Research article Revisiting Bohr's Theory of Hydrogen Atom M.Meyyappan* [Professor of Physics(Retd), Alagappa Government Arts College, Affiliated to Alagappa University,Karaikudi- 630003, Tamilnadu, India, *email id: meydhanam@gmail.com Orchid No:0000 0002 5194 2992] Abstract [Since the velocity of orbital electrons in atoms is of the order of 106 m/s and comparable with the velocity of electromagnetic wave in free space, its relativistic variation of mass cannot be ignored. The Bohr's theory ofhydrogen atom is studied again with and without considering the relativistic variation of mass of orbitalelectron. When a free proton and an electron combine to form a hydrogen atom, half of the loss of its potential energy is converted into its kinetic energy required to move in a stable orbit surrounding the protonand the remaining half is deposited as relativistic increase of its mass. The orbital electron binds with theproton where the atomic binding energy comes from the energy equivalent of the added mass of the system.This energy is radiated out as electromagnetic waves. It is shown that the energy released during the electronic transition from orbit with quantum number n2 to n1 is equal to the energy equivalent of relativisticvariation of mass of the electron. i.e., ΔE = Δm c2 = (mn1- mn2)c2. It gives same value by the difference in total energies of the orbital electron in n1and n2, ΔE = En2- En1]. Key words: Atomic structure- Bohr's theory of Hydrogen atom- spectral feature--Rydberg formula relativistic correction- modified Bohr model-atomic binding energy. Revisiting Bohr's Theory of Hydrogen Atom Introduction Bohr's atomic theory of hydrogen atom [1,2,3], while revolutionary, is primarily limited to mono electron systems such as Hydrogen (1H1) and hydrogen-like ions (2He1- and 3Li2-). This theory revolutionized atomic physics by introducing quantized energy levels, explaining the stability of atoms, and accurately predicting the hydrogen spectral lines and ionization energy. It resolved contradictions in classical physics by proposing stationary orbits where electrons do not radiate energy unless jumping between shells. Unfortunately Bohr's theory of hydrogen atom fails to explain atoms with higher atomic number, where the additional electron electron interactions introduce some corrections in the theory. These interactions depend on the speed of electrons moving in circular paths, its spin, distance between them, and its periodic changes during its continuous orbital motion. Since the radius of the orbit and orbital velocity of the electron are determined by the interaction within the system, any additional interactions disturb the configuration by changing them obviously. The non-relativistic Bohr's theory of hydrogen atom has some limitations [4]. It does not explain the fine structure and the splitting of spectral lines in an external magnetic field (Zeeman effect) or electric field (Stark effect). There is no explanation why some spectral lines in emission spectra are more intense than others. It contradicts the Heisenberg Uncertainty Principle as Bohr assumed electrons move in well-defined circular orbits with a known radius and momentum simultaneously. This model treats electrons solely as particles and fails to account for its wave-particle duality. It fails to explain why atoms form chemical bonds or how they share/transfer electrons to form molecules. All the micro particles have dual nature exhibiting its wave characteristics called matter waves.[5,6].The wavelegth associated with an electron moving with an momentum p is given by h/p. According to de Broglie,the motion of an electron can only be stable if the phase wave is tuned with the length of the path [7]. The wave characteristics of the electron in an atom demand that the circumference of any electronic orbits must be equal to some multiples of electron wavelength. i.e., 2πrn = nλ. where n is a quantum number which can have only some integral values. This condition makes the wave to retrace its circular path with same phase at every point. It is required to keep the atomic structure in the same state