Bohr's Atom Even though AD has many applications on its Construct the most visible and important application is, possibly, to the Bohr’s Atom because AD found the Dirac’s equation without any Wave Mechanics and without developing any complicated mathematical operations. 3-3 Bohr’s Atom. As is known, Dirac’s equation was obtained through the application of Quantum Wave Mechanics to the Bohr atom. Quantum Mechanics as developed by Sommerfeld was also insufficient in describing the experimental results. Magnetic anomalies introduced the magnetic moment hypothesis, known later as Bohr’s Magneton. It was M. M. Uhlenbeck and Goudsmit who introduced the new hypothesis of a magnetic and spinning electron. The theory was never completely developed because the new Wave Mechanics postulated by Schredinger and developed by Pauli and Dirac was successful[1]. The equation from classic Mechanics is transformed to a Relativistic equation. Without detailing all the steps, Dirac introduced, (after a first attempt by Pauli), the complete idea of a magnetic and spinning electron (Uhlenbeck and Goudsmit) as e h / 4p mo c^2 (Bohr’s magneton), and 1/2 * h/2p, as mechanical spin, but always as a wave function. Using a powerful mathematical construct, a complete theory was developed by Dirac with three spatial and later four wave functions, taking the final form of equation (3.67). 3-3-1 The Hydrogen atomic spectrum. Until 1912, when Bohr systematically applied the law of mechanics to the spectral term of the hydrogen atom, the issue was treated in part as empirical, and in part as theoretical. Lord Rutherford’s (1871-1937) suggestion to adopt a planetary model as most physicists followed the model for the atom. With the Balmer series (and later with the Rydberg constant), the work of Lyman in the ultra-violet, and the Pachen series for the infrared, the entire spectrum was successfully covered. Of course, from the experimental point of view, this formalism was insufficient. Bohr’s theory, with its mathematical treatment of the problem, opened the way for further studies. Fig 1 Bohr was first to apply Planck’s idea of quantification. An Electron’s orbit has to be an integer multiple of the Planck quantum of action h/2p. AD’s students must scrutinize this idea carefully. Bohr quantified the energy and momentum of orbital motion. Quantum Mechanics discusses not a quantum of energy, but a quantum of action for electron orbital motion. The various orbits of the electron are quantified, depending on the energy it absorbs or emits. This distinction is crucial, because many professionals who subscribe to conventional wisdom hold that Quantum Mechanics describes a quantum of energy for an independent particle. Of course, the energy that atoms absorb or emit will also be quantified, but only as a result of quantum orbital motion. Bohr, applying classical mechanics, proposed the two following relations: (3.14) m = electron mass, r = radius orbit, n1 = integer number and q = rotation angle. as an expression of the quantification of Planck’s action. The other equation is: (3.15) where e = electron electric charge. The frequency of spectral lines is (3.16) The first term of the right-hand side equals the experimental Rydberg constant. 3-3-2 Wilson and Sommerfeld elliptic orbits. Due to many theoretical considerations, Wilson and Sommerfeld concluded that the Bohr theory was incomplete, proposing the application of Classical Mechanics to the Keplerian elliptical orbiting electrons, where the values are functions of two variables, r and q. The KE in this case is (3.17) and the Lagrange momentum is by definition (3.18) (3.19) The conditions of quantification are (3.20) m r^2q is the momentum of rotation in a central electrical field in agreement with Kepler’s law of the areas, and is coincident with the Bohr condition for circular orbits. To calculate the second integral, we need to take in account the expression for energy (3.21) Using the first equation in (3.14) we have (3.22) This equation shows the variable separation. Sommerfeld calculated the integral by the Cauchy residues method and found the following value (3.23) Equating this to n2 h he found (3.24) Considering that the term ½n1½+n2 could be equal to n (each is an integer that cannot be simultaneously zero), the theory then yields the same level of energy as Bohr’s theory. The introduction of two levels of freedom cannot in itself explain the fine structure of the Balmer series 3-3-3 Sommerfeld fine structure. Sommerfeld applied SR to the Bohr atom assuming that the electron inside the atom is traveling near the speed of light. The KE equation is the known SR KE equation where b is the following expression (3.25) We cannot define the momentum pi conjugate of the variable qi as equal to the derivative of KE with respect to q^·i. We need to introduce the Lagrange relativistic function (3.26) where U is the potential energy. Next we need to define the following expression (3.27) In the hydrogen atom, the potential energy is independent of q^·i, and consequently we can write (3.28) (3.29) The conditions (3.20) for quantification are: (3.30) A detailed study of the trajectory led Sommerfeld to demonstrate that an electron precede around the nucleus. Fig. 3-2 The momentum