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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...
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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

Functional density matrix formulation of quantum statistics.

A Bessa1, C A A de Carvalho, E S Fraga

  • 1Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil. abessa@ect.ufrn.br

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 7, 2010
PubMed
Summary

This study introduces a new functional integral method for quantum statistical physics, simplifying calculations for thermal field theories and quantum systems. The approach offers an alternative to traditional methods, yielding accurate results for free energy and specific heat.

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Area of Science:

  • Quantum Statistical Physics
  • Quantum Field Theory
  • Mathematical Physics

Background:

  • Traditional methods for quantum statistical physics, such as summing over periodic trajectories, can be computationally intensive.
  • A unified formulation for quantum statistical physics is needed to simplify calculations and provide alternative approaches.

Purpose of the Study:

  • To present a unified formulation for quantum statistical physics using functional integrals.
  • To develop an effective theory for boundary field configurations in quantum statistical (thermal) field theory.
  • To apply this method to compute the partition function of a one-dimensional quantum-mechanical system at finite temperature.

Main Methods:

  • Representing the density matrix as a functional integral.
  • Developing an effective theory for stochastic boundary field configurations.
  • Calculating the partition function using a sum over paths with coincident end points and non-vanishing boundary terms.
  • Employing a modified expansion into modified Matsubara modes.

Main Results:

  • The functional integral formulation provides a unified approach to quantum statistical physics.
  • The effective theory for boundary configurations simplifies calculations.
  • The method accurately computes the partition function for an interacting one-dimensional quantum-mechanical system.
  • Results for free energy and specific heat show excellent agreement with established semiclassical methods.

Conclusions:

  • The proposed functional integral method offers a powerful and efficient alternative for quantum statistical physics calculations.
  • This approach simplifies the computation of partition functions for quantum systems at finite temperatures.
  • The method's accuracy is validated by its agreement with existing semiclassical results.