of stability at all time. The first Bohr's postulate states that the angular momentum of the orbital electron is some integral multiples of h/2π.The wave characteristics of the electron make the electronic orbits to be discrete (Fig.1) Fig.1. Discreteness of electronic orbits in an atom If the orbital electron is represented by de-Broglie wave, they should move together, that is the wave velocity should be very same as that of the electron. For any type of waves, the product of frequency and wavelength must be equal to its velocity of propagation. The phase velocity u = ν λ , energy equivalent of the electron of mass m is mc^2 which gives the frequency of its characteristic wave as mc^2/h and λ = h/p = h/mv .Substituting these values u = (mc^2 /h) (h/mv) = c^2/v. Since the particle velocity v cannot equal or exceed the velocity of light in free space c, the de-Broglie wave velocity u is always greater than c. It is vivid that v and u usually are not equal for a moving body. Since u is greater than c , the de Broglie wave must be different in nature from that of the other known waves electromagnetic and sound. It implies that the wave velocity of matter wave has no simple physical significance at all. To avoid this difficulty, the matter wave associated with a moving body is represented as a wave packet. The group velocity of a system of waves is given by w = u- λ (du/dλ). This can be written as w=-λ^2 d/dλ(u/λ) =- λ^2 dν/dλ or 1/w =- (1/λ2) (dλ/dν) = d/dν(1/λ) If E and Vrepresent the total and potential energies of the system, then(1/2) mv^2 = E-V or v = [2(E-v)/m]^1/2 and E=hν.Using these relations, 1/λ = mv/h=(1/h)[2m(hν- V)]1/2 hence 1/w = (1/h)[2m(hν- V)]^1/2 = 1/v or w = v. Thus the wave packet travels with the same velocity as the body. The conceptual difficulty in the interpretation of the wave velocity exists since a single relation is employed to represent the frequencies of radiation and matter waves. Here an attempt [8] is made to derive a relation for the phase velocity of de-Broglie matter wave, where the matter wave is considered as entirely different from that of radiation. When the matter at rest is completely converted into energy, then according to Einstein's theory of relativity νo = moc^2/h . If the matter is moving, then ν = mc^2/h = moc^2/h + K.E/h or ν > νo where ν and νo represent the frequencies of radiation that comes out of complete destruction of matter in motion and at rest respectively. Hence E= moc^2 = hνowave and E = mc^2 = h ν(wave). When the matter is at rest, there is no matter wave at all, but still it has energy equivalent of mass. Hence it is reasonably supposed that matter wave is related with particle's kinetic energy rather than its total energy. The kinetic energy of matter is (m-mo) c^2 and for its wave nature it is h νmatter. With this knowledge,one can derive an expression for ν (matter) νmatter = (m-mo) c^2/h = mo c^2 /h{[1/ √(1-v^2/c^2)]-1} but ν(wave) = E/h = mc^2/h = [(moc^2)/h√(1-v^2/c^2)] which give ν(matter)/ν(wave) =1- √(1-v^2/c^2). Similarly λ (wave)= c/ν(wave) = hc/E and λ(matter)= h/p and its ratio gives λ(matter)/ λ (wave) = E/pc = mo c^2 / √(1-v^2/c^2) x[1/[mov/√(1-v^2/c^2)]c = c/v . When v = 0, λ(matter) → ∞,it corresponds to a particle at rest.When v = c, λ(matter) = λ wave.It corresponds to a specific case where the matter wavelength is exactly equal to radiation wavelength. This is true only for luxons like photons. Let umatter be the velocity of matter waves. Then umatter/uwave=(λmatter/ λ wave) x (νmatter/ νwave)= (c/v)[1- √(1 - v^2/c^2)].Substituting u(wave) = c , u(matter) = (c^2/v)[1- √(1-v^2/c^2)]. At two extreme specific cases, when v = 0 , um → 0 and v =c, um→ c.Thatis de-Broglie wave velocity u(m) can never be greater than c, the velocity of light in free space. This expression accounts for the relation for group velocity. The angular frequency of the matter wave is given by ω=2πν(m)=2πν(wave)[1- √(1-v^2/c^2)] = [2π moc^2/h] {[1- √(1-v^2/c^2)]/[√(1-v^2/c^2)]} or dω/ dν =2πmov/h(1-v^2/c^2)^3/2 and the propagation constant is given by k = 2π/λ = 2πmov/[h √(1-v^2/c^2)] or dk/dv = 2π mo/[h (1-v^2/c^2)^3/2], hence