frame. The basis set for the eigenspinors is then given by the eigenvectors of $\sigma_{3}$; the orientation of each is responsible for the addition of a constant in the differential equation, leaving the overall structure unaffected. Thus the differential equation is separable in all four components, with solutions given by the product of free particle states for $x^{\prime}_{2}$ and $x^{\prime}_{3}$ and eigenstates of the displaced harmonic oscillator, $H_{n}$, for $x^{\prime}_{1}$. The explicit expressions for the eigenspinors are $$\begin{aligned}
\phi_{P \uparrow}^{\prime} &= e^{i  k_{\perp}' x_{\perp}' }  \left( \begin{array}{c}
H_{\vert n \vert} \left( x_{1}'/l_{\mathcal{B}}'+ l_{\mathcal{B}}' k_{2}' \right)  \\
0 \\
\end{array} \right) \, , \label{eige_6a}\\
\phi_{P \downarrow}^{\prime} &= e^{i  k_{\perp}' x_{\perp}' } \left( \begin{array}{c}
0 \\
H_{\vert n \vert +1} \left( x_{1}'/l_{\mathcal{B}}'+ l_{\mathcal{B}}' k_{2}' \right) \\
\end{array} \right) \, , \label{eige_6b}\end{aligned}$$ with the corresponding Dirac particle energies given by $$\label{eige_7}
\upvarepsilon_{n,\pm}^{ \prime } = \pm\left( \frac{\hbar c} { l_{\mathcal{B}}'} \right) \sqrt{2 \vert n \vert + 2  + (l_{\mathcal{B}}' k^{\prime}_{3}  )^{2}} \, ,$$ the eigenvalues of $\hat{p}_{0}^{\prime}$. The quantized motion described by these solutions is caused by the harmonic potential created through the interaction between particle spin and magnetic field. The Dirac spinors are obtained using equation (\[eige\_0\]) and will be given by 