4.2: Measuring Electron Spin- the Stern-Gerlach Experiment

If a particle that has both charge and angular momentum interacts with magnetic fields, and if we know what that charge is through other experiments, then we ought to be able to figure out the angular momentum of that particle by some sort of experiment involving magnetic fields. If a particle with charge and angular momentum moves through a nonuniform magnetic field, it will be pulled along the direction of the nonuniformity based on the projection or component of its angular momentum along the direction of the magnetic field nonuniformity.

Remember that angular momentum is a 3-vector. For a spinning object, the angular momentum 3-vector is oriented along the axis about which the object is spinning. To figure out which direction along that axis the angular momentum points, you use the right-hand-rule: orient your right hand so that if you curl your fingers, they point along the sense of rotation. Then, your thumb points along the direction of the angular momentum 3-vector. For a classical spinning object like a top or a planet, that angular momentum 3-vector can point in any direction. Indeed, the angular momentum 3-vector of the Earth’s rotation is pointed at an angle of 23.5◦ with respect to the angular momentum 3-vector of the Earth’s orbit; they’re not perfectly aligned.

Let’s imagine what a classical physicist, having accepted (somehow) that all electrons have exactly the same angular momentum, would expect to see if he sent a beam of electrons through a nonuniform magnetic field that bent electrons along the z-axis. If an electron’s angular momentum happened to be oriented entirely along the +z-axis, its path would be deflected upwards the maximum amount. If its angular momentum happened to be oriented entirely along the −z axis, its path would be deflected downwards the maximum amount. Most of the electrons would have their angular momentum 3-vector randomly oriented somewhere in between, and so the beam should spread out into a vertical smear as it passed through the nonuniform magnetic field.

In the early 1920’s, two physicists, Otto Stern and Walther Gerlach, performed this experiment.¹ What they observed was not a continuous smear, but rather that the beam split into two different beams.

Think about what this means. This means that when you take a beam of electrons whose angular momenta are all randomly oriented, if you measure the z component of angular momentum you get one of only two different values. The component of spin angular momentum of an electron along the z-axis is either \(\ 5.27 \times 10^{-35} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}\), or \(\ -5.27 \times 10^{-35} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}\). The z-component of the spin angular momentum of the electron is quantized. These values of angular momentum relate Planck’s constant \(\ \hbar\) (pronounced “h-bar”), which has the value \(\ \hbar=1.055 \times 10^{-34} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}\). When the z-spin of an electron is measured, it comes out to either \(\ +\hbar / 2\) or \(\ -\hbar / 2\). Indeed, it becomes much more convenient to measure angular momentum in units of \(\ \hbar\) in quantum mechanics, so we refer to the electron as a “spin\(\ -\frac{1}{2}\) particle”. Remember, however, that whenever somebody says that an electron is measured to have z spin of 1/2, they really mean that the z component of its angular momentum is \(\ +\hbar / 2\).

We define an observable as a quantity that we could, at least in principle, measure. The position of a particle is an observable, as is its momentum. The z component of the spin angular momentum of an electron is an observable. One of the primary features of quantum physics is that many observables have the same property that we see for electron spin: when they are in fact observed, they take on one of a finite number of values. They are quantized. It is this property from which quantum mechanics takes its name.

In Quantum Mechanics, many observables are quantized. That is, when measured, they take on one of a finite number of possible values.

It’s tempting to think of the electrons whose z spins are \(\ +\hbar / 2\) as having their angular momentum oriented entirely along the +z-axis, and those whose z spins are \(\ -\hbar / 2\) as having their angular momentum oriented entirely along the −z-axis. Indeed, physicists will often refer to “spin up” and “spin down” particles. However, the total angular momentum of an electron is actually \(\ (\sqrt{3} / 2) \hbar\). That means that you never observe an electron with its spin oriented entirely along the z axis! There must always be some component of spin oriented along another axis.

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¹Stern and Gerlach did measure the spin of the electron, but at the time they thought they were measuring quantized orbital angular momentum! For the history of this experiment, see Bernstein (2010).