I Introduction (1.29) We can calculate van+ll for high values of n from Equation 1. 26 by substituting Equation 1.27 for the orbital radii. We obtain 职=()[-m+ 1「(2n+1) a0Ln2(n+1)2 4 To apply the correspondence principle we equate orbit and van+ln for high n and obtain (4丌∈0) (1.31) Examination of Equation 1. 25 shows that, because the orbital radii are quantized in accord with Equation 1. 27, the total internal energy of the atom must also be quantized. We may thus write Equation 1. 25, replacing r with n-ao and E with En to indicate the nth energy level. We obtain E (1.32) where n is called the principal quantum number. Substituting for ao we have Note that the minus sign is required since the electron is bound to the proton. The TME must therefore be negative. E=0 corresponds to infinitely separated proton and electron each having zero kinetic energy Equation 1. 33 is called the Bohr energy. Although the Bohr model is not entirely correct, the Bohr energy is correct. It applies to any quantum level of the hydrogen atom as designated by the quantum number n. In fact, it applies to any one-electron atom, for example helium with one electron removed, when the number of protons in the nucleus is included. When compared with Equation 1.9, the Bohr energy yields the value of the Rydberg constant, which is found to be
14 1 Introduction ν2 orbit = 1 4π2n4a2 0 1 me e2 4π 0 1 n2a0 (1.29) We can calculate ν2 (n+1)n for high values of n from Equation 1.26 by substituting Equation 1.27 for the orbital radii. We obtain lim n→∞ν2 (n+1)n = lim n→∞ 1 2 e2 4π 0 1 ha0 1 n2 − 1 (n + 1)2 2 = lim n→∞ 1 2 e2 4π 0 1 ha0 (2n + 1) n2 (n + 1)2 = e2 4π 0 1 ha0 1 n3 (1.30) To apply the correspondence principle we equate ν2 orbit and ν2 (n+1)n for high n and obtain a0 = (4π 0) 2 mee2 (1.31) Examination of Equation 1.25 shows that, because the orbital radii are quantized in accord with Equation 1.27, the total internal energy of the atom must also be quantized. We may thus write Equation 1.25, replacing r with n2a0 and E with En to indicate the nth energy level. We obtain En = − e2 4π 0 1 2n2a0 (1.32) where n is called the principal quantum number. Substituting for a0 we have En = −1 2 e2 4π 0 2 me 2 · 1 n2 (1.33) Note that the minus sign is required since the electron is bound to the proton. The TME must therefore be negative. E = 0 corresponds to infinitely separated proton and electron each having zero kinetic energy. Equation 1.33 is called the Bohr energy. Although the Bohr model is not entirely correct, the Bohr energy is correct. It applies to any quantum level of the hydrogen atom as designated by the quantum number n. In fact, it applies to any one-electron atom, for example helium with one electron removed, when the number of protons in the nucleus is included. When compared with Equation 1.9, the Bohr energy yields the value of the Rydberg constant, which is found to be
=10973731568525×107m (1.34) where the numerical value given is the accepted value today. The agreement be- tween this theoretically obtained value and that empirically determined using atomic pectroscopy was astonishing. while we know today that some of the concepts of the Bohr model are incorrect, it remains a paradigm of clear and creative thinking Bohr's use of known empirical facts together with his statement of the correspon- dence principle led to a breakthrough in physics that gave birth to quantum phys as we know it today. Although physicists know that the wave nature of matter, as ex amplified by, for example, the Davisson-Germer experiment, makes precise location of particles problematic, most nevertheless envision a Bohr-like atom when thinking about atoms(even if they don' t admit it in public). Besides permitting visualization, the Bohr model also gives the correct order of magnitude and scaling with principal quantum number of parameters, such as orbital distances and electronic velocities Most importantly, it also gives the correct quantized energies It also follows from the above analysis that the electronic angular momentum ust be quantized in units of h, the postulate that is incorrectly attributed to Bohr. Indeed, this postulate follows as a consequence of the his two stated postulates and the correspondence principle. Note, however, that h has units of angular momentum (as does h). Before leaving the Bohr energy it is useful to cast this important quantity in terms of other, more revealing, parameters. One of the most convenient ways of writing it is in terms of the fine structure constant, which is a combination of fundamer constants that results in a pure number that is very nearly 1/137. This number is of fundamental importance in quantum physics. It is given by The Greek letter a is universally used for the fine structure constant. Regrettably, it is also universally used for a number of other important quantities. In terms of the fine structure constant the bohr energy is The reason Equation 1.36 is convenient is that most physics students know that he rest mass of the electron is 0.51MeV(IMeV= 10%eV). A simple calculation shows that the lowest energy state of the Bohr atom, and, consequently, hydrogen, is -13.6ev. This energy is also called the ionization potential since it is the minimum
