I Introduction 识g.1.2(a) Schematic iagram of the apparatus used in the Franck-Hertz The cathode, grid ey I △x<△)<A and anode are labeled C. G and A, respectively b)Simulated data Nobel Prize in Physics. The citation for the 1925 prize reads: " for their discovery of the laws governing the impact of an electron upon an atom. " Notice that this is the same Hertz who discovered the photoelectric effect and whose student, Lenard, won the nobel prize for elucidating it Figure 1. 2a shows a schematic diagram of the apparatus used for this experi- ment. It consists of a cathode C from which electrons are emitted by heating with a high current(not shown in the diagram), an anode a to collect the electrons, a grid g between the cathode and anode. The entire apparatus is contained within a glass envelope from which the air has been evacuated and atoms of a given species ntroduced In the original experiments, mercury atoms were used, but any atom will suffice. Electrons emitted from the cathode are accelerated by the grid voltage VG, pass through the grid and are collected at the anode. The anode is kept at a slightly lower potential than the grid to prevent the electrons from acquiring addi- tional kinetic energy. Electrons arriving at the anode are collected and the current iA Data are in the form of graphs of VG versus iA. As expected, the current increases as VG increases, but it decreases at regular intervals as shown in the hypothetical data plotted in Fig. 1. 2b. These data clearly suggest quantized atomic energy levels. For any setting of the grid voltage the maximum electronic kinetic energy in the apparatus is eVG. When e VG is lower than energy separation between the lowest atomic level, referred to as the"ground state", and the next highest level, the"first excited state, none of the electronic kinetic energy can be converted to atomic in- ternal energy. This is because there simply isn't any level to excite between the ground state and the first excited state. The only thing that can occur is elastic scat tering between the electrons and the atoms. When, however, e VG reaches the energy eparation between the ground state and the first excited state some of the atoms "become excited in these inelastic collisions the exciting electrons lose kinetic energy(by an amount equal to the excitation energy) and are thus not collected at he anode. The result is that the current decreases. as VG is further increased the electrons that have already excited the atom once can be reaccelerated and collected he current thus increases again. when these electrons are accelerated to a kinetic energy sufficient to excite the atom again, the anode current again decreases.( For simplicity we are assuming that only the ground and first excited states are important in this experiment. Thus, the peaks in the curve of VG versus iA will be equally
4 1 Introduction Fig. 1.2 (a) Schematic diagram of the apparatus used in the Franck–Hertz experiment. The cathode, grid and anode are labeled C, G, and A, respectively. (b) Simulated data Nobel Prize in Physics. The citation for the 1925 prize reads: “for their discovery of the laws governing the impact of an electron upon an atom.” Notice that this is the same Hertz who discovered the photoelectric effect and whose student, Lenard, won the Nobel Prize for elucidating it. Figure 1.2a shows a schematic diagram of the apparatus used for this experiment. It consists of a cathode C from which electrons are emitted by heating with a high current (not shown in the diagram), an anode A to collect the electrons, and a grid G between the cathode and anode. The entire apparatus is contained within a glass envelope from which the air has been evacuated and atoms of a given species introduced. In the original experiments, mercury atoms were used, but any atom will suffice. Electrons emitted from the cathode are accelerated by the grid voltage VG, pass through the grid and are collected at the anode. The anode is kept at a slightly lower potential than the grid to prevent the electrons from acquiring additional kinetic energy. Electrons arriving at the anode are collected and the current i A measured. Data are in the form of graphs of VG versusi A. As expected, the current increases as VG increases, but it decreases at regular intervals as shown in the hypothetical data plotted in Fig. 1.2b. These data clearly suggest quantized atomic energy levels. For any setting of the grid voltage the maximum electronic kinetic energy in the apparatus is eVG. When eVG is lower than energy separation between the lowest atomic level, referred to as the “ground state”, and the next highest level, the “first excited state,” none of the electronic kinetic energy can be converted to atomic internal energy. This is because there simply isn’t any level to excite between the ground state and the first excited state. The only thing that can occur is elastic scattering between the electrons and the atoms. When, however, eVG reaches the energy separation between the ground state and the first excited state some of the atoms “become excited.” In these inelastic collisions the exciting electrons lose kinetic energy (by an amount equal to the excitation energy) and are thus not collected at the anode. The result is that the current decreases. As VG is further increased, the electrons that have already excited the atom once can be reaccelerated and collected. The current thus increases again. When these electrons are accelerated to a kinetic energy sufficient to excite the atom again, the anode current again decreases. (For simplicity we are assuming that only the ground and first excited states are important in this experiment.) Thus, the peaks in the curve of VG versus i A will be equally
