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The Quantum-Mechanical Model of an Atom

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.

¹H NMR: Interpreting Distorted and Overlapping Signals

¹H NMR: Interpreting Distorted and Overlapping Signals

Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...

Atomic Nuclei: Nuclear Relaxation Processes

Atomic Nuclei: Nuclear Relaxation Processes

In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis, the precessing magnetic moments are randomly oriented around the z-axis.

The de Broglie Wavelength

The de Broglie Wavelength

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...

Entropy Change in Reversible Processes

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

First Law: Particles in One-dimensional Equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton's first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a net external force. It also states that there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This cause is a net external force. For example, consider what happens to an object sliding along a rough horizontal surface. The object quickly grinds to a halt, due to the net force of friction. If...

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Updated: Feb 22, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

非局所的な相互作用を持つ複雑な量子システムにおける異常動力学.

P Trajanovski^1,2, E K Lenzi³, I Petreska¹

¹Institute of Physics, Faculty of Natural Sciences and Mathematics-Skopje, Ss. Cyril and Methodius University in Skopje, Skopje, Macedonia.

Chaos (Woodbury, N.Y.)

|February 20, 2026

まとめ

この要約は機械生成です。

この研究では,偏微分派生と記憶効果を備えた一般化されたシュロディンガー方程式を導入し,異常な輸送と非局所相互作用をモデル化しています. 発見は,新しい量子現象と,時空ダイナミクスにおける独特の局所化振る舞いを明らかにしています.

さらに関連する動画

Study of Protein Dynamics via Neutron Spin Echo Spectroscopy

Study of Protein Dynamics via Neutron Spin Echo Spectroscopy

Published on: April 13, 2022

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Published on: May 27, 2020

関連する実験動画

Last Updated: Feb 22, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Study of Protein Dynamics via Neutron Spin Echo Spectroscopy

Study of Protein Dynamics via Neutron Spin Echo Spectroscopy

Published on: April 13, 2022

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Published on: May 27, 2020

科学分野:

量子力学は,量子力学という
数学物理学の数学物理学について
非線形ダイナミクス非線形ダイナミクス

背景:

異常な輸送現象には,標準微分方程式を超えた高度な数学モデルが必要です.
時間の非局所性と長距離相互作用は,様々な量子システムにおいて極めて重要です.

研究の目的:

微分微分と記憶効果を組み込んだ一般化されたシュロディンガー方程式を提案し,調査する.
量子力学における非局所的潜在力と記憶の影響を分析する.
提案されたモデルの分析的解決策を導き出し,探求する.

主な方法:

異常輸送のための分数リエス導関数の導入.
時間の非局所性のためのメモリカーネルの含有.
積分演算子を使用して長距離の相互作用をモデリングする.
分析解を導き出すためのグリーン関数アプローチの適用.

主要な成果:

グリーン関数法を用いた分析解の導出.
分数ダイナミクス,ノンローカルポテンシャル,およびメモリ効果の相互作用から生じる新しい量子現象の識別.
グリーンの関数の進化における新しい局所最大値の観測.
時空領域における異なる局所化行動の特徴化.

結論:

一般化されたシュレーディンガー方程式は,異常な輸送と非局所的な影響を効果的に捉えます.
分数ダイナミクス,メモリ,および長距離相互作用の相互作用は,ユニークな量子行動につながります.
この発見は,複雑な輸送と相互作用メカニズムを持つ量子力学に関する新しい洞察を提供します.