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Relativistic Correction of the Rydberg Formula Journal of Modern Physics, 2020, 11, 294-303 https://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Relativistic Correction of the Rydberg Formula Koshun Suto Chudai-Ji Temple, Isesaki, Japan How to cite this paper: Suto, K. (2020) Abstract Relativistic Correction of the Rydberg Formula. Journal of Modern Physics, 11, The relationship E = −K holds between the energy E and kinetic energy K of 294-303. the electron constituting a hydrogen atom. If the kinetic energy of the elec- https://doi.org/10.4236/jmp.2020.112018 tron is determined based on that relationship, then the energy levels of the Received: January 10, 2020 hydrogen atom are also determined. In classical quantum theory, there is a Accepted: February 18, 2020 formula called the Rydberg formula for calculating the wavelength of a pho- Published: February 21, 2020 ton emitted by an electron. In this paper, in contrast, the formula for the wa- velength of a photon is derived from the relativistic energy levels of a hydro- Copyright © 2020 by author(s) and Scientific Research Publishing Inc. gen atom derived by the author. The results show that, although the Rydberg This work is licensed under the Creative constant is classically a physical constant, it cannot be regarded as a funda- Commons Attribution International mental physical constant if the theory of relativity is taken into account. License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Keywords Open Access Rydberg Formula, Rydberg Constant, Classical Quantum Theory, Energy-Momentum Relationship in a Hydrogen Atom, Relativistic Kinetic Energy 1. Introduction In the classical quantum theory of Bohr, the energy levels of the hydrogen atom are given by the following formula [1] [2]. 2 11me4 1 =−⋅e EBO,n 22 (1a) 24πε0 n α 22mc = −e , n =1. ,2,⋅⋅⋅ (1b) 2n2 Here, EBO refers to the total mechanical energy predicted by Bohr. Also, α is the following fine-structure constant. e2 α = . (2) 4πε0c DOI: 10.4236/jmp.2020.112018 Feb. 21, 2020 294 Journal of Modern Physics K. Suto Bohr thought the following quantum condition was necessary to find the energy levels of the hydrogen atom. mve nn⋅π2 r =π 2. n (3) The energy of the hydrogen atom is also given by the following formula. 1 11e2 E= − K = Vr( ) = − . (4) 2 24πε0 r If E in Equation (1b) is substituted into Equation (4), then the following for- mula can be derived as the orbital radius of the electron. 2 = πε 2 = ⋅⋅⋅ rn 40 2 nn , 1, 2, . (5) mee The photonic energy emitted during a transition between energy levels (EEBO,nm− BO, ) and wavelength λ for principal quantum numbers m and n can be expressed as follows. hc １ １ EBO,nm− E BO, = hν = = hcR∞ −, m= 1, 2, , n= m + 1, m + 2, . (6) λ mn22 Here, R∞ is the Rydberg constant, which is defined by the following equa- tion. 2 24 α mcee2π me R∞ = = . (7) 2h ch3 The Rydberg formula can be derived from Equation (6) as indicated below. 1 EEBO,nm− BO, １ １ = =R∞ −, m= 1, 2, , nm= + 1, m + 2, . (8) λ hc mn22 2. Relationship Enfolded in Bohr’s Quantum Condition This section to Section 4 are excerpts from another paper, but this material is repeated because it is needed here. The Planck constant h can be written as fol- lows [3]: h mcλ = = eC. (9) 22ππ Here, λC is the Compton wavelength of the electron. When Equation (9) is used, the fine-structure constant α can be expressed as follows. e2 α = 2 . (10) 2ελ0emc C Also, the classical electron radius re is defined as follows. e2 = re 2 . (11) 4πε0emc If re α is calculated here, r λ eC= . (12) α 2π DOI: 10.4236/jmp.2020.112018 295 Journal of Modern Physics K. Suto If Equation (5) is written using re and α, the result is as follows. 2 22e 4πε cr =π=ε 2 0e22= rn 4.0 2 n 22nn 2 (13) mee 4πεα0e mc e Next, if in Equation (9) and rn in Equation (13) are substituted into Eq- uation (3), r mcλ mv⋅π2e n2 =π 2. n eC (14) e n α 2 2π If Equation (12) is also used, then Equation (14) can be written as follows. r m cr mv⋅π2e n2 =π 2. n ee (15) e n α 2 α From this, the following relationship can be derived [4]. v α n = . (16) cn 3. The Relation between Kinetic Energy and Momentum Derived from the STR Relationship The energy-momentum relationship in the special theory of relativity (STR) holds in an isolated system in free space. Here, if m0 is rest mass and m relati- vistic mass, the relationship can be written as follows. 