Complementarity between spin components in quantum mechanics
Spin is a purely quantum phenomenon and arises naturally from the Dirac Equation. However all the properties of spin can be obtained from rationalizing the data from a Stern-Gerlach filter. There is a great simulation of this put out by the U of Colorado Phet Simulations
that can be used to show that only one spin axis can be measured at any instance. That is if you measure the Z component in a Stern-Gerlach (SG) filter, then you cannot measure any other components without introducing error (dispersion). If the Z component has zero dispersion then the X and Y components will have maximum dispersion if measured in the same way as the Z component. A state with maximum dispersion reveals nothing about the system.
Take a beam of spins initially polarized in the +Z direction, pass it through a SG filter oriented in the same Z direction. With 100% certainty the beam is split up. If the initial polarization is in the –Z direction it is deflected down with 100% certainty.
Now change the initial polarization to +X direction and the beam is split into two with a 50% probability of going up or down when the field is in the Z direction:
Finally let’s take the initially polarized in the Z direction and tilt the SG filter to some angle* θ,
It is well known that the probabilities of being split up and down obey Malus’ Law: Hence from the SG experiment we obtain the probabilities of deflection of spins up and down. This actually means that starting with the spins polarized in the Z direction, the expectation value of the Pauli Spin matrix, σZ , is given by the difference in populations of up and down deflections when the SG magnet is oriented at any angle θ,
Differences in populations are proportional to the spin polarization. Using the fact that the sum of the probabilities is unity gives the expectation values in terms of the experimental probabilities*,
Case 1 displays no dispersion, while Case 2 displays a maximum dispersion of 1 Case 1 is a pure state. That is, the expectation value of measuring the spin polarization in the same direction gives us two states with no dispersion, By changing the direction of the initial polarization from +Z to –Z Case 1 gives the polarization as a pure, dispersion free, state in the –Z direction.
Case 2 is a state with maximum dispersion. In other words, the expectation value of measuring the Z component when the initial polarization is in either the +X or –X direction gives no information. It says that the spin polarization is zero, so the number of spins pointing up is equal to the number pointing down. Moreover the dispersion is maximum, being ±1.
Cases 1 and 2 are opposite (called complementary) and confirm the property of spin that is often heard: You can only measure spin in one direction at a time. If you measure in the Z direction you get two dispersion free states if the initial polarization is in the same direction (parallel or anti-parallel) as the SG field.
If you polarize the spin in the X direction, then the outcome gives no information about spin when it is measured in the Z direction. Case 2 basically says that the spin polarization can lie somewhere between +1 and -1, but we have no knowledge from this experiment what that value might be.
Of course there is nothing special about the Z direction, and is generalized to any direction*
Spin ½ states only give two dispersion free states when prepared and measured in the same direction, In contrast, if you prepare in one direction and measure in another, dispersion is always introduced.
Bohr called this restriction on quantum mechanics complementarity: you cannot measure incompatible observables ( those that do not commute).
This example nicely shows several things about quantum mechanics. First quantum mechanics is a statistical theory of measurement. You only get the SG results after many spins have been filtered. Second, Heisenberg’s uncertainty relations tell us that you cannot devise an experiment that will measure both the Z and X polarization simultaneously. You can do it for one, but not the other, and vice versa.
These results do not say, however, that the Z and X components do not exist simultaneously. They only tells us that it is impossible to measure them simultaneously without introducing dispersion in one if the other is dispersion free.
* had a problem with subscripts on the equations. All subscipts should be θ and not φ