ω = dω/dk = (dω/dv)/(dk/dv) = v. That is the de-Broglie wave group associated with a moving particle travels with the same velocity as the body. Very same relation for hydrogen spectrum can be obtained by assuming the orbital electrons as matter waves. When an electron jumps from outer orbit to any one the inner orbits, its radius and potential energy are decreased ,orbital velocity and binding energy are increased and these changes result with a decrease in the wavelength of the matter waves associated with the orbital electron. During this jumping the excess energy is radiated out as electromagnetic radiation of wavelength λ(wave),. As they are all the after-effects of the above electronic transition, one may believe that they may be linked indirectly. Based on this assumption, let us determine the wavelength of em radiation emitted in a transition from orbit with quantum number n2 to orbit with quantum number n1 The energy of transition (ΔE) from n(2)to n(1) = E(2)- E(1) , where E(1),E(2) denote the total energy of the orbital electron in the orbit 1 and 2 respectively. E(2)- E(1)= ΔE =hν=hc/λ(wave) or λ(wave) = hc/ΔE(2→1). The wavelengths of matter wave associated with the electron in the orbit with quantum number n(2)and n(1) are h/mv(2) and h/mv(1) respectively. It gives mv(1) = h/λ(matter1) and mv(2) = h/λ(matter2). With this knowledge one can estimate the kinetic and potential energies of the orbital electron in terms of its wavelength of matter wave. Kinetic energy of the electron in orbit n(2) = (1/2) mv2^2= (1/2m)h^2/λ^2(matter2). Potential energy of the electron in orbit n(2) =- e^2/Kr^2=- mv(2)^2 =- (1/m)h^2/λ^2(matter2). Total energy of the electron in the orbit n(2)= Sum of kinetic and potential energies E(2) =- (1/2m) h^2/λ^2(matter2).Total energy of the electron in the orbit n(1)=E(1)=- (1/2m)h^2/λ^2(matter1) ΔE(2→1) =(h^2/2m) [1/λ^2(matter1)- 1/λ^2(matter2)] The radius of the electronic orbit with quantum number n in hydrogen atom is shown as r(n)= n^2ao , where ao is Bohr's radius, the radius of the innermost orbit of the electron in the hydrogen atom. From this one can derive mv^2 = e^2/Kn^2ao and λ^2(matter) = h^2 Kn^2ao/me^2 = (2πnao)^2. This can be verified by substituting the values for matter wavelength in ΔE(2→1). ΔE(2→1) =(h^2/2m)[1/(2πao)2][1/n2^2- 1/n1^2] = [h^2/8π^2mao2][1/n2^2- 1/n1^2] = [e^4m/8h^2εo^2][1/n2^2- 1/n1^2] and λ(2→1)= 8cεo^2h^3/(1/n(f)^2- 1/n(i)^2)me^4= (1/n(f)^2- 1/n(i)^2)/RH where R H= me^4/8cεo^2h^3 called Rydberg constant. By redoing non-relativistic Bohr's theory of hydrogen atom the energy radiatied out during jumping of the electron from one orbit to another is derived.To understand the mutual dependence of various variables such as orbital velocity v(n), orbital radius r(n) and the order of the orbit (n) is derived. (1) Orbital velocity v(n ) Vs orbital radius (rn) The stability of the electron in the orbit requires that the mutual attractive force between the nucleus and the orbital electron must be equal to the cetrifugal force experienced by it. e^2/(4πεo)m = constant = v(n)^2 r(n) ....... (1) It predicts that v(n) is inversely proportional to r(n)^1/2 .The same result is obtained from the Bohr's postulate on the angular momentum of the electron mv(n)r(n) = nh/2π ........ (2) It gives v(n)r(n)/n = v(n)^2r(n)/n v(n) or n v(n) = constant as v(n)^2r(n) = constant . Squaring equ (2) and dividing by equ (1) one can arrive r(n) = n^2 ao . By substituting this value for r(n) we get v(n)^2 n^2 ao = constant, It shows that the radii of higher orbits in hydrogen atom are 4 ao,9 ao,16 ao.... n^2ao and the spacing between the nth and (n+1)th orbits is (2n+1)ao (2) Orbital velocity v(n) Vs Quantum number(n) Dividing equ (1) by equ (2) we have n v(n) = [e^2/2εoh] = constant ,which indicates that v(n) is inversely proportional to n.i.e, as we go away from the center, the orbital electron moves slower. Very same relationship can be derived from the condition introduced for the circumference of the orbit (3) Orbital readius r(n) Vs Quantum number (n) Again by squaring the relation [v(n)r(n)/n]^2 = v(n)^2 r(n)/[n^2 /r(n)] or