlach apparatus
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Classical mechanics assumes that the position and momentum of a particle have at any time a well defined value. If these values are known at some instant, one can calculate how they evolve and therefore predict the exact result of a further measurement. This deterministic scheme is however inadequate when dealing with incompatible observables. As an example we consider measurements involving the different spatial components of the spin of one atom. Spin is an observable pertaining to any quantum particle. It has the dimensions of an angular momentum and enters all processes involving such physical quantity. Due to their spin, electrons react to a gradient of magnetic field the way a spinning electric charge would do. The particle spin is a vector. However, at variance with ‘external' angular momentum (i.e. the angular momentum associated with the rotational and translational motion of the particle), each spin component (Sx , Sy , Sz) can only take discrete values (which are multiples of the Planck's constant h). This is easily seen in a Stern-Gerlach experiment with silver atoms.

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FIGURE 3

 

FIGURE 2

This result indicates that the probability associated with the different outcomes in two successive measurements of Sz is modified according to whether Sx is measured or not in between. It is important to stress that this modification occurs independently of the set-up used to carry out the measurement, i.e. it is not induced by the particular physical interaction between the atoms and the apparatus. Rather, it is a typical feature of measurements involving two observables that are mutually incompatible. As we show in the section on origins, this odd situation can be easily represented by means of state vectors within the state space.

The contrast between deterministic models inspired to classical physics (see hidden variables) and the quantum formalism involving incompatible observables becomes even sharper when considering the correlations between pairs of entangled particles. Experiments with entangled pairs (and also with triplets and quadruplets of entangled particles) have been carried out using two-level systems analogous to spin up/spin down particles.

The polarization of photons provides for example a close analogous of electron's spin. Directing a light beam through polarizing filters (that play the role of the Stern-Gerlach apparatuses), one obtains results equivqlent to those described in this section. Classical electromagnetism can account for observations in this case. However, when the same optical experiments are carried out with individual photons, a quantum treatement making use of state vectors becomes necessary (see also particle interference).

The same split along the axis parallel to the gradient of the magnetic field is observed if the experiment is repeated by changing the orientation of the magnets. Thus, what we find is that the spin component along any direction can only take two values, that we call conventionally spin up and spin down.

The incompatibility of two different components of spin, for example Sz and Sx, shows up when we try to prepare the atoms in such a way that a joint measurement of Sz and Sx yields up-up with probability 1, i.e. p(Sz= & Sx= )=1. Figure 2 shows the filtering process through which we prepare atoms for which p(Sz=↑). An analogous filter (with the magnets oriented perpendicularly) can be used to prepare atoms for which p(Sx= ↑).

The correlations between the spins of two  entangled  particles are measured by means of two distant Stern-Gerlach apparatuses. The relative orientation of the two apparatuses can be changed. When they are parallel, the spins  of the two particles are always found to point in opposite directions (see 'origins' for further discussion)
Selection of Sz  'up'

The problem is that if we try to implement a double filtering process in order to prepare atoms for which p(Sz= & Sx= )=1 (figure 3) it simply doesn't work! Exiting the Sx filter, the atom displays with equal probability either Sz= or Sz= (and not, as could be expected in a determininstic model, Sz= with probability 1).

Because of their particular electronic structure, silver atoms have a global spin that can be roughly identified with the spin of the electron orbiting in the outer electronic shell. If a beam of silver atoms is directed through an inhomogeneous magnetic field with vertical gradient, it splits in two well-separated components along the z axis (see figure 1). This is exactly the behaviour expected for two ensembles of atoms that have angular momenta equal in modulus but pointing in opposite directions (like tops spinning either clockwise or counterclockwise with the same angular velocity).

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FIGURE 1