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# 3.3 Spin observables

In quantum mechanics measurable quantities are called observables. Spin is an example of an observable because it can be measured in an experiment. (Position and orbital angular momentum are other examples of observables.) Each observable is associated with an operator and, in general, the only possible outcomes of a measurement of an observable are any of the eigenvalues.

When a measurement is performed on a quantum system with spin, the wavefunction collapses into one of the eigenstates of the observable being measured. For example, if we measure the spin of an electron along the z-axis, the quantum state collapses into one of the two basis states: or with probabilities determined by the initial state before measurement (see Equation 9). This collapse means and any superposition that existed before the measurement is lost.

After the measurement, the electron will be in a new well-defined spin state, either spin-up or spin-down depending on the result of the measurement. The general spin state has collapsed into one of the eigenstates due to being measured. As long as the initial general spin state is not an eigenstate, the spin state after the measurement will be different from the spin state before the measurement.

## Exercise 9

Particles are prepared in the spin state

1. If a single particle is prepared in the state , what prediction can be made about the result of measuring for this particle?

2. If a million particles are prepared identically, all in the state , what prediction can be made about the results of measuring for this collection of particles?

### Answer

1. No definite prediction can be made for a single particle in the given state, but a measurement of will give either or; see Equations 7 and 8. In given state and using Equation 9, and so the probability of getting is and the probability of getting is . As expected these two probabilities sum to unity because for any measurement either one or the other outcome will be obtained. This shows that the value is more likely, but the value would not be that surprising.

2. For a million particles, the expected outcome is that close to three-quarters or 750,000 measurements will give , and the remainder will give .

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