n^2 /r(n) = constant as v(n)^2 r(n) = constant. It shows that r(n) is directly proportional to n^2. (4) Rate of variation of orbital velocity with respect to radius dv(n)/dr(n) Differentiating v^2 r = constant with respect to r, v^2 + 2vr (dv/dr) = 0, which gives dv(n)/dr(n) =-v(n)/2r(n). By substituting the values of r(n)[r(n) = n^2 ao] and the corresponding v(n)[mv(n) r(n) = mv(n)n^2ao = nh/2π or v(n) = h/2π mnao which states nv(n) = constant] one can find out the rate with which the orbital velocity changes with radius of the orbit, dv(n)/dr(n) =- v(n)/2r(n) =- h/4πmao^2 n^3. i.e., dvn/drn is negative and varies inversely proportional to n^3. The standard Bohr model works well for hydrogen, but the relativistic model is necessary for heavy hydrogen-like ions where the orbital electrons travel at a significant fraction of the speed of light, rendering non-relativistic calculations inaccurate. Objective of the study The objective of the Study is to predict how the non-relativistic theory gets modified when the relativistic increase of mass of the orbital electron is incorporated in the Bohr's theory of hydrogen atom. When Bohr's atomic theory was reviewed with the aim of extending Bohr's atomic theory to atoms other than hydrogen, the first thing that came to mind was taking into the account of the relativistic variation of mass of the orbital electrons moving rapidly in circular paths. This correction is assumed to be necessary because the speed of an electron moving in an atomic orbit is of the order of 10^6 m/s. As the velocity of the orbital electron increases, its mass increases relative to its rest mass. Therefore, the relativistic effects on the core electrons are to give them larger masses that gives more kinetic energy to the particle and this shrinks the Bohr radius. For the expandable utility, the present non-relativistic Bohr's theory of hydrogen atom is studied again by incorporating the relativistic variation of mass of the orbital electron. The present relativistic Bohr's theory of hydrogen atom provides appropriate guidance to minimize its limitations. With this focus in mind the theoretical study based on famous Niels Bohr atom model is undertaken. Related Work/Literature Review The standard Bohr model was studied by many researchers (9-12) by introducing relativistic variation of mass of the orbital electron that improves the Bohr model by giving an account to one or more limitations. The relativistic energy of an electron, moving at a significant fraction of the speed of light c, is defined by its total energy En = moc^2/(1- v(n)^2/c^2)^1/2 , which includes both rest energy moc^2 approximately 0.511 MeV and kinetic energy. The total energy of the electron is E = (p^2c^2 + mo^2c^4)^1/2 , where p is the momentum of the electron. The relativistic energy levels for an electron of charge e and rest mass mo in a hydrogenic atom with atomic number Z are given by: En = [mo^2 c^4 + mo^2c^4α^2Z^2/n^2]^1/2 where α = 2πe^2/hc is the fine-structure constant, which approximates to E(n)= moc^2[1 +α^2Z^2/2m ]. The relativistic model shows that the energy depends not only on the principal quantum number n but also includes relativistic corrections that affect the fine (structure of hydrogenic spectral lines. Again the radius of the orbits [r(n)= n^2(h/2π)^2/meke^2] technically changes because the electron's mass m(e) increases with velocity, making the orbits slightly smaller than predicted by the non-relativistic model, especially for low n. Relativistic Bohr's Theory of Hydrogen atom The hydrogen atom with a central proton and an orbiting electron is a two body problem. When a free electron enters the active space it gets accelerated towards the proton, thereby its kinetic energy is increased gradually due to accelerated motion towards the nucleus and simultaneously it gains some mass due to relativistic variation of its velocity. The energy required for this comes from the source that accelerates the electron [13]. The work done by the electron reduces its potential energy. In this motion of the electron within the atom, the energy must be conserved at every point