[Paper Review] Derivation of Nonlinear Schrödinger Equation

Xiang-Yao Wu, B. Zhang+6 more|arXiv (Cornell University)|Apr 1, 2011

Nonlinear Photonic Systems2 references3 citations

TL;DR

This paper derives nonlinear Schrödinger equations by adding higher-order terms to the Lagrangian density of the Schrödinger field, naturally yielding the Gross-Pitaevskii and logarithmic nonlinear Schrödinger equations. The key result is that the nonlinear terms are extremely weak due to Planck-scale suppression, implying that nonlinearity in quantum systems is negligible under standard conditions.

ABSTRACT

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.

Motivation & Objective

• To explore the origin of nonlinear Schrödinger equations within a field-theoretic framework using higher-order Lagrangian terms.
• To derive the Gross-Pitaevskii and logarithmic nonlinear Schrödinger equations from a consistent variational principle.
• To assess the strength of nonlinearity in such equations and determine whether it is observationally significant.
• To establish a theoretical basis for weak nonlinearity in quantum fields, relevant to condensed matter and nonlinear physics.

Proposed method

• Add higher-order terms of the form $(\psi^*\psi)^m$ to the standard Schrödinger field Lagrangian density.
• Apply the Euler-Lagrange equation to the modified Lagrangian density to derive the corresponding field equations.
• Use dimensional analysis to express coupling constants in terms of fundamental constants ($\hbar$, $m$, $c$) to assess the magnitude of nonlinearity.
• Compare derived equations with known forms: Gross-Pitaevskii (m=2) and logarithmic (m=1 with log term).
• Analyze the dimensionality of coupling constants to show they are suppressed by powers of $\hbar^3/(m^2c)$ or $\hbar^6/(m^5c^4)$, indicating weak nonlinearity.
• Verify that the resulting equations reduce to the standard GP and logarithmic forms under appropriate parameter choices.

Experimental results

Research questions

• RQ1 Can the Gross-Pitaevskii equation be derived from a fundamental Lagrangian by adding higher-order terms to the Schrödinger field Lagrangian?
• RQ2 Can the logarithmic nonlinear Schrödinger equation emerge from a similar variational principle with a logarithmic interaction term?
• RQ3 How large is the coefficient of the nonlinear term in such derived equations, and is it consistent with observational constraints?
• RQ4 What is the physical origin and magnitude of nonlinearity in the Schrödinger equation when higher-order terms are included?
• RQ5 Are the resulting nonlinear equations consistent with the principles of quantum field theory and second quantization?

Key findings

• The Gross-Pitaevskii equation is derived as a special case (m=2) of a generalized nonlinear Schrödinger equation from a higher-order Lagrangian term $(\psi^*\psi)^2$.
• The logarithmic nonlinear Schrödinger equation arises naturally from a Lagrangian term proportional to $\psi^*\psi \ln(\psi^*\psi)$.
• The coupling constant for the m=2 case is $e = g \hbar^3 / (m^2 c)$, where $g$ is dimensionless, leading to a nonlinear term suppressed by $\hbar^3 / (m^2 c)$.
• For m=3, the coupling constant is $e = g' \hbar^6 / (m^5 c^4)$, further suppressing the nonlinearity by higher powers of $\hbar$.
• The nonlinear terms are extremely small—on the order of $\hbar^3 / (m^2 c) \ll 1$—indicating that nonlinearity in the Schrödinger equation is weak in physical systems.
• The derived equations include known models (GP and logarithmic) as special cases, confirming consistency with established nonlinear quantum field theories.