Electron Wavelength and Hydrogen Atom Structure:

Soumendra Nath Thakur

27-10-2023

DOI:10.13140/RG.2.2.35955.91685

Abstract: 

This research explores the intricate relationship between the de Broglie wavelength of an electron and the fundamental components of atomic hydrogen, namely the nucleus and proton. We delve into the implications of this relationship for the understanding of atomic structure. With the nucleus and proton both measuring approximately 1 femtometer (1 fm), and the de Broglie wavelength of an electron at roughly 0.1 nanometers (0.1 nm), we examine the minute differences between these sizes.

Our findings reveal that the de Broglie wavelength of an electron is just shy of the size of the nucleus and proton in atomic hydrogen, signifying that the electron's core cannot approach the nucleus. Furthermore, we discuss the effect of electron energy changes on its de Broglie wavelength and the resulting alterations in electron orbits. This research sheds light on the wave-particle duality of electrons and its impact on atomic structure, providing valuable insights into the behavior of electrons in the microscopic world.

Soumendra Nath Thakur

Tagore’s Electronic Lab, India

ORCID iD: 0000-0003-1871-7803

Keywords: De Broglie wavelength, Atomic hydrogen, Nucleus size, Electron behavior, Energy dependent orbits,

1. Introduction:

The world of quantum physics is a realm of fascinating and often perplexing phenomena. One of the central principles that underlie the behavior of particles on this microscopic scale is the wave-particle duality, as first introduced by Louis de Broglie. According to this concept, particles, such as electrons, exhibit both particle-like and wave-like characteristics. A fundamental parameter that helps us grasp this wave-like aspect of particles is the de Broglie wavelength.

In this research, we delve into the intriguing relationship between the de Broglie wavelength of an electron and the components of atomic hydrogen, particularly the nucleus and proton. The nucleus and proton, both with a size on the order of 1 femtometer (1 fm), are the building blocks of atomic hydrogen, while the de Broglie wavelength of an electron is approximately 0.1 nanometers (0.1 nm). These minuscule measurements lead us to investigate the fine differences between these sizes and their implications.

Our exploration seeks to unravel the significance of these size differentials and their impact on the structure of an atom. The outcome of this research unveils an intriguing revelation: the de Broglie wavelength of an electron hovers very close to the size of the atomic nucleus and proton in hydrogen. This proximity suggests that the electron's core cannot draw near to the nucleus, raising questions about the dynamics of electrons within atoms.

Additionally, we examine the connection between changes in electron energy, expressed as hf (where h is the Planck constant and f is the frequency), and the alterations in the de Broglie wavelength of the electron. These energy-related transformations play a vital role in influencing the electron's orbits, adding another layer of complexity to the study of atomic structure.

This research takes us on a journey into the subtle intricacies of quantum mechanics, offering valuable insights into the behavior of electrons in the microscopic world. It illustrates how wave-particle duality is a fundamental concept that governs the behavior of particles like electrons and emphasizes the relevance of size differentials within the subatomic realm. Through these explorations, we aim to contribute to a deeper understanding of atomic physics and its profound implications for the world of quantum science.

2. Method:

Defining the Components:

Begin by defining the fundamental components involved in the research: the atomic nucleus, the proton, and the electron.

Establish their respective sizes, with a particular focus on the nucleus and proton, both at approximately 1 femtometer (1 fm).

Introducing the De Broglie Wavelength:

Present the concept of the de Broglie wavelength and its significance in quantum mechanics. Highlight that the de Broglie wavelength of an electron is approximately 0.1 nanometers (0.1 nm).

Calculating Size Differences:

Calculate the numerical difference between the size of the de Broglie wavelength of an electron and the size of the atomic nucleus and proton. Express the calculations in nanometers for clarity and relevance.

Analyzing Implications:

Explore the implications of these size differentials in the context of atomic hydrogen's structure. Consider the restrictions placed on the electron's core in relation to the nucleus.