of its path. It means that the loss of its potential energy must be equal to gain in its kinetic energy and the energy used for the relativistic increase of mass. That is a fraction of the loss of potential energy is converted into its kinetic energy and remaining is stored as its relativistic increase of mass, so that the energy is conserved all along it path. As the accelerated electron is abruptly stopped at any one of the allowed orbits, it takes up a curved path until it attains stability in a stable orbit. It revolves round the proton in circular orbit, where the electrostatic force of attraction is exactly counter-balanced with its centrifugal force The conservation of energy of the electron at all of its position requires that the loss of its potential energy must be equal to gain in its kinetic energy and the energy used for the relativistic increase of mass. e^2/Kr(n) = (1/2)m(n) v(n)^2 + Δmc^2 ,where Δm = (mn- mo) .....(3) When the electron is attracted by the nucleus it gets accelerated. Since the force is inversely proportional to intermediate distance, the acceleration experienced by the electron is not uniform. Let mn, r(n) , v(n) be the mass, radius and velocity of the electron in its n th orbit F = m(n)a(n)=e^2/Kr(n)^2 = m(n)v(n)^2/r(n) or a(n) = v(n)^2/r(n).......(4) It shows that e^2/K = m(n)a(n) r(n)^2 = m(n) v(n)^2 r(n) = [mo/(1- v(n)^2/c^2)^1/2]v(n)^2 r(n) . From the discreteness of the electronic orbits in atom 2πr(n) = nh/m(n)v(n). Squaring both sides and substituting the value for mnvn, 4π2rn2 = n2h2/mn2vn2 = n^2 h^2 K r(n)/m(n)e^2. It gives r(n) = n^2ao(1-v(n)^2/c^2]^1/2. By substituting the value of rn we get e^2/K = mo v(n)^2 n^2 ao = constant or v(n) n = constant irrespective of the relativistic variation of mass of electron.The variable acceleration a(n) = v(n)^2/r(n) = [v(n)^2/(1-v(n)^2/c^2)^1/2][1/n^2ao]. It helps to study how the relativistic approach on the Bohr's theory of hydrogen atom makes changes in the dependency among the various dependent variables (1).Orbital velocity v(n) Vs Orbital radius r(n) Let us suppose that an electron in the hydrogen atom is in its nth orbit having radius rn.The nuclear attractive force is counter-balanced with the centrifugal force. Relativistic mass m(n) increases with velocity v(n) according to Einstein’s Special Theory of Relativity, defined by m(n)= m(o)/ [1- v(n)^2/c^2]^1/2.Incorporating this value in equ (1) e^2/moK = v(n)^2r(n)/(1-v(n)^2/c^2)^1/2 = v(n)^2n^2ao = constant ....... (5) The relativistic variation of mass of the orbital electron makes no correction in the dependency of vn on rn.It implies that the added mass of the electron in the orbital motion is converted into atomic binding energy which holds the orbital electron with the nucleus. (2).Orbital velocity v(n) Vs Quantum number(n) Dividing equ (1) by equ (2) , v(n) = e^2 /2nhεo ........(6) As the orbital velocity vnis independent of mass of the electron, the dependency of v(n) on n is not altered due to the relativistic variation of mass of the orbital electron. (3) Orbital radius r(n)Vs Quantum number (n) From equ (2) m(n)^2v(n)^2r(n)^2 = n^2h^2/4π^2 Dividing one by equ (1), m(n)r(n) = n^2h^2/4π^2x(K/e^2) [mo/(1-v(n)^2/c^2)^1/2]r(n) = n^2h^2εo/πe^2= n^2moao r(n) = [n^2h^2εo/moπe^2][1-v(n)^2/c^2]^1/2= n^2ao(1-v(n)^2/c^2]^1/2 ........(7) The relativistic variation of mass of the orbital electron makes the orbit to shrink about its center. Approximately the change in the radius is h^2/8π^2m0^2aoc^2irrespective of n.Substituting this value of r(n) in m(n)v(n)r(n) = nh/2π [mo/(1-v(n)^2/c^2)^1/2 ] x v(n)xn^2ao(1-v(n)^2/c^2]^1/2 = nh/2π , v(n)= h/n2π moao .........