Quantum Physics

28.3 The Bohr Theory of Hydrogen 9072. Only certain electron orbits are stable. These are orbits in which the hydrogenatom doesn’t emit energy in the form of electromagnetic radiation. Hence, thetotal energy of the atom remains constant, and classical mechanics can be usedto describe the electron’s motion.3. Radiation is emitted by the hydrogen atom when the electron “jumps” from amore energetic initial state to a less energetic state. The “jump” can’t be visualizedor treated classically. In particular, the frequency f of the radiation emitted in thejump is related to the change in the atom’s energy and is independent of the frequencyof the electron’s orbital motion. The frequency of the emitted radiation is given byE i E f hf [28.3]where E i is the energy of the initial state, E f is the energy of the final state, h isPlanck’s constant, and E i E f .4. The size of the allowed electron orbits is determined by a condition imposedon the electron’s orbital angular momentum: the allowed orbits are those forwhich the electron’s orbital angular momentum about the nucleus is an integralmultiple of (pronounced “h bar”), where :m e vr nn 1, 2, 3, . . . [28.4]With these four assumptions, we can calculate the allowed energies and emissionwavelengths of the hydrogen atom. We use the model pictured in Figure 28.5,in which the electron travels in a circular orbit of radius r with an orbital speed v.The electrical potential energy of the atom isPE k eq 1 q 2r k e(e)(e)rh/2k ewhere k e is the Coulomb constant. Assuming the nucleus is at rest, the total energyE of the atom is the sum of the kinetic and potential energy:E KE PE 1 2 m ev 2 e 2 k e[28.5]rWe apply Newton’s second law to the electron. We know that the electric forceof attraction on the electron, k e e 2 /r 2 , must equal m e a r , where a r v 2 /r is the centripetalacceleration of the electron. Thus,e 2k e[28.6]r 2 m v 2erFrom this equation, we see that the kinetic energy of the electron is12 m ev 2 k ee 2[28.7]2rWe can combine this result with Equation 28.5 and express the energy of the atom asE k ee 2[28.8]2rwhere the negative value of the energy indicates that the electron is bound to theproton.An expression for r is obtained by solving Equations 28.4 and 28.6 for v andequating the results:v 2 n2 m 2 e r 2 k ee 2m e rr n n2 2n 1, 2, 3, . . . [28.9]m e k e e 2This equation is based on the assumption that the electron can exist only in certainallowed orbits determined by the integer n.e 2rNIELS BOHR, Danish Physicist(1885 – 1962)Bohr was an active participant in the earlydevelopment of quantum mechanics andprovided much of its philosophical framework.During the 1920s and 1930s, heheaded the Institute for Advanced Studiesin Copenhagen. The institute was a magnetfor many of the world’s best physicistsand provided a forum for the exchange ofideas. When Bohr visited the United Statesin 1939 to attend a scientific conference,he brought news that the fission of uraniumhad been observed by Hahn andStrassman in Berlin. The results were thefoundations of the atomic bomb developedin the United States during WorldWar II. Bohr was awarded the 1922 NobelPrize for his investigation of the structureof atoms and of the radiation emanatingfrom them. Energy of the hydrogen atom The radii of the Bohr orbitsare quantizedPrinceton University/Courtesy of AIP Emilio Segre Visual Archives