Editing Geometric phase (section)

=== Geometric phase and quantization of cyclotron motion ===

An electron subjected to magnetic field <math>B</math> moves on a circular (cyclotron) orbit.{{ref|plan}} Classically, any cyclotron radius <math>R_c</math> is acceptable. Quantum-mechanically, only discrete energy levels ([[Landau quantization|Landau levels]]) are allowed, and since <math>R_c</math> is related to electron's energy, this corresponds to quantized values of <math>R_c</math>. The energy quantization condition obtained by solving Schrödinger's equation reads, for example, <math>E = (n + \alpha)\hbar\omega_c,</math> <math>\alpha = 1/2</math> for free electrons (in vacuum) or <math display="inline">E = v \sqrt{2(n + \alpha)eB\hbar},\quad \alpha = 0</math> for electrons in [[graphene]], where <math>n = 0, 1, 2, \ldots</math>.{{ref|cyclo}} Although the derivation of these results is not difficult, there is an alternative way of deriving them, which offers in some respect better physical insight into the Landau level quantization. This alternative way is based on the semiclassical [[Bohr–Sommerfeld quantization]] condition
<math display="block">
\hbar\oint d\mathbf{r} \cdot \mathbf{k} - e\oint d\mathbf{r}\cdot\mathbf{A} + \hbar\gamma = 2 \pi \hbar (n + 1/2),
</math>
which includes the geometric phase <math>\gamma</math> picked up by the electron while it executes its (real-space) motion along the closed loop of the cyclotron orbit.<ref>For a tutorial, see Jiamin Xue: "[https://arxiv.org/abs/1309.6714 Berry phase and the unconventional quantum Hall effect in graphene]" (2013).</ref> For free electrons, <math>\gamma = 0,</math> while <math>\gamma = \pi</math> for electrons in graphene. It turns out that the geometric phase is directly linked to <math>\alpha = 1/2</math> of free electrons and <math>\alpha = 0</math> of electrons in graphene.