1.2 Early Theory 15 RH = e2 4π 0 2 me 4πc3 = 1.0973731568525× 107 m−1 (1.34) where the numerical value given is the accepted value today. The agreement between this theoretically obtained value and that empirically determined using atomic spectroscopy was astonishing. While we know today that some of the concepts of the Bohr model are incorrect, it remains a paradigm of clear and creative thinking. Bohr’s use of known empirical facts together with his statement of the correspondence principle led to a breakthrough in physics that gave birth to quantum physics as we know it today. Although physicists know that the wave nature of matter, as exemplified by, for example, the Davisson–Germer experiment, makes precise location of particles problematic, most nevertheless envision a Bohr-like atom when thinking about atoms (even if they don’t admit it in public). Besides permitting visualization, the Bohr model also gives the correct order of magnitude and scaling with principal quantum number of parameters, such as orbital distances and electronic velocities. Most importantly, it also gives the correct quantized energies. It also follows from the above analysis that the electronic angular momentum must be quantized in units of , the postulate that is incorrectly attributed to Bohr. Indeed, this postulate follows as a consequence of the his two stated postulates and the correspondence principle. Note, however, that has units of angular momentum (as does h). Before leaving the Bohr energy it is useful to cast this important quantity in terms of other, more revealing, parameters. One of the most convenient ways of writing it is in terms of the fine structure constant, which is a combination of fundamental constants that results in a pure number that is very nearly 1/137. This number is of fundamental importance in quantum physics. It is given by α = e2 (4π 0) c 1 137 (1.35) The Greek letter α is universally used for the fine structure constant. Regrettably, it is also universally used for a number of other important quantities. In terms of the fine structure constant the Bohr energy is En = −1 2 α2 mec2 n2 (1.36) The reason Equation 1.36 is convenient is that most physics students know that the rest mass of the electron is 0.51MeV (1MeV = 106eV). A simple calculation shows that the lowest energy state of the Bohr atom, and, consequently, hydrogen, is −13.6 eV. This energy is also called the ionization potential since it is the minimum
16 I Introduction energy required to liberate the electron from the hydrogen atom, leaving behind a hydrogen ion. "A hydrogen ion is simply a proton, but the term"ionization poten- tial"is applied to all atoms and molecules. It is also convenient to remember the Bohr energy in electron-volts. For this purpose we may rewrite Equation 1.33 as 3.6056923eV from which it is clear that the ionization potential of hydrogen is 13.6 eV. From the Bohr energy as given in Equation 1. 37 it is a simple matter to calculate energy differences between any pair of levels. We have △En=13605693( ) (1.38) where it is assumed that n n. If we let n 2 and use the relation E hc/i e immediately recover the Balmer formula, Equation 1. 9, the formula that pre- dicts the wavelengths of emitted radiation for which the lower state is n=2, the Balmer series. There are, however, other series that are observed. For example, if we let n= l we obtain a formula that predicts the wavelengths of the Lyman series. Because the ground state lies much lower than n 2 these energy differences are considerably greater than those of the Balmer series. Consequently, transitions in the Lyman series yield radiation in the ultraviolet region of the spectrum. Let us now investigate the relationship between the quantum number n and the angular momentum From Equations 1.24, 1.31, and 1. 27 the electronic velocity in the nth orbit so the angular momentum of the electron in the nth orbit is L We have therefore resurrected the"postulate"that the orbital angular momentum is quantized in units of h. Interestingly, this result is incorrect because, as we will learn later, the states of hydrogen can have any integer multiple of h or zero as long as it is less than the principal quantum number n. This means that the electronic angular momentum in the ground state is zero, not unity as predicted by Equation 1.40. Nonetheless, the Bohr model of the atom provides us with quantities that give the correct order of magnitude of actual atomic parameters. For this reason it is
