An electron rotates and creates a magnetic field.In an attempt to explain the electron spin in a classical model, one can assume that it is just a rotating charged particle causing a magnetic moment. However, such an approach results in contradictions which renders a classical explanation of the intrinsic magnetic moment of an electron obsolete.

Problem Statement

One way to find a contradiction in the classical explanation of spin: the needed angular velocity of an electron.One way to find contradictions in such a classical argumentation is to find a lower estimation for the angular velocity at which an electron would have to spin to cause its magnetic moment:

  1. Find a classical electron radius \(r_{e}\): Regard the electron made of a thin charged shell with radius \(r_{e}\). What is the electrostatic energy of such a configuration? Determine \(r_{e}\) by equating this energy with the rest energy of the electron, \(E_0=m_{e}c^{2}\).
  2. Calculate a lower limit for the electron's classical angular velocity \(v_{\varphi}\): To keep things as simple as possible, assume the electron now as a rotating charged loop with radius \(r_{e}\). Compare the resulting magnetic moment to the (approximate) magnetic moment of the electron, the Bohr magneton\[\mu_{B} = \frac{e\hbar}{2m_{e}c}\ .\]Express your result in terms of the fine-structure-constant \[\alpha = \frac{e^{2}}{4\pi\epsilon_{0} \hbar c} \approx \frac{1}{137}\ .\]Physically, does your result for \(v_{\varphi}\) make sense?


Background: The Magnetic Moment of the Electron

The electron is characterized by three different numbers: charge, mass, and spin. Since the famous oil-drop-experiment by Fletcher 1909 under the supervision of Millikan one knew that the electron has both fixed charge and mass. The latter one follows because of known results from charge to mass ratio experiments performed by Thomson at the end of the 19th century. What about the spin and the magnetic moment of the electron? Here, the Stern-Gerlach experiment in 1922 showed that quantum particles have an intrinsic quantized magnetic moment. All these findings had a deep impact on fundamental physics and were extremely important for the formulations of relativistic quantum mechanics (Dirac) and quantum electrodynamics (Tomonaga, Schwinger, Feynman).

So we know that the intrinsic magnetic moment has to be quantized. Indeed, quantum mechanics tells us that the total magnetic moment has to be related to the spin \(\mathbf{s}\) of an electron and of course to its actual motion e.g. orbiting a hydrogen atom with angular momentum \(\mathbf{L}=\mathbf{r}\times\mathbf{p}\). Starting from the spin-free Hamilton of an electron in a (electro)magnetic field with vector potential \(\mathbf{A}\left(\mathbf{r}\right)\),\[\begin{eqnarray*}H&=&\frac{1}{2m}\left(\mathbf{p}-e\mathbf{A}\left(\mathbf{r}\right)\right)^{2}+\phi\left(\mathbf{r}\right)\\&\approx&\frac{\mathbf{p}^{2}}{2m}-\mu_{B}\mathbf{B}\cdot\frac{\mathbf{L}}{\hbar}+\phi\left(\mathbf{r}\right)\ ,\end{eqnarray*}\]we find that the magnetic moment should be something like \(\mathbf{m}_{L}=\mu_{B}\mathbf{L}/\hbar\). Now, we may simply replace the electron angular momentum (operator) \(\mathbf{L}\) with the intrinsic angular momentum (spin) \(\mathbf{s}\). Such an approach, however, is not justified since angular momentum and spin behave differently. This difference is known as the Landé-factor \(g\),\[\mathbf{m}_{s}=\mu_{B}g\frac{\mathbf{s}}{\hbar}\ .\]The Landé-factor can be calculated extremely precise within quantum electrodynamics as \(g\approx-2.002319\dots\). The difference to \(-2\) comes from the interaction of the electron with its surrounding vacuum. So the measurement of \(g\) gives a lot of insight into the applicability of quantum electrodynamics and is of great fundamental interest. If you are interested in how such highly nontrivial measurements can be done and how their results impact physics, you may want to have a look at the elaborated report “Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment” by Hanneke et al., Physical Review A 83 (2011) (or the more concise Physical Review Letter 100 (2008) by the same authors, arXiv, PDF).

Journal Page | arXiv | PDF

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