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.

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.

Zero Strength of the Lorentz Force and Lack of a Classical Energy Radiation in the Bohr's Hydrogen Atom?

It is demonstrated that the Lorentz force acting on the electron particle in the Bohr's hydrogen atom, as well as the classical radiation energy of that atom, tends to be zero on condition both the electric and magnetic fields in the atom are considered in a definite quantum state. The ratio of the mentioned fields becomes of importance for discussion of the occurence of the electron spin.

Phase-Space Areas of the Body Motion in the Solar System Deduced from the Bohr-Sommerfeld Atomic Theory and Approximate Invariance of Their Ratios for the Pairs of Planets and Satellites

Energy-time and momentum-position phase spaces defined by the electron orbits in the hydrogen-like atom exhibit special properties of equivalence. It is demonstrated that equivalence of the same kind can be obtained for the phase-space areas defined by the orbit pairs of planets, or satellites, which compose the solar system. In the choice of the examined areas it is useful to be guided by the Bohr-Sommerfeld atomic theory.

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.

Emission Intensity in the Hydrogen Atom Calculated from a Non-Probabilistic Approach to the Electron Transitions

Quantum aspects of the Joule-Lenz law for the transmission of energy allowed us to calculate the time rate of energy transitions between the quantum states of the hydrogen atom in a fully non-probabilistic way. The calculation has been extended to all transitions between p and s states having main quantum numbers not exceeding 6. An evident similarity between the intensity pattern obtained from the Joule-Lenz law and the corresponding quantum-mechanical transition pro-babilities has been shown.

Two Problems of Time Entering Respectively the Relativistic Mechanics and Electron Transport in Quantum Theory

In the relativistic mechanics, we calculate a minimal distance between the time scale of a one-dimensional motion having a larger velocity and the time scale of a similar motion with a lower velocity. Concerning the quantum theory, we demonstrate that mechanical parameters entering the electron motion in the Bohr hydrogen atom can provide us with a correct size of the time interval entering the Joule-Lenz law for the emission energy between two neighbouring quantum levels of the atom.

Model of the Pulsing Atom

In this work, we reanalyzed the movement of an electron in the electrostatic field of nucleus. The trajectory of the electron’s motion is an ellipse with a minor semiaxis, tending towards zero. From a mathematical point of view the movement of an electron in such an orbit will be equivalent to the oscillation of an electron. The action produced by electrons in movement between stationary points is discrete and proportional to a Planck constant. This condition sets the allowable values of the electron energy and the radius of their orbit. Electrons on the same shell perform symmetric synchronous oscillations. Their frequency is of the order of 1016 Hz. Most of the time the electrons are located on the periphery of the atom, periodically they simultaneously rush to the nucleus, the atom rapidly compresses and immediately decompresses, i.e. pulsates. The model gives Bohr formula for the energy of single-electron atom and suitable values of ionization potentials of the atoms of the second period of the Periodic Table.

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.

Relativized Quantum Physics Generating <i>N</i>-Valued Coulomb Force and Atomic Hydrogen Energy Spectrum

Though not well-known, Einstein endeavored much of his life to general-relativize quantum mechanics, (rather than quantizing gravity). Albeit he did not succeed, his legacy lives on. In this paper, we begin with the general relativistic field equations describing flat spacetime, but stimulated by vacuum energy fluctuations. In our precursor paper, after straightforward general relativistic calculations, the resulting covariant and contravariant energy-momentum tensors were identified as n-valued operators describing graviton excitation. From these two operators, we were able to generate all three boson masses (including the Higgs mass) in precise agreement as reported in the 2010 CODATA (NIST);moreover local, as-well-as large-scale, accelerated spacetimes were shown to naturally occur from this general relativized quantum physics approach (RQP). In this paper, applying the same approach, we produce an n-valued Coulombs Force Law leading to the energy spectrum for atomic hydrogen, without assuming quantized atomic radii, velocity and momentum, as Bohr did.

Hydrogen Bonding Character Between the Glycine and BF4^-

氢在 BF4- 和 glycine 之间的结合的特性理论上在 B3LYP/6 的水平被学习； 31+G* ，单个点精力在 B3LYP/6311++G * 的水平被执行 * 。intramolecular 氢契约的相关几何特征，精力性质，以及人物被学习了。在拓扑的分析显示了的分子理论的原子(3,1 ) 为氢契约的批评的点。另外，电子密度和拉普拉斯算符在为氢契约建议的范围。特别，在氢契约形成之上的氢的原子充电，偶极子时刻， enegry 以及体积的变化 systemitically 被讨论。

Investigation of the Antioxidant and UV Absorption Properties of 2-(2’-hydroxy-5’-methylphenyl)-benzotriazole and Its Ortho-Substituted Derivatives via DFT/TD-DFT

