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Related Concept Videos

Complex Numbers01:29

Complex Numbers

The real number system cannot represent the square root of a negative number, which restricts solutions for certain equations, such as quadratics with negative discriminants. To address this, the complex number system was developed, introducing the imaginary unit i, where i = √(-1). This extension allows for the representation of all roots, including those involving negative radicands.A complex number is written in the form x + yi, where x and y are real numbers. Here, x represents the real...
The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...
Complex Zeros01:29

Complex Zeros

Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
¹H NMR: Complex Splitting01:13

¹H NMR: Complex Splitting

A proton M that is coupled to a proton X results in doublet signals for M. However, NMR-active nuclei can be simultaneously coupled to more than one nonequivalent nucleus. When M is coupled to a second proton A, such as in styrene oxide, each peak in the doublet is split into another doublet.
Splitting diagrams or splitting tree diagrams are routinely used to depict such complex couplings. While drawing splitting diagrams, the splitting with the larger coupling constant is usually applied first.
Quadratic Equations in the Complex Number System01:29

Quadratic Equations in the Complex Number System

A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of a...
RLC Circuit as a Damped Oscillator01:30

RLC Circuit as a Damped Oscillator

An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
Consider a series RLC circuit. Here, the presence of resistance in the circuit leads to energy loss due to joule heating in the resistance. Therefore, the total electromagnetic energy in the circuit is no longer constant and decreases with time. Since the magnitude of charge, current, and potential difference continuously decreases, their oscillations are said to be damped. This is...

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Related Experiment Video

Updated: May 7, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with

Philipp Bader1, Sergio Blanes, Fernando Casas

  • 1Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain.

The Journal of Chemical Physics
|October 5, 2013
PubMed
Summary
This summary is machine-generated.

Researchers developed new numerical methods for solving the Schrödinger eigenvalue problem using imaginary time propagation. These advanced techniques improve efficiency for diffusive and near-integrable systems, offering better computational performance.

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Area of Science:

  • Computational Quantum Mechanics
  • Numerical Analysis
  • Physical Chemistry

Background:

  • The Schrödinger eigenvalue problem is fundamental in quantum mechanics.
  • Imaginary time propagation is a standard technique for solving such problems.
  • Existing splitting methods face limitations with high-order fractional time steps for diffusive systems.

Purpose of the Study:

  • To develop and analyze improved numerical methods for the Schrödinger eigenvalue problem.
  • To overcome limitations of existing high-order time-stepping schemes.
  • To enhance computational efficiency for various quantum systems.

Main Methods:

  • Application of splitting methods to separable Hamiltonians.
  • Analysis and development of fractional complex time step schemes with positive real parts.
  • Exploration of methods utilizing potential gradients, including fourth and sixth-order schemes.
  • Proposal of a time-stepping variable order algorithm.

Main Results:

  • New fractional complex time step schemes outperform existing methods for diffusive problems.
  • Novel fourth-order methods are proposed for cases with available potential gradients.
  • Highly optimized sixth-order schemes are presented for near-integrable systems.
  • The variable order algorithm demonstrates enhanced efficiency.

Conclusions:

  • The developed numerical methods offer significant improvements in efficiency and accuracy.
  • Fractional complex time steps provide a viable alternative for high-order time propagation.
  • The proposed schemes are particularly effective for near-integrable and diffusive quantum systems.