Energy-Related Investigations:

Investigate the role of changes in electron energy, represented as hf, where h is the Planck constant and f is the frequency. Discuss how variations in energy impact the de Broglie wavelength of the electron and its resulting orbital changes.

Theoretical Framework:

Employ relevant theoretical frameworks in quantum mechanics to interpret the findings. Connect the size differentials and energy variations to established quantum principles.

Data Visualization:

Utilize diagrams, charts, or illustrations to visually represent the size differences between the components. Present any relevant mathematical equations or formulae used in the calculations.

Discussion and Conclusion:

Summarize the research findings and their significance in understanding atomic structure. Offer insights into the implications of the de Broglie wavelength's proximity to the nucleus size. Discuss how energy-related changes influence electron behavior and orbital dynamics.

3. Mathematical Presentation:

In this section, we will delve into the mathematical aspects of the research to provide a quantitative understanding of the size differentials between the components of atomic hydrogen and the implications for electron behavior. By utilizing these mathematical expressions and calculations, the research elucidates the size disparities between fundamental atomic components and how these differences influence electron behavior and energy-related orbital dynamics in atomic hydrogen.

3.1. Defining Component Sizes:

 Atomic Nucleus Size (R_nucleus) = 1 femtometer (1 fm)

Proton Size (R_proton) = 1 femtometer (1 fm)

De Broglie Wavelength of Electron (λ_electron) ≈ 0.1 nanometers (0.1 nm)

3.2. Size Difference Calculations:

Difference between Electron Wavelength and Nucleus/Proton Sizes:

Δλ = λ_electron - R_nucleus = 0.1 nm - 1 fm = 0.1 nm - 0.000001 nm = 0.099999 nm

Difference between Electron Wavelength and Hydrogen Atom Size (R_atom):

 Δλ = λ_electron - R_atom = 0.1 nm - 0.1 nm = 0 nm (or very close to 0 nm)

3.3. Implications of Size Differences:

The size of the De Broglie wavelength of an electron (0.099999 nm) is greater than the size of the atomic nucleus or proton in atomic hydrogen (1 fm or 0.000001 nm).

This indicates that the electron's core cannot reach or approach the nucleus or proton in an atom, suggesting a fundamental spatial limitation in atomic hydrogen.

4.3. Energy-Related Changes:

Changes in electron energy (E) are associated with changes in frequency (f), given by Planck's relation: E = hf.

As energy (hf) increases, the de Broglie wavelength of the electron (λ_electron) decreases, reaching down to approximately 0.1 nm.

These changes in energy are directly connected to alterations in electron orbits and behavior, corresponding to energy loss or gain by the electron.

4. Discussion:

The research presented here explores the intriguing size differentials and their implications for electron behavior within atomic hydrogen. We have observed significant insights into the spatial relationships between key atomic components and how they correlate with the behavior of electrons. The discussion delves into the profound implications of these findings.

4.1. Core Limitation in Atomic Hydrogen:

The research reveals that the De Broglie wavelength of an electron, which is approximately 0.1 nanometers, is significantly larger than the size of the atomic nucleus or a proton in atomic hydrogen, both of which measure 1 femtometer (0.000001 nanometers). This size differential highlights a fundamental limitation – the electron's core cannot reach or closely approach the nucleus or proton within the atom.

4.2. Energy-Dependent Behavior:

It is essential to recognize that changes in electron energy (E) result in alterations in its frequency (f), as defined by Planck's equation, E = hf. Consequently, these changes in energy influence the De Broglie wavelength of the electron. As the energy (hf) increases, the electron's wavelength decreases, reaching down to approximately 0.1 nanometers.

4.3. Implications for Electron Orbits:

The De Broglie wavelength of an electron, being greater than the size of the atomic nucleus or proton, suggests that the electron's spatial distribution is diffused and wave-like. This leads to a core limitation, making it improbable for the electron to exist within the nucleus. Hence, electron orbits are determined by energy changes, which cause shifts in the De Broglie wavelength and, consequently, the electron's orbital behavior.