(8) Comparing equs (6) and (8) we get the same result for the Bohr's radius ao = εoh^2/ π moe^2 By estimating the orbital velocity of the electron in the hydrogen atom and the corresponding change in its relativistic mass, one can show that the energy equivalent to the relativistic change of mass is equal to its binding energy with the nucleus. The velocity of the electron in the nth orbit in hydrogen atom is given by v(n)= e^2/2nεoh = (1.602 x 10^-19)^2/2nx(8.85 x 10^-12)x(6.626 x 10^-34) =[2.188 x 106/n] m/s v(n) = h/n 2πmoao= 7x6.626 x10^-34/n x 22 x 9.11 x 10^-31x 5.29 x 10^-11 = [2.187 x 106/n] m/s v(n)^2/c^2 = (2.188 x 10^6)^2/n^2(2.998 x 10^8)^2=4.787 x 10^12/n^2x 8.988 x 10^16 =[0.5326/n^2] x 10^-4 The mass of the orbital electron in the nthorbit mo /[1- v(n)^2/c^2]^1/2 ≃ mo[1 + v(n)^2/2c^2] The relativistic increase of mass Δm = mov(n)^2/2c^2and its equivalent energy Δm c^2= [mov(n)^2/2e] eV Δmc^2 =9.108 x10^-31x(2.188 x 10^6)^2/2 x (1.602 x 10^-19) n^2 = 13.6 /n^2 eV It is in agreement with the practical value of ionization energy of hydrogen atom from its various energy levels. The wavelength of various spectral lines in hydrogen spectrum can be studied from the knowledge of difference in the binding energy of the electron in different orbits, which can be worked out from the relativistic mass of the orbital electron in various orbits. The relativistic increase of mass of the electron in the orbits with orbital quantum number n1 and n2 approximately is given by Δm1 = mov1^2/2c^2 and Δm2 = mov2^2/2c^2 respectively and its equivalent energy is [mov1^2/2] and [mov2^2/2] respectively. When the electron jumps from n(2)to n(1) ,the difference of energy due to relativistic change of its mass is emitted out as em radiation. Its wavelength λ(2→1)= hc/ΔE(2→1) =2 hc/mo(v(2)^2- v(1)^2) where v(n)= e^2/2nεoh. Substituting the values for v(1)and v(2),we get the same relation for λ(2→1) as shown in non-relativistic Bohr's theory λ(2→1) = [8h^3εo^2c]/e^4mo[1/n(2)^2- 1/n(1)^2] . Results The relativistic variation of mass of the electron makes no changes in the Bohr's theory of hydrogen atom except the orbital radius, r(n) = r(n)o (1- v(n)^2/c^2)^1/2 . The revision on Bohr's theory of hydrogen atom provides an acceptable explanation in classical way for the non-existence of hydrogen negative ion and for the limiting reachability of the attracted electron towards the nucleus. The first (1s) orbit can accommodate a maximum of two electrons, but in the hydrogen atom its first orbit cannot be filled with two electrons. Two electrons can stay in the 1s orbit only when one more proton is present in the nucleus. It is confirmed with the existence of Helium atom. If one more electron is allowed in the first orbit of hydrogen, its total energy becomes positive and hence it moves away from the nucleus and the system transforms into a less potentially stable state. Electrostatic attractive force due to the nucleus is e^2/Kr^2. Since both the electrons are in the same orbit there must be mutual interaction between them that is why they attain stability by staying exactly diametrically opposite. Electrostatic repulsive force due to the presence of second electron is e^2/4Kr^2. The resultant force experienced by the electron is (3/4)e^2/Kr^2 . As it is counter-balanced by centrifugal force mv^2/r,the kinetic energy of both the electrons which are identical in all respect is m v^2= (3/4) e^2/Kr. The potential energy of the first electron due to the presence of the nucleus only is- e^2/Kr. When the second electron moves towards the neutral hydrogen atom, its electrostatic potential energy is zero until it reaches the electronic orbit of the first electron. Due to additional electron-electron interaction the potential energy is increased by e^2/2Kr. The total energy associated with the system = (3/4) e^2/Kr- e^2/Kr + e^2/2Kr = (e^2/4Kr) . Since there is no binding energy, the system gets transformed into normal hydrogen atom by expelling out the additional electron. Same result is obtained in relativistic approach of Bohr's theory. Let us suppose that both the electrons are placed in the 1s orbit to make negative hydrogen ion. The electrostatic attractive force experienced by an electron due to the presence of nucleus is e^2/Kr^2 and the electrostatic repulsive force experienced by it due to the presence of other electron is e^2/4Kr^2.The resultant force is balanced by centrifugal force required for its orbital motion .It gives the kinetic energy acquired by the electron as (3/8) e^2/Kr. When a negative hydrogen ion is formed, there is a drop of potential energy e^2/Kr