1. I Early Experiments spaced. From the hypothetical data we would conclude that the energy separation between the ground state and the first excited state is eAV. This experiment clearly demonstrates that the atomic energy levels are quantized, for if they werent the current would simply rise continuously and then level off(saturate)when all the electrons were collected While energies in the SI system are measured in joules, this is a rather large unit for measurement and discussion of atomic energies. It is frequently more convenient to use the electron-volt, abbreviated eV. One electron-volt is the kinetic energy ac- quired by a particle of charge e when it is accelerated through a potential difference of one volt Thus leV=(1.602 =1.602×10-19J (16) It is often convenient to write plancks constant in terms of ev rather than j in which case h=6.58x 10-ev.. In the original Franck-Hertz experiment the separation between peaks along the abscissa was roughly 4.9V. 1.1.3 Atomic Spectroscopy Emission Spectroscopy Perhaps the most important experiments for the development of quantum theory were those using atomic spectroscopy. There are two general types of atomic spec oscopy, absorption and emission spectroscopy. In emission spectroscopy a sample of atoms is"excited, usually with an electrical discharge such as a spark. This has the effect of exciting the atoms, not just to the first excited state, but to a va- riety of excited states. In general, however, these states have finite lifetimes. When they decay to lower states, not just the ground state, they do so by emitting light (Whether visible or not, physicists generally refer to electromagnetic radiation as light. )Because the energy levels are uniquely quantized for each atom, the en- ergy of the emitted light is quantized and hence, in accord with Equation 1.3, the wavelengths that are emitted are unique. Emission spectroscopy is routinely used for identification and trace analysis. In the early days of spectroscopy, the latter part of the nineteenth century and the beginning of the twentieth century, the detector common use(aside from the human eye)was a photographic plate Using prisms or diffraction gratings, the light in an emission spectroscopy experiment was dispersed nto its constituent wavelengths and focused on a photographic plate. Because only in discrete wavelengths w ed most of the plat dark. that is. no exposed. The portion that was exposed exhibited lines at the discrete wavelengths emitted by the atoms. These atomic spectra were thus known as"line spectra"and ly, Figure 1.3 shows a schematic diagram of a photographic plate of an emi sion spectrum of atomic hydrogen. Never mind that hydrogen occurs naturally as
1.1 Early Experiments 5 spaced. From the hypothetical data we would conclude that the energy separation between the ground state and the first excited state is eV . This experiment clearly demonstrates that the atomic energy levels are quantized, for if they weren’t the current would simply rise continuously and then level off (saturate) when all the electrons were collected. While energies in the SI system are measured in joules, this is a rather large unit for measurement and discussion of atomic energies. It is frequently more convenient to use the electron-volt, abbreviated eV. One electron-volt is the kinetic energy acquired by a particle of charge e when it is accelerated through a potential difference of one volt. Thus, 1eV = 1.602 × 10−19C (1V) = 1.602 × 10−19J (1.6) It is often convenient to write Planck’s constant in terms of eV rather than J in which case = 6.58×10−16 eV·s. In the original Franck-Hertz experiment the separation between peaks along the abscissa was roughly 4.9 V. 1.1.3 Atomic Spectroscopy Emission Spectroscopy Perhaps the most important experiments for the development of quantum theory were those using atomic spectroscopy. There are two general types of atomic spectroscopy, absorption and emission spectroscopy. In emission spectroscopy a sample of atoms is “excited,” usually with an electrical discharge such as a spark. This has the effect of exciting the atoms, not just to the first excited state, but to a variety of excited states. In general, however, these states have finite lifetimes. When they decay to lower states, not just the ground state, they do so by emitting light. (Whether visible or not, physicists generally refer to electromagnetic radiation as “light.”) Because the energy levels are uniquely quantized for each atom, the energy of the emitted light is quantized and hence, in accord with Equation 1.3, the wavelengths that are emitted are unique. Emission spectroscopy is routinely used for identification and trace analysis. In the early days of spectroscopy, the latter part of the nineteenth century and the beginning of the twentieth century, the detector in common use (aside from the human eye) was a photographic plate. Using prisms or diffraction gratings, the light in an emission spectroscopy experiment was dispersed into its constituent wavelengths and focused on a photographic plate. Because only certain discrete wavelengths were emitted most of the plate was dark, that is, not exposed. The portion that was exposed exhibited lines at the discrete wavelengths emitted by the atoms. These atomic spectra were thus known as “line spectra” and the transitions are known, even today, as lines. Figure 1.3 shows a schematic diagram of a photographic plate of an emission spectrum of atomic hydrogen. Never mind that hydrogen occurs naturally as
