Relativistic Reduction of the Electron-Nucleus Force in Bohr’s Hydrogen Atom and the Time of Electron Transition between the Neighbouring Quantum Energy Levels

The aim of the paper is to get an insight into the time interval of electron emission done between two neighbouring energy levels of the hydrogen atom. To this purpose, in the first step, the formulae of the special relativity are applied to demonstrate the conditions which can annihilate the electrostatic force acting between the nucleus and electron in the atom. This result is obtained when a suitable electron speed entering the Lorentz transformation is combined with the strength of the magnetic field acting normally to the electron orbit in the atom. In the next step, the Maxwell equation characterizing the electromotive force is applied to calculate the time interval connected with the change of the magnetic field necessary to produce the force. It is shown that the time interval obtained from the Maxwell equation, multiplied by the energy change of two neighbouring energy levels considered in the atom, does satisfy the Joule-Lenz formula associated with the quantum electron energy emission rate between the levels.

The Quantum Condition That Should Have Been Assumed by Bohr When Deriving the Energy Levels of a Hydrogen Atom

Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. This famous quantum condition was not derived logically, but it beautifully explained the energy levels of the hydrogen atom. Therefore, Bohr’s quantum condition was accepted by physicists. However, the energy levels predicted by the eventually completed quantum mechanics do not match perfectly with the predictions of Bohr. For this reason, it cannot be said that Bohr’s quantum condition is a perfectly correct assumption. Since the mass of an electron which moves inside a hydrogen atom varies, Bohr’s quantum condition must be revised. However, the newly derived relativistic quantum condition is too complex to be assumed at the beginning. The velocity of an electron in a hydrogen atom is known as the Bohr velocity. This velocity can be derived from the formula for energy levels derived by Bohr. The velocity v of an electron including the principal quantum number n is given by αc/n. This paper elucidates the fact that this formula is built into Bohr’s quantum condition. It is also concluded in this paper that it is precisely this velocity formula that is the quantum condition that should have been assumed in the first place by Bohr. From Bohr’s quantum condition, it is impossible to derive the relativistic energy levels of a hydrogen atom, but they can be derived from the new quantum condition. This paper proposes raising the status of the previously-known Bohr velocity formula.

Time of the Energy Emission in the Hydrogen Atom and Its Electrodynamical Background

The time of the energy emission between two neighbouring electron levels in the hydrogen atom has been calculated first on the basis of the quantum aspects of the Joule-Lenz law, next this time is approached with the aid of the electrodynamical parameters characteristic for the electron motion in the atom. Both methods indicate a similar result, namely that the time of emission is close to the time period of the electromagnetic wave produced in course of the emission. As a by-product of calculations, the formula representing the radius of the electron microparticle is obtained from a simple combination of the expressions for the Bohr magnetic moment and a quantum of the magnetic flux.

Bohr’s Spectrum of Quantum States in the Atomic Hydrogen Deduced from the Uncertainty Principle for Energy and Time

A modified uncertainty principle coupling the intervals of energy and time can lead to the shortest distance attained in course of the excitation process, as well as the shortest possible time interval for that process. These lower bounds are much similar to the interval limits deduced on both the experimental and theoretical footing in the era when the Heisenberg uncertainty principle has been developed. In effect of the bounds existence, a maximal nuclear charge Ze acceptable for the Bohr atomic ion could be calculated. In the next step the velocity of electron transitions between the Bohr orbits is found to be close to the speed of light. This result provides us with the energy spectrum of transitions similar to that obtained in the Bohr’s model. A momentary force acting on the electrons in course of their transitions is estimated to be by many orders larger than a steady electrostatic force existent between the atomic electron and the nucleus.

Left Chiral Solutions for the Hydrogen Atom of the Wave Equation for Electron + Neutrino

The resolution of our wave equation for electron + neutrino is made in the case of the H atom. From two non-classical potentials, we get chiral solutions with the same set of quantum numbers and the same energy levels as those coming from the Dirac equation for the lone electron. These chiral solutions are available for each electronic state in any atom. We discuss the implications of these new potentials.

