Invariant Eigen-Structure in Complex-Valued Quantum Mechanics

TL;DR

This paper explores complex-valued quantum mechanics, revealing that eigen-structures of quantum systems are invariant under complex transformations, unifying different quantum system types within a complex domain framework.

Contribution

It demonstrates the invariance of eigen-structures across real and complex quantum systems, unifying Hermitian, PT-symmetric, and non PT-symmetric systems through complex transformations.

Findings

Eigen-structures are invariant under linear complex mapping.

Real and complex quantum systems share identical eigen-structures.

Different quantum system classes can be unified in the complex domain.

Abstract

The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the complex-valued quantum mechanics in that the eigen-structures of real and complex quantum systems, such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant under linear complex mapping. In other words, there is indeed no distinction between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems when viewed from a complex domain. Their eigen-structures can be made coincident through linear transformation of complex coordinates.