6 I Introduction a photographic plate of the f atomic hydrogen in the visible region of the electromagnet spectrum Shown are the lines fthe Balmer series diatomic molecules. When an electrical discharge occurs, most of the molecules dissociate and become atoms, so the observed spectrum is predominantly that of atomic hydrogen. The first lines of atomic hydrogen to be discovered were those of the balmer series. so named because in 1885 a Swiss school teacher. j.j. balmer without any physical explanation, set forth a formula that accurately predicted the bserved wavelengths of the known lines of atomic hydrogen The wavelengths of these Balmer lines had been known for many years, but it was Balmer who first related them through his now-famous formula. There are many other lines in the spectrum of atomic hydrogen, but the lines of the Balmer series were discovered first because the strongest of these lines lie in the visible region of the spectrum. The Balmer series actually terminates in the near-ultraviolet region of the spectrum at a wavelength of about 365 nm(see Problem 3). Because Balmer was unaware of the origination of these lines he designated them Ho, HB and so on, meaning the first hydrogen line, the second line, and so on. The lines of series that were discovered later employ a similar designation, but using the first letter of the discoverer's name. For example, the first line of the lyman series is La The wavelengths of the Balmer lines Ag are given by the relation 入B=36456m2-2m where n is an integer that is greater than 2. Thus, for example, the wavelength of H is 656.2 nm. Equation 1.7 can, however, be put in a more convenient form for later Ise by writing the inverse of the wavelength R where RH is called the Rydberg constant because Johannes Rydberg was instrumen tal in developing a generalized version of Equation 1.8 that predicted the wavelength anm between any two states, m and n, of hydrogen. In this generalized formula the 2- was replaced by the square of another integer. Thus, 1
6 1 Introduction Fig. 1.3 Schematic diagram of a photographic plate of the emission spectrum of atomic hydrogen in the visible region of the electromagnetic spectrum. Shown are the lines of the Balmer series diatomic molecules. When an electrical discharge occurs, most of the molecules dissociate and become atoms, so the observed spectrum is predominantly that of atomic hydrogen. The first lines of atomic hydrogen to be discovered were those of the Balmer series, so named because in 1885 a Swiss school teacher, J. J. Balmer, without any physical explanation, set forth a formula that accurately predicted the observed wavelengths of the known lines of atomic hydrogen. The wavelengths of these Balmer lines had been known for many years, but it was Balmer who first related them through his now-famous formula. There are many other lines in the spectrum of atomic hydrogen, but the lines of the Balmer series were discovered first because the strongest of these lines lie in the visible region of the spectrum. The Balmer series actually terminates in the near-ultraviolet region of the spectrum at a wavelength of about 365 nm (see Problem 3). Because Balmer was unaware of the origination of these lines he designated them Hα, Hβ and so on, meaning the first hydrogen line, the second line, and so on. The lines of series that were discovered later employ a similar designation, but using the first letter of the discoverer’s name. For example, the first line of the Lyman series is Lα. The wavelengths of the Balmer lines λB are given by the relation λB = 364.56 n2 n2 − 22 nm (1.7) where n is an integer that is greater than 2. Thus, for example, the wavelength of Hα is 656.2 nm. Equation 1.7 can, however, be put in a more convenient form for later use by writing the inverse of the wavelength: 1 λB = RH 1 22 − 1 n2 (1.8) where RH is called the Rydberg constant because Johannes Rydberg was instrumental in developing a generalized version of Equation 1.8 that predicted the wavelength λnm between any two states, m and n, of hydrogen. In this generalized formula the 22 was replaced by the square of another integer. Thus, 1 λnm = RH 1 m2 − 1 n2 (1.9)
