High-Dimensional Schrödinger Equation Solver Without Basis Functions
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Solution Overview
Problem
Existing methods for solving the Schrödinger equation, particularly in high-dimensional quantum mechanical problems, face computational challenges due to the 'curse of dimensionality' and reliance on basis functions, leading to high computational demands and systematic errors, which limits the size of systems that can be simulated effectively.
Innovation Solution
A new numerical method transforms the eigenvalue problem into a non-linear differential equation by replacing the wave function with a product of a numerical function and a modulation function, allowing for iterative solutions without requiring boundary conditions and reducing computational effort by focusing on significant sub-spaces, and optionally using quantum computers for enhanced performance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If established methods like DFT or HF with basis functions are used to solve the Schrödinger equation, then computational complexity is reduced, but systematic errors increase and manufacturing precision deteriorates
Solution Approach 1:
The patent extracts and removes the basis function approximation component from the solution process. By directly representing the wave function without basis functions, the method eliminates the source of systematic errors while maintaining computational feasibility through the iterative optimization approach.
Solution Approach 2:
The patent replaces the traditional basis function expansion mechanism with a direct numerical optimization mechanism. Instead of expanding the wave function in terms of predetermined basis functions, the method directly optimizes the wave function parameters to satisfy the Schrödinger equation, substituting the mathematical approximation mechanism with a numerical optimization mechanism.
2Manufacturing precision
If numerical methods like QR decomposition are used to solve the Schrödinger equation, then solution precision is improved, but computational demand increases excessively
Solution Approach 1:
The patent segments the high-dimensional Schrödinger equation solution into manageable iterative steps. By breaking down the problem into successive approximations where the wave function is optimized incrementally, the method makes the computationally intractable problem solvable while maintaining high precision.
Solution Approach 2:
The patent introduces dynamics into the solution process by using iterative optimization. The wave function is not solved statically but evolves through successive iterations, allowing the system to converge to the accurate solution dynamically. This dynamic approach replaces static high-demand numerical methods with an adaptive iterative process.
3Measurement precision
If the wave function is extended to describe multiple electron positions simultaneously, then measurement precision is improved, but the curse of dimensionality increases computational complexity
Solution Approach 1:
The patent addresses the dimensionality problem by changing the approach to handling high-dimensional wave functions. Instead of directly computing in the full high-dimensional space which creates the curse of dimensionality, the method uses iterative optimization that works efficiently even in high dimensions by focusing on the essential degrees of freedom.
Data Source
AI summary
The invention refers to a method for solving the Schrödinger equation (Ψ=EΨ), wherein a Hamilton operator () describes the physical problem underlying the eigenvalue problem, the method being adapted to find a corresponding wave function (Ψ) and an energy (E) to solve the eigenvalue problem, the method comprising the following steps: transforming the eigenvalue problem (Ψ=EΨ) into a non-linear differential equation by replacing the wave function (Ψ) byΨ=g·efEQN001and inserting EQN011 into the eigenvalue problem, where f is a numerical function which needs to be computed and g is a modulation function, finding a solution for the numerical function (f) in an iterative process, aborting the iterative process when an abort condition is fulfilled, and with the found solution for the numerical function (f), calculating the wave function (Ψ) from EQN011.