This paper discusses the concept of PT-symmetric Hamiltonians in quantum mechanics, which are non-Hermitian but still describe physical systems with real energy spectra and unitary time evolution. The traditional requirement that the Hamiltonian be Hermitian is replaced by the condition of PT symmetry, which involves space-time reflection symmetry. The paper presents examples of PT-symmetric Hamiltonians, such as $ H = \hat{p}^{2} + i\hat{x}^{3} $ and $ H = \hat{p}^{2} - \hat{x}^{4} $, and shows that their energy levels are real and positive. It also discusses the Lee Model, where the non-Hermitian Hamiltonian is shown to be PT-symmetric, and the C operator is used to construct a positive-definite inner product. The paper explores the implications of PT symmetry, including the possibility of complex extensions of quantum mechanics and the role of boundary conditions in ensuring real eigenvalues. It also addresses the challenges in verifying the physical existence of PT-symmetric Hamiltonians and the importance of rigorous mathematical proofs in establishing their validity. The paper concludes with a discussion of the broader implications of PT symmetry in quantum mechanics and its potential applications in various physical systems.This paper discusses the concept of PT-symmetric Hamiltonians in quantum mechanics, which are non-Hermitian but still describe physical systems with real energy spectra and unitary time evolution. The traditional requirement that the Hamiltonian be Hermitian is replaced by the condition of PT symmetry, which involves space-time reflection symmetry. The paper presents examples of PT-symmetric Hamiltonians, such as $ H = \hat{p}^{2} + i\hat{x}^{3} $ and $ H = \hat{p}^{2} - \hat{x}^{4} $, and shows that their energy levels are real and positive. It also discusses the Lee Model, where the non-Hermitian Hamiltonian is shown to be PT-symmetric, and the C operator is used to construct a positive-definite inner product. The paper explores the implications of PT symmetry, including the possibility of complex extensions of quantum mechanics and the role of boundary conditions in ensuring real eigenvalues. It also addresses the challenges in verifying the physical existence of PT-symmetric Hamiltonians and the importance of rigorous mathematical proofs in establishing their validity. The paper concludes with a discussion of the broader implications of PT symmetry in quantum mechanics and its potential applications in various physical systems.