1. I Early Experiments From Equation 1.7 and the known Balmer wavelengths, RH 1.097 m. while the Balmer formula was deduced on purely empirical grounds it was, as we shall see, crucial to the development of the Bohr theory of hydrogen There is a very useful relation between the wavelength of light A and the energy E of a photon of that wavelength. This relation is easily obtained from Equation 1.3 using convenient units, nm and ev. we have λ(innm)E(ineV)=hc 1240 (1.10) For example, according to this simple formula, the energy per photon of red light of wavelength 620 nm is 2eV. On the other hand, photons having energy of 5eV correspond to a wavelength of 248 nm. Absorption Spectroscopy In absorption spectroscopy a continuous source of light such as light from an incan descent bulb(blackbody radiation)irradiates an atomic sample. The light passin through the sample is detected. Again a photographic plate may be used as the de tector. In this case the background is the continuous bright incident light, but there are"holes"in the continuum due to absorption at specific wavelengths by the atomic sample. This might be thought of as a Frank-Hertz experiment with photons. One of the earliest such experiments was performed in 1824 by Fraunhofer. He dispersed the light from the sun. His continuous source was the solar interior and the atomic sample was the solar atmosphere. There are also molecules in the solar atmosphere but let us concentrate on the atomic constituents fraunhofer observed an abundance of lines which he labeled alphabetically from the red end of the spectrum. Because the solar atmosphere contains hydrogen it would be surprising if lines of the Balmer series were not present. Indeed, C and F are Ha and HB, respectively. Interestingly, the fourth line from the red end, a strong"hole"in the yellow portion of the spec- trum, was, of course, labeled D. We now know this line(actually a pair of lines) to be the result of absorptions by atomic sodium. Observation of the"D-line"is favorite test for the presence of sodium in elementary chemistry. In that test, the heat from the flame from the bunsen burner excites sodium atoms to the first excited state from which they decay, emitting yellow light, the D-line 1.1.4 Electron Diffraction Experiments Two seminal experiments were reported in 1925. One of these, commonly re ferred to as the Davisson-Germer experiment, was performed in the United States The other was performed by G. P. Thomson and his coworkers in Great Britain These experiments are complementary to the photoelectric effect because, while he explanation of the photoelectric effect relied on the particle nature of light, the
1.1 Early Experiments 7 From Equation 1.7 and the known Balmer wavelengths, RH ≈ 1.097 m−1. While the Balmer formula was deduced on purely empirical grounds it was, as we shall see, crucial to the development of the Bohr theory of hydrogen. There is a very useful relation between the wavelength of light λ and the energy E of a photon of that wavelength. This relation is easily obtained from Equation 1.3 using convenient units, nm and eV. We have λ(in nm)E(in eV) = hc = 1240 (1.10) For example, according to this simple formula, the energy per photon of red light of wavelength 620 nm is 2eV. On the other hand, photons having energy of 5eV correspond to a wavelength of 248 nm. Absorption Spectroscopy In absorption spectroscopy a continuous source of light such as light from an incandescent bulb (blackbody radiation) irradiates an atomic sample. The light passing through the sample is detected. Again a photographic plate may be used as the detector. In this case the background is the continuous bright incident light, but there are “holes” in the continuum due to absorption at specific wavelengths by the atomic sample. This might be thought of as a Frank–Hertz experiment with photons. One of the earliest such experiments was performed in 1824 by Fraunhofer. He dispersed the light from the sun. His continuous source was the solar interior and the atomic sample was the solar atmosphere. There are also molecules in the solar atmosphere, but let us concentrate on the atomic constituents. Fraunhofer observed an abundance of lines which he labeled alphabetically from the red end of the spectrum. Because the solar atmosphere contains hydrogen it would be surprising if lines of the Balmer series were not present. Indeed, C and F are Hα and Hβ , respectively. Interestingly, the fourth line from the red end, a strong “hole” in the yellow portion of the spectrum, was, of course, labeled D. We now know this line (actually a pair of lines) to be the result of absorptions by atomic sodium. Observation of the “D-line” is a favorite test for the presence of sodium in elementary chemistry. In that test, the heat from the flame from the Bunsen burner excites sodium atoms to the first excited state from which they decay, emitting yellow light, the D-line. 1.1.4 Electron Diffraction Experiments Two seminal experiments were reported in 1925. One of these, commonly referred to as the Davisson–Germer experiment, was performed in the United States. The other was performed by G. P. Thomson and his coworkers in Great Britain. These experiments are complementary to the photoelectric effect because, while the explanation of the photoelectric effect relied on the particle nature of light, the