4.4. Energy Loss or Gain:

Energy loss or gain by the electron is closely related to orbital changes. Lower energy states correspond to longer De Broglie wavelengths, allowing electrons to occupy higher energy orbits farther from the nucleus. Conversely, higher energy states result in shorter wavelengths, leading to electrons being closer to the nucleus in lower energy orbits. This energy-dependent behavior underscores the importance of energy considerations in atomic hydrogen.

The findings in this research illustrate the intricate interplay between electron size, energy, and orbital behavior in atomic hydrogen. The De Broglie wavelength's interaction with the size of atomic components informs us about the fundamental limitations and energy-driven dynamics that govern the behavior of electrons in the microscopic world of quantum physics. These insights provide a more comprehensive understanding of atomic structure and electron behavior in hydrogen and offer valuable implications for broader applications in quantum mechanics and atomic physics.

5. Conclusion:

This research delves into the intriguing relationship between the De Broglie wavelength of electrons and the size of atomic components within atomic hydrogen. We have explored how the size differentials between electrons, the atomic nucleus, and protons affect the behavior of electrons in atomic hydrogen and how changes in energy play a crucial role in determining electron orbits. The key findings and their implications can be summarized as follows:

5.1. Core Limitation and Electron Behavior:

The De Broglie wavelength of an electron is approximately 0.1 nanometers, significantly larger than the size of the atomic nucleus or a proton, both measuring 1 femtometer. This difference indicates a fundamental core limitation – electrons cannot approach or exist within the nucleus. As a result, electron behavior in atomic hydrogen is inherently wave-like and diffuse.

5.2. Energy-Dependent Orbit Changes:

Changes in electron energy directly influence the De Broglie wavelength, and thus, the electron's orbital behavior. Higher energy states lead to shorter wavelengths, causing electrons to occupy lower energy orbits closer to the nucleus, while lower energy states correspond to longer wavelengths and electrons residing in higher energy orbits farther from the nucleus.

5.3. Energy Dynamics in Atomic Hydrogen:

The research highlights the significance of energy considerations in understanding electron orbits and behavior within atomic hydrogen. Energy loss or gain directly impacts the De Broglie wavelength, which, in turn, governs electron positions and orbits within the atom.

This research provides valuable insights into the fundamental limitations of electron behavior within atomic hydrogen and the pivotal role that energy plays in determining electron orbits. The De Broglie wavelength's interplay with atomic sizes offers a profound understanding of the complex dynamics at the atomic scale. These findings not only contribute to our knowledge of atomic hydrogen but also have broader applications in quantum mechanics and atomic physics. They underscore the intricate relationship between size, energy, and electron behavior in the microscopic realm of quantum physics, further enriching our comprehension of atomic structures and the behavior of electrons.

 6. References:

[1] Principles of Quantum Mechanics by R. Shankar:

[2] Introduction to Quantum Mechanics" by David J. Griffiths:

[3] Modern Physics" by Kenneth S. Krane:

[4] Atomic Physics" by Christopher J. Foot:

[5] Thakur, S. N. (2023, August 24). Relativistic effects and photon-mirror interaction -energy absorption and time delay. ResearchGate. https://doi.org/10.13140/RG.2.2.20928.71683

[6] Thakur, S. N., Bhattacharjee, D., & Frederick, O. (2023, September 22). Photon Interactions in Gravity and Antigravity: Conservation, Dark Energy, and Redshift Effects. ResearchGate. https://doi.org/10.13140/RG.2.2.31280.94720

Summary of Electron Wavelength and Hydrogen Atom Structure:

Abstract:

This research paper investigates the relationship between the De Broglie wavelength of electrons and the atomic components in hydrogen, with a focus on its implications for atomic structure. We explore how the electron wavelength influences the behavior of electrons within atomic hydrogen, shedding light on the core interactions within the atom.

Introduction:

The De Broglie wavelength is a fundamental concept in quantum mechanics, representing the wave-like nature of particles, such as electrons. In the context of atomic hydrogen, understanding the De Broglie wavelength of electrons and its relationship with the atomic components is essential for comprehending atomic structure and behavior. This paper delves into the intricacies of this relationship.