16 1 Introduction energy required to liberate the electron from the hydrogen atom, leaving behind a “hydrogen ion.” A hydrogen ion is simply a proton, but the term “ionization potential” is applied to all atoms and molecules. It is also convenient to remember the Bohr energy in electron-volts. For this purpose we may rewrite Equation 1.33 as En = −13.6056923 eV n2 (1.37) from which it is clear that the ionization potential of hydrogen is 13.6 eV. From the Bohr energy as given in Equation 1.37 it is a simple matter to calculate energy differences between any pair of levels. We have En n = 13.6056923 eV 1 n2 − 1 n2 (1.38) where it is assumed that n < n . If we let n = 2 and use the relation E = hc/λ we immediately recover the Balmer formula, Equation 1.9, the formula that predicts the wavelengths of emitted radiation for which the lower state is n = 2, the Balmer series. There are, however, other series that are observed. For example, if we let n = 1 we obtain a formula that predicts the wavelengths of the Lyman series. Because the ground state lies much lower than n = 2 these energy differences are considerably greater than those of the Balmer series. Consequently, transitions in the Lyman series yield radiation in the ultraviolet region of the spectrum. Let us now investigate the relationship between the quantum number n and the angular momentum. From Equations 1.24, 1.31, and 1.27 the electronic velocity in the nth orbit is vn = 1 me e2 4π 0 1 n2a0 = 1 me n2a0 (1.39) so the angular momentum of the electron in the nth orbit is Ln = mevnrn = n (1.40) We have therefore resurrected the “postulate” that the orbital angular momentum is quantized in units of . Interestingly, this result is incorrect because, as we will learn later, the states of hydrogen can have any integer multiple of or zero as long as it is less than the principal quantum number n. This means that the electronic angular momentum in the ground state is zero, not unity as predicted by Equation 1.40. Nonetheless, the Bohr model of the atom provides us with quantities that give the correct order of magnitude of actual atomic parameters. For this reason it is
1. 2 Early Theory extremely useful. For example, a bit of algebra permits us to write Un in the form (see Problem 5) Notice that Equation 1. 41 tell us that the highest orbital velocity for an electron occurs in the ground state, but, even then, this velocity is more than two orders of magnitude smaller than the speed of light, thus justifying the nonrelativistic treat- ment. It is often convenient to express Bohr parameters in terms of the fine structure constant, so we present in Table 1. I a partial listing Before leaving the subject of the Bohr atom we discuss another of the conve- niences afforded by it. Since the model is that of an electron circling a proton, the electric current that is the result of the electronic motion is the source of a magnetic field. Therefore, a Bohr atom has a magnetic dipole moment associated with the orbital motion of the electron about the proton and the atom generates a magnetic field identical to that of a bar magnet The magnitude of this magnetic moment for the ground state of the Bohr atom is referred to as the Bohr magneton, and is designated by the symbol uB. Magnetic moments are often measured in terms of the bohr magneton so we calculate its value. Figure 1.5 is a schematic diagram of the bohr atom with the magnetic field lines due to the orbital motion of the electron Also indicated in this figure are the relevant parameters. The magnetic moment of a current-carrying loop is given by the product of the area of the loop and the current. The current is the electronic charge divided by the period of the motion, T=2Tao/u. Therefore, the magnetic moment is Table 1.1 Quantities from the bohr model of the atom in terms of the fine structure magnetic field lines generated
