The Klein-Gordon and Dirac equations in a semi-infinite lab ($x > 0$), in thebackground metric $\ds^2 = u^2(x) (-\dt^2 + \dx^2) + \dy^2 + \dz^2$, areinvestigated. The resulting equations are studied for the special case $ u(x) =1 + g x$. It is shown that in the case of zero transverse-momentum, the squareof the energy eigenvalues of the spin-1/2 particles are less than the squaresof the corresponding eigenvalues of spin-0 particles with same masses, by anamount of $mg\hbar c$. Finally, for nonzero transverse-momentum, the energyeigenvalues corresponding to large quantum numbers are obtained, and theresults for spin-0 and spin-1/2 particles are compared to each other.
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机译：半无限实验室（$ x> 0 $）中的Klein-Gordon和Dirac方程，背景度量$ \ ds ^ 2 = u ^ 2（x）（-\ dt ^ 2 + \ dx ^ 2）+ \ dy ^ 2 + \ dz ^ 2 $被调查。对于特殊情况$ u（x）= 1 + g x $，研究了所得方程。结果表明，在横向动量为零的情况下，spin-1 / 2粒子的能量特征值的平方小于相同质量的spin-0粒子的对应特征值的平方，其乘积为$ mg \ hbar c $。最后，对于非零的横向动量，获得了对应于大量子数的能量特征值，并将自旋-0和自旋1/2粒子的结果进行了比较。
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