due to nucleus only in the case of first electron as it is moved in potential field and it is zero for the second electron as it is moved in potential free field outside the orbit .When it reaches the orbit there is an increase in its potential energy by an amount e^2/2Kr due to the additional electron in the same orbit. Both the electrons in the orbit are identical in all respect and hence they have its own kinetic energy.Total kinetic energy acquired by the electrons is (3/4) e^2/Kr . Conservation of energy gives e^2/Kr - e^2/2Kr = e^2/2Kr = (3/4)e^2/Kr + dm c^2 where dm = 2(m - mo) dm c^2 = - (1/4) e^2/Kr Since the binding energy is negative, the negative hydrogen is practically impossible. The revised Bohr's theory of hydrogen atom gives an acceptable explanation for the question 'Why the atomic electron under normal circumstances cannot reach beyond the innermost orbit?'. When an electron is attracted by the proton in hydrogen atom, it gets accelerated towards the nucleus until it reaches a point where its velocity is exactly equal to the orbital velocity required to keep the electron stable in the orbit. It is predetermined by the conservation of energy, loss of potential energy = gain in kinetic energy + relativistic increase of mass. Since the gain in kinetic energy of electron is half of the loss of its potential energy, the energy equivalent of relativistic increase of mass of the electron comes from the remaining half of the loss of potential energy (1/2) e^2/Kr(n) = (1/2)m(n)v(n)^2 or v(n)^2 = e^2/Km(n)r(n), where r(n) = n^2ao(1-v(n)^2/c^2)^1/2 and m(n)= mo(1-v(n)^2/c^2)^-1/2 or v(n)^2 =(1/n^2) [e^2/Kmoao] The highest value of v(n) is v(1) with n = 1; v(1)=[e^2/Kmoao] For stability of the electron in the orbit, electrostatic force = centrifugal force e^2/Kr(n)^2 = m(n)v(n)^2/r(n) or v(n)^2= e^2/Kr(n)m(n) = (1/n^2) [e^2/ Kmoao] Whenanaccelerated electron moves towards the nucleus, usually it will not make a straight line motion, If so, it cannot be stopped abruptly in an allowed orbit The accelerated electron moves along a curved path and ultimately it attains stable orbital motion with uniform velocity. The innermost orbit has lowest possible radius in hydrogen atom and is equal to ao, Bohr's radius. That is when hydrogen atom is formed, the orbital electron cannot be brought closer to nucleus with an intermediate distance less than ao. This can be proved form the Bohr's postulates. From the first Bohr's postulate on angular momentum of the orbital electron m(n)^2v(n)^2 r(n)^2 = n^2h^2/4π^2 and from the second Bohr's postulate stating the equivalence of nuclear attractive force with the centrifugal force mn v(n)^2 r(n) = e^2/K. Dividing one by the other, m(n)r(n) = n^2h^2εo/πe^2 or m(n)= n^2h^2εo/π e^2 r(n).From m(n)r(n)= [mo/(1- v(n)^2/c^2)^1/2][n^2ao(1- v(n)^2/c^2)^1/2] = n^2 mo ao or m(n) = n^2 moao/r(n) and from Bohr radius ao=h^2εo/moπ e^2, or mo = h^2εo/πe^2ao. The relativistic increase of mass cannot be less than zero. i.e., dm = m(n)- mo ≃(1/2)mo v(n)^2 /c^2 = [h^2 εo/π e^2] [n^2/r(n)- 1/ao] . The highest possible value of n^2/r(n) = 1/r1 , since dm cannot be negative r1= ao . The lowest possible radius of the orbital electron in hydrogen atom is ao. Conclusion and Future Scope This theoretical work based on non-relativistic Bohr's theory of hydrogen atom is independently carried out without any financial assistance from any funding agencies. I take this opportunity to acknowledge the love and patience of my family and friends. Their belief in me and constant support provided me the creative strength required to persevere. The present work introduces a new line of thought about the source of relativistic increase of mass of the electron, and its atomic binding energy with the nucleus. During an electron jump from outer orbit to any one of the inner orbits there is a mutual conversion between its potential energy with kinetic energy and relativistic mass energy in accordance with the law of conservation energy. It avoids the continuous release of energy in jumping from its initial to final position. 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