Inspired by the recent work of Filho et al., a Hermitian momentum operator is introduced in a general curved space with diagonal metric. The modified Hamiltonian associated with this new momentum is calculated and discussed. Furthermore, granting the validity of the Heisenberg equation in a curved space, the Ehrenfest theorem is generalized and interpreted with the new position-dependent differential operator in a curved space. The modified Hamiltonian leads to a modified time-independent Schr\"odinger equation, which is solved explicitly for a free particle in the Poincar\'e upper half-plane geometry. It is shown that a "free particle" does not behave as it is totally free due to curved background geometry.

中文翻译：

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自由量子粒子在 Poincaré 上半平面几何中的行为

受 Filho 等人最近工作的启发，在具有对角度量的一般弯曲空间中引入了 Hermitian 动量算子。计算和讨论了与这个新动量相关的修正哈密顿量。此外，为了证明 Heisenberg 方程在弯曲空间中的有效性，在弯曲空间中使用新的位置相关微分算子对 Ehrenfest 定理进行了推广和解释。修正的哈密顿量导致修正的与时间无关的 Schr\"odinger 方程，该方程明确求解 Poincar\'e 上半平面几何中的自由粒子。结果表明，“自由粒子”并不像它那样表现由于弯曲的背景几何形状，完全免费。