I Introduction explanation of these results relied on the wave nature of particles(electrons). This made clear that the same wave-particle duality associated with photons exists for material particles. Just how particles behave as waves is the subject of this book and, indeed, quantum physics. More about that later in this chapter. Davisson and Germer were studying electron scattering from metallic surfaces when an experimental accident forced them to subject a nickel surface to a high temperature. The effect was to crystallize the nickel and make it, in effect, a diffrac tion grating for electrons. The data were explainable as a diffraction pattern. That is, the electrons were interfering with each other in the same way that light waves were known to interfere to produce familiar diffraction patterns. This experiment was performed after Louis de broglie's hypothesis(see below) that ascribed wave properties to matter. The explanation of the data was consistent with de broglie hypothesis. Thomsons experiments were also consistent with de broglie's hypoth- esis. They were similar in nature to the Davisson-Germer experiment, but Thomson used thin metal foils as the"diffraction gratin In 1937 Clinton Joseph Davisson and George Paget Thomson shared the Nobel Prize in Physics"for their experimental discovery of the diffraction of electrons by crystals "Interestingly, George Thomson was the son of yet another Nobel laureate Joseph John Thomson, who was awarded the Nobel Prize in 1906"in recognition of the great merits of his theoretical and experimental investigations on the conduction of electricity by gases In summary, while the wave-particle duality was hard to understand for ph ns, it was virtually incomprehensible for material particles. The question to be answered was: what is it that is doing the waving? This is the subject of this book 1.1.5 The Compton Effect The Compton effect was studied in 1922 and was additional evidence of the wave particle duality of photons. It was performed using x-rays, high-frequency elec tromagnetic radiation, scattered from electrons that are bound in atoms. For this work Arthur Holly Compton was awarded the Nobel Prize in Physics in 1927 the citation for which read"for his discovery of the effect named after him Because the Compton effect is of considerable importance we will derive the Figure 1.4 shows a schematic diagram of the scattering process. A photon of frequency v is incident on an electron at rest. The electron is not actually at rest, but its kinetic energy is small compared with the energy of the x-rays. The ini tial momentum of the photon is Pp hv/c where c is the speed of light. The photon is assumed scattered at an angle 0 with momentum pp hv/e where v is the frequency of the scattered photon. The momentum of the scattered elec tron is P a vector. We wish to find the wavelength 2= c/v of the scattered
8 1 Introduction explanation of these results relied on the wave nature of particles (electrons). This made clear that the same wave–particle duality associated with photons exists for material particles. Just how particles behave as waves is the subject of this book and, indeed, quantum physics. More about that later in this chapter. Davisson and Germer were studying electron scattering from metallic surfaces when an experimental accident forced them to subject a nickel surface to a high temperature. The effect was to crystallize the nickel and make it, in effect, a diffraction grating for electrons. The data were explainable as a diffraction pattern. That is, the electrons were interfering with each other in the same way that light waves were known to interfere to produce familiar diffraction patterns. This experiment was performed after Louis de Broglie’s hypothesis (see below) that ascribed wave properties to matter. The explanation of the data was consistent with de Broglie’s hypothesis. Thomson’s experiments were also consistent with de Broglie’s hypothesis. They were similar in nature to the Davisson–Germer experiment, but Thomson used thin metal foils as the “diffraction grating.” In 1937 Clinton Joseph Davisson and George Paget Thomson shared the Nobel Prize in Physics “for their experimental discovery of the diffraction of electrons by crystals.” Interestingly, George Thomson was the son of yet another Nobel laureate, Joseph John Thomson, who was awarded the Nobel Prize in 1906 “in recognition of the great merits of his theoretical and experimental investigations on the conduction of electricity by gases”. In summary, while the wave–particle duality was hard to understand for photons, it was virtually incomprehensible for material particles. The question to be answered was: what is it that is doing the waving? This is the subject of this book. 1.1.5 The Compton Effect The Compton effect was studied in 1922 and was additional evidence of the wave particle duality of photons. It was performed using x-rays, high-frequency electromagnetic radiation, scattered from electrons that are bound in atoms. For this work Arthur Holly Compton was awarded the Nobel Prize in Physics in 1927 the citation for which read “for his discovery of the effect named after him.” Because the Compton effect is of considerable importance we will derive the result. Figure 1.4 shows a schematic diagram of the scattering process. A photon of frequency ν is incident on an electron at rest. The electron is not actually at rest, but its kinetic energy is small compared with the energy of the x-rays. The initial momentum of the photon is pp = hν/c where c is the speed of light. The photon is assumed scattered at an angle θ with momentum p p = hν /c where ν is the frequency of the scattered photon. The momentum of the scattered electron is p , a vector. We wish to find the wavelength λ = c/ν of the scattered photon