Non-Unitary Behavior of Three-Dimensional Complex Numbers in Quantum Evolution: Numerical Experiments and Controlled Verification 三维复数在量子演化中的非幺正行为:数值实验与对照验证

This report systematically investigates the numerical behavior of the three-dimensional complex algebra under the standard quantum mechanical evolution framework. The three-dimensional complex number is a commutative and associative real algebra spanned by , satisfying . By replacing the conventional complex-valued wave function with a -valued wave function while preserving the formal structure of the Schrödinger equation, numerical evolution under the explicit Euler scheme exhibits continuous norm drift and a directional component transfer from to . To isolate algebraic effects from numerical artifacts, we design controlled experiments with standard two-dimensional complex numbers under identical computational conditions. The results show that both 2D complex and 3D complex systems produce nearly identical norm drift magnitudes and growth patterns. Both systems exhibit exponential numerical divergence after approximately 600 iteration steps under the explicit Euler method. We conclude that the observed non-unitary drift primarily originates from the inherent instability of the explicit Euler discretization, rather than from the algebraic structure of . Further high-precision verification using unitary-preserving algorithms such as the Crank–Nicolson scheme is required to confirm whether possesses intrinsic non-unitary quantum evolution behavior.