Quantum Physics

910 Chapter 28 Atomic PhysicsRemarks The first wavelength is in the red region of the visible spectrum. We could also obtain the energy of thephoton by using Equation 28.3 in the form hf E 3 E 2 , where E 2 and E 3 are the energy levels of the hydrogenatom, calculated from Equation 28.13. Note that this is the lowest energy photon in the Balmer series, because it involvesthe smallest energy change. The second photon, the most energetic, is in the ultraviolet region.Exercise 28.1(a) Calculate the energy of the shortest wavelength photon emitted in the Balmer series for hydrogen. (b) Calculatethe wavelength of a transition from n 4 to n 2.Answers(a) 3.40 eV (b) 486 nmInvestigate transitions between various states by logging into PhysicsNow at www.cp7e.com and goingto Interactive Example 28.1.Bohr’s Correspondence PrincipleIn our study of relativity in Chapter 26, we found that Newtonian mechanics cannotbe used to describe phenomena that occur at speeds approaching the speed oflight. Newtonian mechanics is a special case of relativistic mechanics and appliesonly when v is much smaller than c. Similarly, quantum mechanics is in agreementwith classical physics when the energy differences between quantized levels arevery small. This principle, first set forth by Bohr, is called the correspondenceprinciple.For example, consider the hydrogen atom with n 10 000. For such large valuesof n, the energy differences between adjacent levels approach zero and the levelsare nearly continuous, as Equation 28.13 shows. As a consequence, the classicalmodel is reasonably accurate in describing the system for large values of n. Accordingto the classical model, the frequency of the light emitted by the atom is equalto the frequency of revolution of the electron in its orbit about the nucleus. Calculationsshow that for n 10 000, this frequency is different from that predicted byquantum mechanics by less than 0.015%.28.4 MODIFICATION OF THE BOHR THEORYThe Bohr theory of the hydrogen atom was a tremendous success in certain areasbecause it explained several features of the hydrogen spectrum that had previouslydefied explanation. It accounted for the Balmer series and other series; it predicteda value for the Rydberg constant that is in excellent agreement with the experimentalvalue; it gave an expression for the radius of the atom; and it predictedthe energy levels of hydrogen. Although these successes were important to scientists,it is perhaps even more significant that the Bohr theory gave us a model ofwhat the atom looks like and how it behaves. Once a basic model is constructed,refinements and modifications can be made to enlarge on the concept and to explainfiner details.The analysis used in the Bohr theory is also successful when applied to hydrogenlikeatoms. An atom is said to be hydrogen-like when it contains only one electron.Examples are singly ionized helium, doubly ionized lithium, triply ionized beryllium,and so forth. The results of the Bohr theory for hydrogen can be extended to hydrogen-likeatoms by substituting Ze 2 for e 2 in the hydrogen equations, where Z is theatomic number of the element. For example, Equations 28.12 and 28.15 becomeandE n m ek e 2 Z 2 e 42 21 1 n 2 m ek e 2 Z 2 e 44c 3 1n f2 1n i2n 1, 2, 3, . . . [28.18][28.19]