The demand and pursuit of chemical entities with UV filtration and antioxidant properties for enhanced photoprotection have been driven in recent times by acute exposure of humans to solar ultraviolet radiations. The structural, electronic, antioxidant and UV absorption properties of drometrizole (PBT) and designed ortho-substituted derivatives are reported via DFT and TD-DFT in the gas and aqueous phases. DFT and TD-DFT computations were performed at the M062x-D3Zero/6-311++G(d,p)//B97-3c and PBE0-D3(BJ)/def2-TZVP levels of theory respectively. Reaction enthalpies related to hydrogen atom transfer (HAT), single-electron transfer followed by proton transfer (SET-PT), and sequential proton loss electron transfer (SPLET) mechanisms were computed and compared with those of phenol. Results show that the presence of -NH2 substituent reduces the O-H bond dissociation enthalpy and ionization potential, while that of -CN increases the proton affinity. The HAT and SPLET mechanisms are the most plausible in the gas and aqueous phases respectively. The molecule with the -NH2 substituent (PBT1) was identified to be the compound with the highest antioxidant activity. The UV spectra of the studied compounds are characterized by two bands in the 280 - 400 nm regions. Results from this study provide a better comprehension antioxidant mechanism of drometrizole and present a new perspective for the design of electron-donor antioxidant molecules with enhanced antioxidant-photoprotective efficiencies for applications in commercial sunscreens.

The dynamical properties of Rydberg hydrogen atom near a metal surface

The dynamical properties of Rydberg hydrogen atom near a metal surface are presented by using the methods of phase space analysis and closed orbit theory. Transforming the coordinates of the Hamiltonian, we find that the phase space of the system is divided into vibrational and rotational region. Both the Poincaré surface of section and the closed orbit theory verify the same conclusion clearly. In this paper we choose the atomic principal quantum number as n = 20. The dynamical character of the exited hydrogen atom depends sensitively on the atom-surface distance d. When d is sufficiently large, the atom-surface potential can be expressed by the traditional van der Waals force and the system is integrable. When d becomes smaller, there exists a critical value dc. For d > dc, the system is near-integrable and the motion is regular. While chaotic motion appears for d < dc, and the system tends to be non-integrable. The trajectories become unstable and the electron might be captured onto the metal surface.

Quantum Mechanics of Moving Bound States

The quantum mechanics of bound states with discrete energy levels is well understood. The quantum mechanics of scattering processes is also well understood. However, the quantum mechanics of moving bound states is still debatable. When it is at rest, the space-like separation between the constituent particles is the primary variable. When the bound state moves, this space-like separation picks up the time-like separation. The time-separation is not a measurable variable in the present form of quantum mechanics. The only way to deal with this un-observable variable is to treat it statistically. This leads to rise of the statistical variables such entropy and temperature. Paul A. M. Dirac made efforts to construct bound-state wave functions in Einstein’s Lorentz-covariant world. In 1927, he noted that the c-number time-energy relation should be incorporated in the relativistic world. In 1945, he constructed four-dimensional oscillator wave functions with one time coordinate in addition to the three-dimensional space. In 1949, Dirac introduced the light-cone coordinate system for Lorentz transformations. It is then possible to integrate these contributions made by Dirac to construct the Lorentz-covariant harmonic oscillator wave functions. This oscillator system can explain the proton as a bound state of the quarks when it is at rest, and explain the Feynman’s parton picture when it moves with a speed close to that of light. While the un-measurable time-like separation becomes equal to the space-like separation at this speed, the statistical variables become prominent. The entropy and the temperature of this covariant harmonic oscillator are calculated. It is shown that they rise rapidly as the proton speed approaches that of light.

Theoretical Investigation on Interaction between Guanine and Luteolin

luteolin 和 guanine 的交往的模式被使用有 631+G* 基础的功能的理论 B3LYP 方法设置了的密度调查了。为 luteolinguanine 建筑群的十八稳定的结构分别地被发现了。结果显示建筑群被结合的氢主要稳定相互作用。同时，氢债券的数字和力量在决定能形成二的建筑群或更多的氢契约的稳定性起重要作用。在分子和天赋的原子的理论结合轨道也被利用了调查涉及所有系统的氢契约。被基础集合重叠错误改正的所有建筑群的相互作用精力是 6.0456.94 kJ/mol。计算结果显示在 luteolinguanine 建筑群有强壮的氢结合相互作用。我们比较了在 luteolin 和 DNA 的四个底之间的相互作用，并且发现 luteolinthymine 是最强壮并且 luteolinadenine 最弱。在 luteolin 和 DNA 底之间的相互作用都比 luteolinwater 强壮。

Ni_(1)/g-C_(3)N_(4)催化乙炔加氢反应的密度泛函理论研究

乙烯是石化产业的主要原料之一,广泛运用于工业生产和基本生活。乙烯中少量的乙炔杂质从根本上会影响下游产品,那么乙炔选择性加氢就可以提高乙烯产量,减少杂质产生。本文中构建了Ni_(1)/g-C_(3)N_(4)催化剂模型,以密度泛函理论为依据,从B3LYP泛函基组出发,计算过程中采用DFT-D3校正。对Ni原子使用LAN2DZ赝势,对于C,H,N非金属原子使用6-31g**基组。通过分析静电势、态密度和在催化剂上的吸附行为,系统性研究了Ni_(1)/g-C_(3)N_(4)催化剂催化乙炔加氢的反应机理,详细阐述了选择性和活性。结果表明,Ni_(1)/g-C_(3)N_(4)催化乙炔加氢的最优反应路径是乙炔加氢生成乙烯,能垒是20.05 kcal·mol^(-1);乙烯加氢生成乙烷,能垒是90.29 kcal·mol^(-1)。Ni_(1)/g-C_(3)N_(4)催化剂具有低活性、高选择性的特征,展现了其良好的催化性能,为乙炔高效催化加氢领域提供了帮助。

