The Schrödinger equation in the complex plane and quantum entanglement

• URL: http://arxiv.org/abs/2606.28640v1
• Date: Fri, 26 Jun 2026 22:52:29 GMT
• Title: The Schrödinger equation in the complex plane and quantum entanglement
• Abstract summary: This momentum is a quantum analog of the classical, kinematic momentum analytically continued into the complex plane.<n>The zeros of the wavefunction give rise to simple poles in the momentum.<n>The zero-point vibrations within bound states are solely due to the fluctuations of the momentum and manifest as rigid-body flows at infinity.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We formulate a continuity equation for the Schrödinger equation in the complex space. We define a complex momentum by normalizing the complex current by the particle density. This momentum is a quantum analog of the classical, kinematic momentum analytically continued into the complex plane. The kinematic momentum and the gradient of the wavefunction's phase each represent a fluid-like flow in the complex plane; the phase-gradient flow is incompressible. The zeros of the wavefunction give rise to simple poles in the momentum. The poles manifest as irrotational vortexes in the phase-gradient flow, while critical points of the wavefunction present as rigid body-like rotational flows of the kinematic momentum. A discrete nature of elementary excitations comes about inherently because the quantity of the poles is automatically integer. An exact quantization condition is subsequently formulated, which reduces to the Bohr-Sommerfeld condition in the semiclassical limit. We establish a priori that the Bohr-Sommerfeld condition must be exact for the Harmonic Oscillator. We show that the kinetic energy is a sum of contributions of the average value and fluctuations, respectively, of the kinematic momentum. The zero-point vibrations within bound states are solely due to the fluctuations of the momentum and manifest as rigid-body flows at infinity. The momentum poles -- and hence the wavefunction's zeros -- can be viewed as emergent, consistent with the remarkable property of quantum entanglement exhibited by standing wave solutions of the Schrödinger equation.

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