Figure 1
We can think of the spin of an electron as a 3-vector attached to the electron which behaves like a bar magnet. If an electron whose spin is pointing in a direction at an angle \(θ\) to the vertical is placed in the magnetic field of an apparatus \(A\), its spin will align with the magnetic field of \(A\). In doing so the electron will emit radiation and lose all of its potential energy. As the angle \(θ\) increases, the electron’s potential energy increases. Therefore one might expect that the bigger \(θ\) is, the more energy the electron will radiate away. However, this is not what happens. Experimentally, it has been demonstrated that no matter what direction the spin is initially pointing in, either one of two things will happen when the electron’s spin aligns with the magnetic field: either the electron will emit no radiation or it will emit exactly one photon whose energy \(E_γ\) equals the potential energy \(PE(180°)\) when the spin is pointing down. The electron’s initially prepared spin can be in any direction but, oddly, when you measure the spin it is always only up or down.
The apparatus \(A\) in Figure (1) is used to create a magnetic field. The field lines are straight and start at the north pole and end at the south pole as illustrated in Figure (1). To simplify things we’ll draw \(A\) as a box with an arrow on it where the arrow represents the direction of the magnetic field. If we imagine “turning \(A\) off,” then rotating \(A\) by \(90°\) without effecting the electron, and then turning \(A\) back on we’ll be measuring the x component of spin \(\hat{σ}_x\). If we do the same thing as before and turn \(A\) off, then rotate \(A\) by \(90°\) along the xy-plain, and turn \(A\) back on we’ll be measuring the y-component of spin \(\hat{σ}_y\). If we follow the same procedure as before except rotate \(A\) by an arbitrary angle so that it is pointing in an arbitrary direction, the axis along which \(A\) is pointing will be the component of the measured spin which we’ll represent as \(\hat{σ}_r\).
There are two lights attached to the bottom of \(A\). Suppose that the spin of the electron is initially prepared in any arbitrary direction, if we measure \(\hat{σ}_z\) and no photon is emitted we will have measured \(σ_z=+1\) and the spin is now up. If we measure \(\hat{σ}_z\) and a photon is emitted then we will have measured \(σ_z=-1\) and the spin is now down. (It’ll take some time getting used to this notion but before the component of spin \(\hat{σ}_z\) is measured the z-component of spin could be anything. But when you go to measure \(\hat{σ}_z\) there are only two possible values of the z-component of spin you can measure which are .) This entire discussion applies to whenever we measure the components of spin \(\hat{σ}_x\), \(\hat{σ}_y\), and in general \(\hat{σ}_r\) along any arbitrary axis: before the component of spin with respect to some axis is measured it can be anything; but when we go to measure that component of spin it can only be \(σ_r=±1\).
Imagine repeating over and over again the following experiment:
1) Initially prepare the electron’s spin in any arbitrary direction.
2) Turn \(A\)’s magnetic field off and then rotate \(A\) until its “up arrow” is pointing along the z-axis as in fig. #.
3) Turn \(A\) back on and measure the component of the electron’s spin \(\hat{σ}_z\).
4) Turn \(A\)’s magnetic field off and then, after that, prepare/reset the electron’s spin to be the same as it was in step 1) before measuring \(\hat{σ}_z\).
5) Repeat
If you perform this procedure many times where the electron’s spin is initially prepared pointing entirely along the positive x-axis, then in 50% of the experiments the electron will emit no radiation and in the other 50% the electron will emit a photon. If you go through this procedure for \(θ=45°\), in 75% of the experiments the electron will emit no radiation (measured spin is up) and in 25% of the experiments the electron will emit a photon (measured spin is down). If you go through this procedure for \(θ=135°\), 75% of the time you will measure the electron’s spin to be pointing down and 25% of the time to be pointing up. There are, however, two special cases where if the electron’s spin is prepared at \(θ=135°\) the spin will be measured to be up 100% of the time and if the electron’s spin is prepared at \(θ=180°\) the spin will be measured to be down 100% of the time.