Dark Matter and the Energy-Momentum Relationship in a Hydrogen Atom

Einstein derived the energy-momentum relationship which holds in an isolated system in free space. However, this relationship is not applicable in the space inside a hydrogen atom where there is potential energy. Therefore, in 2011, the author derived an energy-momentum relationship applicable to the electron constituting a hydrogen atom. This paper derives that relationship in a simpler way using another method. From this relationship, it is possible to derive the formula for the energy levels of a hydrogen atom. The energy values obtained from this formula almost match the theoretical values of Bohr. However, the relationship derived by the author includes a state that cannot be predicted with Bohr’s theory. In the hydrogen atom, there is an energy level with n = 0. Also, there are energy levels where the relativistic energy of the electron becomes negative. An electron with this negative energy (mass) exists near the atomic nucleus (proton). The name “dark hydrogen atom” is given to matter formed from one electron with this negative mass and one proton with positive mass. Dark hydrogen atoms, dark hydrogen molecules, other types of dark atoms, and aggregates made up of dark molecules are plausible candidates for dark matter, the mysterious type of matter whose true nature is currently unknown.

Ehrenfest Approach to the Adiabatic Invariants and Calculation of the Intervals of Time Entering the Energy Emission Process in Simple Quantum Systems

In the first step, the Ehrenfest reasoning concerning the adiabatic invariance of the angular orbital momentum is applied to the electron motion in the hydrogen atom. It is demonstrated that the time of the energy emission from the quantum level n+1 to level n can be deduced from the orbital angular momentum examined in the hydrogen atom. This time is found precisely equal to the time interval dictated by the Joule-Lenz law governing the electron transition between the levels n+1 and n. In the next step, the mechanical parameters entering the quantum systems are applied in calculating the time intervals characteristic for the electron transitions. This concerns the neighbouring energy levels in the hydrogen atom as well as the Landau levels in the electron gas submitted to the action of a constant magnetic field.

Theoretical Prediction of Negative Energy Specific to the Electron

If an electron emits all of its rest mass energy mec2, the relativistic energy of the electron will become zero. According to the special theory of relativity, an electron whose relativistic energy is zero does not have photon energy. In this paper, however, an electron is regarded as having photon energy mec2 and negative energy −mec2, even when its relativistic energy is zero. The state where relativistic energy is zero is achieved due to the positive energy and negative energy canceling each other out. Relativistic energy becomes zero for an electron in a hydrogen atom when the principle quantum number n is zero. The author has already pointed out the existence of an energy level with n=0. If this model is used, it is possible for an electron in the state with n=0 to emit additional photons, and transition to negative energy levels. The existence of negative energy specific to the electron has previously been nothing more than a conjecture. However, this paper aims to theoretically show the existence of negative energy based on a discussion using an ellipse. The results show that the electron has latent negative energy.

Relativistic Correction of the Rydberg Formula

The relationship E = −K holds between the energy E and kinetic energy K of the electron constituting a hydrogen atom. If the kinetic energy of the electron is determined based on that relationship, then the energy levels of the hydrogen atom are also determined. In classical quantum theory, there is a formula called the Rydberg formula for calculating the wavelength of a photon emitted by an electron. In this paper, in contrast, the formula for the wavelength of a photon is derived from the relativistic energy levels of a hydrogen atom derived by the author. The results show that, although the Rydberg constant is classically a physical constant, it cannot be regarded as a fundamental physical constant if the theory of relativity is taken into account.

Statistics and Correlation Properties of Diamagnetic High Rydberg Hydrogen Atom

在磁场的 Rydherg 氢原子的系列与 B 花键基础功能用线性变化方法被计算了[Acta Phys。罪恶。55 (2006 ) 3380 ] 。把我们关于高 Rydberg 精力层次做了一些统计分析的计算基于这些。近邻居的精力间距分发和 3 统计与是的磁场关于抗磁的 Rydberg 氢原子被显示出 0.6 T 和 6 T。多样地交叉的现象，多样地反生气，并且交叉混合了并且精力层次反生气出现在这份报纸。为两个盒子，在更低的精力的范围，精力水平统计性质关门到泊松分发。与增加精力，精力水平统计性质离开到泊松分发并且一步一步地趋于到 Wigner 分发。

Fine-Structure Splittings of Nitrogen Isoelectronic Sequence： Competitions among Spin-Orbit Interactions, Breit Interactions and Electron Correlations

Using the multi-configuration Dirac-Fock self-consistent field method and the relativistic configuration interaction method with quantum-electrodynamics corrections performed by the GRASP code, we calculate the fine-structure energy levels of the ground-state configuration （1s^22s^22p^3） of the nitrogen isoelectronic sequence, according to the L-S coupling scheme with atomic number Z up to 22. Based on the calculated results, we elucidate the mechanism of the orderings of fine-structure energy levels of 2^ D3/2,5/2 and 2^P1/2,3/2 respectively, i.e. for 2^D3/2,5/2 orderings, the competition between the spin-orbit interactions and the Breit interactions; for 2^P1/2,3/2 orderings, the electron correlations, especially the electron correlations owing to the 2p^5 configuration interactions.

