Gravitational well is a widely used system for the verification of the quantum weak equivalence principle (WEP). We have studied the quantum gravitational well under the shed of noncommutative (NC) space so that the results can be utilized for testing the validity of WEP in NC-space. To keep our study widely usable, we have considered both position–position and momentum–momentum noncommutativity. Since coherent state (CS) structure provides a natural bridge between the classical and quantum domain descriptions, the quantum domain validity of purely classical phenomena like free fall under gravity might be verified with the help of CS. We have constructed CS with the aid of a Lewis–Riesenfeld phase-space invariant operator. We deduced the uncertainty relations from the expectation values of the observables and showed that the solutions of the time-dependent Schrödinger equation are squeezed coherent states.

引力阱是一种广泛用于验证量子弱当量原理（WEP）的系统。我们已经研究了非可交换（NC）空间下的量子引力井，以便将结果用于测试WEP在NC空间中的有效性。为了使我们的研究广泛可用，我们考虑了位置-位置和动量-动量的非对易性。由于相干态（CS）结构提供了经典域和量子域描述之间的自然桥梁，因此可以借助CS验证纯经典现象（如在重力作用下自由落体）的量子域有效性。我们借助Lewis-Riesenfeld相空间不变算子构造了CS。