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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: vertical line up arrow mathematical right angle bracket or vertical line down arrow mathematical right angle bracket 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

absolute value of cap a mathematical right angle bracket equals Square root of three divided by two up arrow mathematical right angle bracket prefix plus of one divided by two vertical line down arrow mathematical right angle bracket

  1. If a single particle is prepared in the state vertical line cap a mathematical right angle bracket, what prediction can be made about the result of measuring cap s sub z for this particle?

  2. If a million particles are prepared identically, all in the state vertical line cap a mathematical right angle bracket, what prediction can be made about the results of measuring cap s sub z 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 cap s sub z will give eitherprefix plus of italic h over two pi solidus twoornegative italic h over two pi solidus two; see Equations 7 and 8. In given state vertical line cap a mathematical right angle bracket and using Equation 9, a sub one equals Square root of three solidus two and a sub two equals one solidus two so the probability of getting prefix plus of italic h over two pi solidus two is left parenthesis Square root of three solidus two right parenthesis squared equals three solidus four and the probability of getting negative italic h over two pi solidus two is left parenthesis one solidus two right parenthesis squared equals one solidus four. 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 prefix plus of italic h over two pi solidus two is more likely, but the value negative italic h over two pi solidus two 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 cap s sub z equals prefix plus of italic h over two pi solidus two, and the remainder will give cap s sub z equals negative italic h over two pi solidus two.

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