A real expectation value of the time-dependent non-Hermitian Hamiltonians

Abstract

With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent $\mathcal{P}\mathcal{T}$-symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the $\mathcal{C}(t)\mathcal{P}\mathcal{T}$-inner product, where $\mathcal{C}(t)$ is a obtained from the charge conjugation operator $\mathcal{C}$ through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the $\mathcal{C}(t)\mathcal{P}\mathcal{T}$ normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

arXiv comment12 pages

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Cited by in corpus (2)

• Exact solutions for time-dependent complex symmetric potential well
• Uncertainty Relation for Pseudo-Hermitian Quantum Systems