1.2 Early Theory 17 extremely useful. For example, a bit of algebra permits us to write vn in the form (see Problem 5) vn = α n c (1.41) Notice that Equation 1.41 tell us that the highest orbital velocity for an electron occurs in the ground state, but, even then, this velocity is more than two orders of magnitude smaller than the speed of light, thus justifying the nonrelativistic treatment. It is often convenient to express Bohr parameters in terms of the fine structure constant, so we present in Table 1.1 a partial listing. Before leaving the subject of the Bohr atom we discuss another of the conveniences afforded by it. Since the model is that of an electron circling a proton, the electric current that is the result of the electronic motion is the source of a magnetic field. Therefore, a Bohr atom has a magnetic dipole moment associated with the orbital motion of the electron about the proton and the atom generates a magnetic field identical to that of a bar magnet. The magnitude of this magnetic moment for the ground state of the Bohr atom is referred to as the Bohr magneton, and is designated by the symbol μB. Magnetic moments are often measured in terms of the Bohr magneton so we calculate its value. Figure 1.5 is a schematic diagram of the Bohr atom with the magnetic field lines due to the orbital motion of the electron. Also indicated in this figure are the relevant parameters. The magnetic moment of a current-carrying loop is given by the product of the area of the loop and the current. The current is the electronic charge divided by the period of the motion, T = 2πa0/v. Therefore, the magnetic moment is Table 1.1 Quantities from the Bohr model of the atom in terms of the fine structure constant α α = e2 (4π 0) c En = −1 2 α2 me c2 n2 a0 = mecα vn = α n c Fig. 1.5 The Bohr model of the atom shown with the magnetic field lines generated by the orbiting electron
A=e (rao) (1.42) which may be written in terms of the orbital angular momentum L=mevao as 2m where the vector nature of the angular momentum has been taken into account. Because the electronic charge is negative, the angular momentum and the mag netic moment are in opposite directions. From Equation 1. 43 it is clear that there is a direct relationship between the magnetic moment and the angular momen tum. Because the angular momentum is quantized in units of h(see Equation 1. 40), the magnitude of the magnetic moment in the first Bohr orbit, the bohr μB-2me 44) 1.2.2 The de broglie Wavelength In 1923 Louis de Broglie, in his doctoral thesis at the Sorbonne in Paris, proposed hat material particles, that is, particles having nonzero mass such as electrons, ex- hibit a wave-particle duality as had been established for light. At first this notic was met with skepticism, but after some encouragement from notable scientists, particularly Einstein, it gained credibility. A few years later, the experiments of Davisson and Germer validated the idea as did other experiments performed in other laboratories. In 1929 de Broglie was awarded the Nobel Prize in Physics"for his discovery of the wave nature of electrons de Broglie set forth a relationship between the momentum of a particle and the wavelength"of matter waves, as they were called. Today we simply refer to the de Broglie wavelength. He deduced a relation between photons and their momentum and proposed the same relation for particles. The relativistic relation between energy nd momentum for a particle of rest mass, mo, is given by p2c2+moc+4 which, for the massless photon, reduces to
18 1 Introduction μ = e v 2πa0 πa2 0 = eva0 2 (1.42) which may be written in terms of the orbital angular momentum L = meva0 as μ = − e 2me L (1.43) where the vector nature of the angular momentum has been taken into account. Because the electronic charge is negative, the angular momentum and the magnetic moment are in opposite directions. From Equation 1.43 it is clear that there is a direct relationship between the magnetic moment and the angular momentum. Because the angular momentum is quantized in units of (see Equation 1.40), the magnitude of the magnetic moment in the first Bohr orbit, the Bohr magneton, is μB = e 2me (1.44) 1.2.2 The de Broglie Wavelength In 1923 Louis de Broglie, in his doctoral thesis at the Sorbonne in Paris, proposed that material particles, that is, particles having nonzero mass such as electrons, exhibit a wave–particle duality as had been established for light. At first this notion was met with skepticism, but after some encouragement from notable scientists, particularly Einstein, it gained credibility. A few years later, the experiments of Davisson and Germer validated the idea as did other experiments performed in other laboratories. In 1929 de Broglie was awarded the Nobel Prize in Physics “for his discovery of the wave nature of electrons.” de Broglie set forth a relationship between the momentum of a particle and the “wavelength” of matter waves, as they were called. Today we simply refer to the de Broglie wavelength. He deduced a relation between photons and their momentum and proposed the same relation for particles. The relativistic relation between energy and momentum for a particle of rest mass, m0, is given by E = p2c2 + m2 0c4 (1.45) which, for the massless photon, reduces to E = pc (1.46)