22 2 22 2 (m0 c) += p c( mc ) . (17) What is the relationship between relativistic kinetic energy and momentum if this relationship holds? Incidentally, Sommerfeld once defined kinetic energy as the difference be- 2 2 tween the relativistic energy mc and rest mass energy mc0 of an object [5]. That is, 22 21 v K= mc − mc00 = mc −1, β = . (18) 2 12 c (1− β ) Sommerfeld believed that Equation (18), which can be derived from Equation (17), can also be applied to the electron in a hydrogen atom. First, it is clear that the following formula holds [4]. 2 2 222+− = 2 m00 c( mc m c) ( mc ) . (19) Expanding the left side of this equation yields the following. 24 24 24 22 2 2 2 mc0+( mc − mc 00) =( mc) ++( m m 0)( mc − mc 0) c. (20) Using this, Equation (19) becomes as follows. 2222 22 2 (m0 c) ++( m m 00)( mc − m c) c =( mc ) . (21) Since this equation and Equation (17) are equal, the following relationship must hold when Equation (18) is taken into account. DOI: 10.4236/jmp.2020.112018 296 Journal of Modern Physics K. Suto 2 22 p=+−=+( mmmcmcmmK0)( 00) ( ) . (22) The following formula is obtained from this. 2 pre Kre = . (23) mm0 + Here, Kre is relativistic kinetic energy and pre relativistic momentum. The “re” in Kre and pre stands for “relativistic”. Equation (23) is the formula for relativistic kinetic energy. Classical (non-relativistic) kinetic energy, in contrast, is defined as follows. 2 1 2 pcl Kcl= mv 0 = . (24) 22m0 In classical theory, mass does not depend on velocity. That is, Equation (23) and Equation (24) are the same if mm= 0 . 4. Energy-Momentum Relationship of the Electron Derived with Another Method The author has previously derived the following relationships applicable to the electron constituting a hydrogen atom [6]. 22 2 22 2 (mce ) −= pcnn( mc ) . (25) Here, 22 mcnn= mce − K re, . (26) mn is the mass of an electron in a state where the principal quantum number is n. These energy relationships can be illustrated as follows (Figure 1). In this paper, Equation (25) will be derived more simply by using a method different from that used previously. The logic of Equations (19) to (23) is bor- rowed to accomplish that purpose. Now, it is clear that the following equation holds. 2 2 2+− 22 = 2 mcnn( mcee mc) ( mc ) . (27) Expanding and rearranging this equation, the following equation is obtained. 2222 22 2 (mcn) ++( mee m nn)( mc − mc) c =( mce) . (28) Next, the relativistic kinetic energy of the electron can be defined as follows by referring to Equation (23). 2 22pre,n Kre,nn=−= mc e mc . (29) mme + n From this, 2 22 (me + mn)( mce −= mc nn) pre, . (30) Finally, Equation (28) matches Equation (25). DOI: 10.4236/jmp.2020.112018 297 Journal of Modern Physics K. Suto Figure 1. Energy levels of a hydrogen atom derived from Bohr’s classical quantum theory and this paper: According to the virial theorem, EKBO,nn= − cl, and EKre,nn= − re, . An electron at rest in free space emits a photon when it is taken into a hydrogen atom. Also, the electron acquires the same amount of kinetic energy as the energy of the emitted photon. If EEBO=0, re = 0 are described using an absolute energy scale, then the elec- tron is at rest in free space, and this corresponds to the state of having a rest mass energy 2 of mce . Incidentally, pre,n= mv nn [4], and thus it is clear that the following equation holds. pre,n c= mvc nn. (31) Here, if we substitute pcre,n in Equation (31) into Equation (25) and rear- range, then the following value is obtained. −12 v2 = + n mmn e 1.2 (32) c If the relation in Equation (16) is used here, Equation (32) becomes as follows. −12 12 α 22 n =+= mmn ee1.2 m 22 (33) nn +α Hence, the energy levels of a hydrogen atom Ere,n are: −12 α 2 = − =22 − = 2 + − = Ere,n K re, nn mc mce mc e 12 1 ,n 0,1,2, . (34) n DOI: 10.4236/jmp.2020.112018 298 Journal of Modern Physics K. Suto 5. Relativistic Energy of a Hydrogen Atom Derived from Equation (16) When both sides of Equation (16) are squared, and then multiplied by me 2 , 11mv22 mα een = . (35) 22cn22 Hence, 1 α 22mc E=−=− mv2 e . (36) BO,nn2 e 2n2 If Equation (16) is taken as a departure point, the energy levels of the hydro- gen atom derived by Bohr can be derived immediately. Equation (16) has tre- 2 mendous power. However, from a relativistic perspective, (12)mve n is an ap- proximation of the kinetic energy of the electron.
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