基于高斯基展开法研究氢原子能级和波函数

本文采用高斯基展开法精确求解氢原子能级,并且研究高斯基展开法中变分参数的选取对于结果精确度的影响.程序采用Python语言编写,这一求解方法为计算物理、量子力学等课程的教学提供了参考.

The atomic structure and the properties of Ununbium (Z=112) and Mercury (Z=80)

A super heavy element Uub (z = 112) has been studied theoretically in conjunction with rela-tivistic effects and the effects of electron correlations.The atomic structure and the oscillator strengths of low-lying levels have been calculated,and the ground states have also been determined for the singly and doubly charged ions. The influence of relativity and correlation effects to the atomic properties of such a super heavy element has been investigated in detail. The results have been compared with the properties of an element Hg. Two energy levels at wave numbers 64470 and 94392 are suggested to be of good candidates for experimental observations.

反氢原子能级及光谱的经典定性研究

利用一种带相位因子的经典屏蔽Coulomb势,采用经典分析方法对反氢原子的能级和光谱给出了预言性的描述,结果发现其能级与光谱同氢原子有较大区别。在反氢原子中主量子数n小于1的分数值且正电子由基态跃迁到激发态时才产生光谱,与氢原子发光机制正好相反。

氢原子的量子理论

阐述了表征氢原子内在属性的各种物理量的微观本质,证明氢原子系统的量子能量、系统内部电子的量子轨道动量及原子核和电子的量子相对距离均与原子系统所处的量子状态有关.当原子系统处于不同的量子状态时,上述量子物理量的取值完全不同.首次建立适合氢原子特性的量子算符代数理论.根据氢原子的量子哈密顿量表示,结合创新的量子算符代数理论,得到氢原子的能量、氢原子的基态能量、电子轨道角动量、氢原子的光谱常数等各种物理量的理论值.结果表明,氢原子的能量、氢原子的基态能量、氢原子的光谱常数均与氢原子中的原子核及电子的量子尺寸有关.氢原子的光谱常数与实验测定值完全符合.

氢原子一级Stark效应能级的解析式

根据简并态微扰论、球坐标系中氢原子波函数的性质,得到任意能级下氢原子一级Stark效应能级计算的简单方法,并计算了n=5的能级修正值,由此得到了氢原子一级Stark能级的解析式.

氢原子斯塔克效应的能级分裂规律

根据简并情况下的微扰理论和n=1,2,3,4能级对应的久期方程的规律,推出任意状态下氢原子斯塔克效应中的久期方程行列式中各矩阵元的分布规律,并找出了氢原子一级斯塔克效应在电场中的能级分裂规律.

氢原子的量子尺寸效应新理论

根据实么正变换理论 ,并结合SU ( 1,1)Lie代数 ,对氢原子的量子尺寸效应进行详细研讨。结果表明 ,氢原子的能级和波函数随时间变化的规律较复杂 。

氢原子斯塔克效应中久期方程的简化公式

根据简并态微扰理论和氢原子波函数的性质,得到久期方程中微扰矩阵元的分布规律。导出了任意能级下氢原子斯塔克效应中久期方程的简化公式。并给出了氢原子n=5能级的一级斯塔克效应。

微扰法计算中等强磁场中氢原子的能级

本文在考虑氢原子轨道运动磁矩与磁场之间、自旋磁矩与磁场之间和感生磁矩与外磁场之间的相互作用的基础上，根据角动量和球谐函数的性质，应用简并态微扰方法研究了在中等强磁场中氢原子的能级，给出了计算中等强磁场中氢原子的一级近似能级的方法，具体计算了 T范围内氢原子 的各能级的数值，结果与有关文献给出的理论计算值是相近的，表明本文所给出的方法是简单的、计算结果是正确的。