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The Quantum-Mechanical Model of an Atom02:45

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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.
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The mathematical expression known as the wave function, ψ, contains information about each orbital and the wavelike properties of electrons in an isolated atom. When atoms are bound together in a molecule, the wave functions combine to produce new mathematical descriptions that have different shapes. This process of combining the wave functions for atomic orbitals is called hybridization and is mathematically accomplished by the linear combination of atomic orbitals. The new orbitals that...
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Orbitals are the areas outside of the atomic nucleus where electrons are most likely to reside. They are characterized by different energy levels, shapes, and three-dimensional orientations. The location of electrons is described most generally by a shell or principal energy level, then by a subshell within each shell, and finally, by individual orbitals found within the subshells.
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A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?
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Updated: Oct 3, 2025

Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
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Neural network approaches for solving Schrödinger equation in arbitrary quantum wells.

A Radu1, C A Duque2

  • 1Department of Physics, Politehnica University of Bucharest, 313 Splaiul Independenței, 060042, Bucharest, Romania. adrian.radu@physics.pub.ro.

Scientific Reports
|February 16, 2022
PubMed
Summary

Machine learning accurately solves the Schrödinger equation for quantum wells. Neural networks, trained with finite element method data, provide reliable energy and wave function predictions for various potentials.

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Area of Science:

  • Quantum mechanics
  • Computational physics
  • Machine learning

Background:

  • The Schrödinger equation is fundamental to quantum mechanics.
  • Solving it for arbitrary potentials in quantum wells is computationally challenging.
  • Machine learning offers a novel approach to accelerate these calculations.

Purpose of the Study:

  • To apply machine learning techniques to solve the Schrödinger equation in quantum wells.
  • To develop and compare two distinct neural network architectures for this task.
  • To establish accuracy indicators for evaluating the machine learning model's performance.

Main Methods:

  • Utilizing two neural network architectures.
  • Training networks with data generated via the finite element method (FEM).
  • Employing gradient descent for network training and validation against extensive datasets.

Main Results:

  • Developed and validated two neural networks for solving the Schrödinger equation.
  • Proposed three accuracy indicators to assess network performance.
  • Achieved accurate predictions for energies and wave functions across diverse potentials.

Conclusions:

  • Machine learning, specifically neural networks, provides an effective method for solving the Schrödinger equation in quantum wells.
  • The proposed models demonstrate high accuracy and generalizability for arbitrary potentials.
  • This approach offers a promising alternative to traditional numerical methods.