Method:

We employ the De Broglie wavelength formula, λ = h / p, to calculate the wavelengths of electrons within atomic hydrogen. By examining the interplay between the electron's momentum, kinetic energy, and velocity, we gain insights into how the electron wavelength influences its behavior within the atom.

Mathematical Presentation:

We present a mathematical representation of the De Broglie wavelength and its implications for electron behavior in atomic hydrogen. This presentation highlights the significance of the De Broglie wavelength in quantum mechanics and atomic physics.

Discussion:

Our discussion explores how the De Broglie wavelength of electrons in atomic hydrogen relates to the size of the atomic components, including the nucleus and the atom itself. We investigate the energy-dependent nature of the electron wavelength and its impact on electron orbits within the atom.

Conclusion:

Our research underscores the importance of the De Broglie wavelength in understanding the behavior of electrons in atomic hydrogen. We conclude that the electron wavelength is closely linked to the size of the atomic components and plays a pivotal role in shaping atomic structure. This relationship has implications for energy changes in electrons, leading to variations in their orbits within the atom.

This study offers valuable insights into the wave-particle duality of electrons and its profound impact on atomic hydrogen, enriching our comprehension of atomic structures and quantum phenomena.

References:

[1] Principles of Quantum Mechanics" by R. Shankar: 

[2] Introduction to Quantum Mechanics by David J. Griffiths:

[3] Modern Physics by Kenneth S. Krane: 

[4] Atomic Physics by Christopher J. Foot: 

Summary of Analytical Representation of the Degrees of Freedom: Referencing Quantum Cosmology:

This research paper, titled, ”Analytical Representation of the Degrees of Freedom: Referencing Quantum Cosmology, DOI: 10.13140/RG.2.2.25774.89923”,  explores the profound implications of dimensional analysis and its applications in the context of fundamental physics. Dimensions, including length (L), width (W), height (H), and time (T), serve as the foundational attributes of our physical world. The powers of these dimensions are fundamental in understanding how physical quantities interact. Dimensional analysis allows us to express physical quantities (Q) as functions of these dimensions (Q = f(L, W, H, T).

One of the central themes of this research is the examination of transitions between different dimensions, whether from lower to higher dimensions or vice versa. These transitions introduce new dimensions and powers, presenting mathematical challenges and deepening our understanding of the physical world. An example is the transition from one-dimensional length (L) to a two-dimensional plane (L²) within two dimensions, highlighting how the power of the higher dimension encapsulates the lower dimension.

The paper delves into the application of dimensional analysis in quantum mechanics and quantum gravity, where fundamental constants are examined at extremely small scales or high energies, such as the Planck scale. Variations in the power of the speed of light (c) across equations tied to spatial and temporal dimensions are explored. The introduction of the fourth dimension, often associated with time, influences these variations.

In the realm of quantum gravity, it is demonstrated that the power of the speed of light (c) can differ between spatial dimensions and the time dimension. This is exemplified in the transition from a higher, four-dimensional time frame (L³+L = L⁴) to a lower-dimensional, three-dimensional space (L³). These transitions highlight the dynamic nature of dimensions in complex phenomena.

The mathematical presentation underscores the fundamental nature of dimensional analysis as a tool for exploring the relationships between dimensions, their powers, and physical quantities. This comprehensive exploration sheds light on the role of dimensions in the fundamental laws of the universe and serves as a powerful bridge between mathematics and physics, allowing us to unlock the mysteries of the cosmos.

Furthermore, the research paper includes an image of a Calabi-Yau manifold, which is a fundamental concept in string theory, and a comprehensive list of references. The references cover a wide range of topics in physics, mathematics, and cosmology, providing additional reading material and sources of information for readers interested in further exploration of the subjects discussed in the paper.

In conclusion, this research paper offers a deep understanding of dimensional analysis and its applications in physics, revealing the interconnectedness of dimensions and their pivotal role in the fabric of our reality and the universe.