pq is merely the momentum of rotation around the center and is constant - that is - the theorem of areas is valid The first equality in (3.30) yields (3.31) The total energy is the electron rest mass plus the KE and the potential energy given by (3.32) Taking into account equations (3.25), (3.28) and (3.29) it is easy to verify that W can be written as W = c (3.33) The total energy is conceptually equal to that of classic mechanics, that is, W = E + mo c^2, where E is the kinetic energy and mo c^2is the rest mass energy. Solving equation (3.33) with respect to pr, we have pr = (3.34) Where A, B and C are written (3.35) (3.36) (3.37) Sommerfeld introduced, for the first time, what is called the fine-structure constant (3.38) With this we can write C as follows (3.39) Applying the theorem of residues we have (3.40) Equating the second member to n2 h according to equation (3.30) and replacing A, B and C with their values, Sommerfeld found, after a simple calculation 1 + (3.41) This equation yields the energy E of the ground state defined by the quantum number n1 and n2. Since we are not making a detailed and specialized study of this issue, we will only add that the first approximation with no degree greater than a^2 will only yield equation (3-24). A better result is obtained if the term in a^2 is developed to the second degree. If c is replaced by its values from the a equation (3.38) and reworking the term as a sum, the final equation for E is (3.42) The third term inside the brackets explains the fine structure of Balmer’s Series, because the term depends on and n2 independently, not only on + n2. Sommerfeld called n1 + n2 the total quantum number n. Setting n1 = k and taking (3.43) where R is the Rydberg constant, equation (3.42) is (3.44) 3-3-4 Autodynamics’ approach. Total energy in AD is conceptually different from SR. We saw that in SR W = mo c^2 + E, where E is electron KE. In AD, E could be the KE or simply the Energy, absorbed or emitted by an electron. This is possible for two reasons. First: The principle of energy absorption-energy emission or decay. Second: An atom only absorbs the same quantum of energy that it can emit. E preferably represents this type of quantified energy. AD enables the momentum equation to be written directly in polar coordinates, (3.45) (3.46) or to create a Lagrange function, taking the partial derivative as a function of b^2 in polar coordinates. (3.47) The Lagrange function is written as (3.48) Taking the partial derivative we have (3.49) (3.50) As will be seen later, using equations (3.45) and (3.46) is equivalent to using equations (3.49) and (3.50), because the terms are equal in sum and are eliminated. The total internal energy W is equal to the electron rest mass energy plus the electron potential energy W = mo c^2 - (3.51) As it will be demonstrated later, this equation showing the variable separation, can be written as follows W = c (3.52) As is shown in Appendix A10, the second part of (3.52) is equal to mo c^2. Rearranging, squaring both sides, we have (3.53) Working out the equation we find (3.54) As we will see later, the q term introduces a new quantum number that leads to the four quantum numbers of the Dirac’s equation. Making (3.55) (3.56) (3.57) The equation showing the variable separation is (3.58) Remembering that W = mo c^2 + E (3.59) and introducing the Sommerfeld fine structure constant a, A, B and C take the following form: (3.60) (3.61) (3.62) Applying the theorem of residues, a similar equation to equation (3.40) is found, with the only difference being that q divides the result. (3.63) Equating the right-hand term to n2 h and replacing the constants by its values in equations (3.60), (3.61) and (3.62) and working out the equation it yields (3.64) Remembering that (3.65) and taking into account equation (3.45) and (3.46) (3.66) We have now four quantum numbers, n, n1, n2 and r. Changing the notation, defining n1 / r = p and n = n1 + n2 = p + l +1 and l + 1 = j + 1/2, the equation is (3.67) This is the original Dirac equation, found through wave mechanics with four wave functions in an electromagnetic field, taking into account the spinning magnetic electron. Developing a^2to a second degree, the final equation, equivalent to equation (3.44) is (3.68) It is interesting that this simple application of AD directly yields the Dirac’s equation, originally obtained by a complex mathematical derivation. Apparently, the AD treatment of the Bohr atom needs to take into account the electron magnetic moment. We say apparently, because the participation of q, that represents the vector momentum of the electron around its orbit, could be equivalent to Dirac’s equation. In this case, it should only be necessary to introduce the spin of the electron into the equations. [1] .- The famous equation is Dy + (8 p^2 m) / h^2 [E-U (x,y,z)] y = 0 If U = 0, the solution of the equation is: y = a e^[(2^p I) / h [Et - (2 m E)^1/2 (ax +^b^y +^g^z)] a = constant amplitude, a, b, g the cosine regarding the direction of the wave propagation., E = h n , l = h / (2 m E)^1/2 = h / m v and m = rest mass, v = electron velocity, E = Energy. From “L’Electron Magnetique” (Theorie de Dirac) par Louis de Broglie, Hermann et Cie. Editeiurs, Paris, 1934, page 60.