Exact Solution of a Spin-$\frac{1}{2}$ Particle for a Linear Potential

The problem of a spin-$\frac{1}{2}$ particle moving in a linear potential field in two-dimensions is searched to obtain for nonzero energy eigenvalues and the corresponding normalized eigenfunctions. The zero-mode ($E=0$) eigenfunctions are also studied and it is seen that they are normalizable. The variation of the non-zero eigenfunctions and also eigenvalues are according to the position and potential parameter $\gamma$, respectively, given in the text.