单原子Ge助剂修饰Cu(111)晶面上CO_(2)加氢制甲醇的机理研究

针对CO_(2)热催化转化制甲醇过程中CO_(2)吸附、活化较困难及副产物较多的问题,提出采用单原子Ge助剂修饰Cu(111)晶面的解决思路,通过密度泛函理论(DFT)计算研究了CO_(2)在Ge-Cu(111)晶面上加氢合成甲醇的反应机理。结果表明,单原子Ge助剂的电子调控增加了与其相邻的Cu原子的电子云密度,使CO_(2)分子在含Ge活性界面上的吸附能力显著增强:CO_(2)在GeCu(111)晶面上的吸附能约为Cu(111)晶面的1.5倍,约为Pd改性Cu(111)晶面的2.4倍,进而使逆水煤气变换(RWGS)反应路径速控步骤的活化能降低了近20 kJ·moL^(-1),同时衍生出3条生成甲醇的RWGS新路径;此外,Ge-Cu(111)晶面上甲酸盐路径由于速控步骤活化能大幅上升而被禁阻,进而CO及烃类等副产物选择性大幅降低,Ge-Cu(111)晶面上CO_(2)加氢制甲醇选择性升高。

氢原子n=4能级的一级和二级斯塔克效应

文章利用简并态微扰理论,计算了弱外电场中氢原子n=4能级的一级和二级Stark效应.计算结果表明,外电场中,对于一级Stark效应,原来是16重简并的能级分裂成了7个能级;对于二级Stark效应,氢原子的16个态所形成16维子空间,且同一子空间的能级下移相同的量值,而不同子空间的能级下移量值不同.计算结果对弱电场作用下氢原子的光谱分析具有重要的意义.

多元胺-TFSA型质子化离子液体吸收CO_(2)的理论分析

开展了以质子化的正己胺(HHexam^(+))、己基乙二胺(HHexen^(+))及己基二亚乙基三胺(HHexdien^(+))为阳离子的TFSA[==(CF_(3)SO_(2))_(2)N^(-)]型质子化离子液体(PILs),即[HHexam][TFSA]、[HHexen][TFSA]及[HHexdien][TFSA]型PILs吸收CO_(2)的研究。首先,选择密度泛函理论,在M06-2X/6-311G(d,p)水平下,对上述3种PILs的构型进行优化,分别得到了其较稳定构象,结果显示,PILs的阳离子中N—H和阴离子中N原子间主要形成N—H···N型较强氢键。然后,分别利用其中最稳定构象,创建并优化PILs-nCO_(2)构型,PILs和nCO_(2)分子间主要形成N—H···O型弱或中等强度氢键。主要氢键部位N—H···O中N—H键的振动频率的变化值、电子密度值及二阶微扰能的计算结果显示,[HHexam][TFSA]、[HHexen][TFSA]及[HHexdien][TFSA]分别与2、3、4分子CO_(2)结合时将不再形成氢键网络。采用COSMOtherm软件计算的CO_(2)在3种PILs中的亨利常数(kPa)大小为[HHexam][TFSA](1.91×10^(4))>[HHexen][TFSA](1.68×10^(4))>[HHexdien][TFSA](1.51×10^(4)),即3种PILs对CO_(2)的溶解能力大小为极性头部具有3个氨基的[HHexdien][TFSA]>2个氨基的[HHexen][TFSA]>1个氨基的[HHexam][TFSA]。以上结果中可以看出,PILs结构中氨基数目的多少对其吸收CO_(2)的能力有较显著影响,即随着PILs结构中氨基数目的增多,其对CO_(2)的溶解能力随之增大。

Theoretical Study on the Hydrogen Bonding Interactions in Complexes of 5-Hydroxytryptamine with Water

The energies, geometries and harmonic vibrational frequencies of 1 ： 1 5-hydroxytryptamine-water （5-HT-H20） complexes are studied at the MP2/6-311 ＋ ＋ G（d,p） level. Natural bond orbital （NBO）, quantum theory of atoms in molecules （QTAIM） analyses and the localized molecular orbital energy decomposition analysis （LMO-EDA） were performed to explore the nature of the hydrogen-bonding interactions in these complexes. Various types of hydro- gen bonds （H-bonds） are formed in these 5-HT-H20 complexes. The intermolecular C4H55HT＇＂Ow H-bond in HTW3 is strengthened due to the cooperativity, whereas no such cooperativity is found in the other 5-HT-H20 complexes. H-bond in which nitrogen atom of amino in 5-HT acted as proton donors was stronger than other H-bonds. Our researches show that the hydrogen bonding interaction plays a vital role on the relative stabilities of 5-HT-H20 complexes.