《Foundations of Quantum Physics》Burkhardt Foundations of quantum physics，Charles E. Burkhardt · Jacob J. Leventhal

Contents 5.6.2 The Transition Rate 470 15.6.3 The Einstein Coefficients-Spontaneous Emission 473 5.6.4 Selection rules 15.6.5 Transition Rates and lifetimes 15.7 References 483 Problems 483 A Answers to Problems 485 Chapter 3 Chapter 5 ter 6 Chapter 8 Chapter 9 Chapter 10 492 Chapter 11 493 Chapter 12 493 Chapter 13 494 Chapter 14 496 B Useful Constants 497 C Energy Unit D Useful formulas e Greek Alphabet F Acronyms G -Functions Integral -functio ons G 2 Half-Integral l-functions H Useful Integrals I Useful Series Taylor Series 2 Binomial Expansion 511 Trick 51

xvi Contents 15.6.2 The Transition Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 470 15.6.3 The Einstein Coefficients—Spontaneous Emission 473 15.6.4 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 15.6.5 Transition Rates and Lifetimes . . . . . . . . . . . . . . . . . 480 15.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 A Answers to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 B Useful Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 C Energy Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 D Useful Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 E Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 F Acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 G -Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 G.1 Integral -Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 G.2 Half-Integral -Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 H Useful Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 I Useful Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 I.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 I.2 Binomial Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 I.3 Gauss’ Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

J Fourier Integrals K Commutator dentities 51 K1 General Identities K2 Quantum Mechanical Identities L Miscellaneous Operator Relations 521 L 1 Baker-Campbell-Hausdorff (BCH) Formula L.2

Contents xvii J Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 K Commutator Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 K.1 General Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 K.2 Quantum Mechanical Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 L Miscellaneous Operator Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 L.1 Baker–Campbell–Hausdorff (BCH) Formula . . . . . . . . . . . . . . . . 521 L.2 Translation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Chapter 1 Introduction As students begin their study of quantum physics they are usually bombarded with descriptions of experiments and theoretical innovations from the early part of the twentieth century that demonstrated and attempted to clarify the inadequacy of the physics at that time. We will describe only a few of the experiments and some of the oncepts that are particularly pertinent to the material to be presented in this book. We take some liberties with chronology to dramatize the detail 1.1 Early Experiments L.I. 1 The Photoelectric effect The photoelectric effect was discovered in 1887 by Gustav Ludwig Hertz while performing experiments directed toward confirming Maxwell,s theory of electro- magnetic waves. He observed that charged particles(electrons)were ejected from metal surfaces when the surface was illuminated by light. The electron flux was strongly dependent upon the wavelength of the light. Although Hertz did not follow up on his discovery, one of his students, Philipp Eduard Anton von Lenard, reported quantitative measurements of the effect in 1902. For this work Lenard received the Nobel Prize in 1905. The citation reads: for his work on cathode rays. " Subse- quently, in 1925, Hertz shared the Nobel Prize for a different body of work, a subject hat will be discussed later in this chapter The origin of the photoelectric effect remained a mystery until, in one of his three remarkable papers published in 1905, Albert Einstein, using Max Planck's treatment of blackbody spectra, explained the effect. Subsequently, in 1916, Robert Andrews Milliken performed detailed experiments that confirmed Einsteins explanation. Ein- stein received the Nobel Prize in 1921 for this work, although many think that his work on relativity also deserves a prize. The citation for Einstein's prize reads: " for his services to Theoretical Physics, and especially for his discovery of the law of he photoelectric effect Milliken was also awarded a Nobel Prize, his in 1923, the citation for which reads: "for his work on the elementary charge of electricity and on the photoelectric effect. C.E. Burkhardt, J.J. Leventhal, Foundations of Quantum Physics DOI: 10.1007/978-0-387-77652-1-L, O Springer Science+Business Media, LLC 2008

Chapter 1 Introduction As students begin their study of quantum physics they are usually bombarded with descriptions of experiments and theoretical innovations from the early part of the twentieth century that demonstrated and attempted to clarify the inadequacy of the physics at that time. We will describe only a few of the experiments and some of the concepts that are particularly pertinent to the material to be presented in this book. We take some liberties with chronology to dramatize the details. 1.1 Early Experiments 1.1.1 The Photoelectric Effect The photoelectric effect was discovered in 1887 by Gustav Ludwig Hertz while performing experiments directed toward confirming Maxwell’s theory of electro￾magnetic waves. He observed that charged particles (electrons) were ejected from metal surfaces when the surface was illuminated by light. The electron flux was strongly dependent upon the wavelength of the light. Although Hertz did not follow up on his discovery, one of his students, Philipp Eduard Anton von Lenard, reported quantitative measurements of the effect in 1902. For this work Lenard received the Nobel Prize in 1905. The citation reads: “for his work on cathode rays.” Subse￾quently, in 1925, Hertz shared the Nobel Prize for a different body of work, a subject that will be discussed later in this chapter. The origin of the photoelectric effect remained a mystery until, in one of his three remarkable papers published in 1905, Albert Einstein, using Max Planck’s treatment of blackbody spectra, explained the effect. Subsequently, in 1916, Robert Andrews Milliken performed detailed experiments that confirmed Einstein’s explanation. Ein￾stein received the Nobel Prize in 1921 for this work, although many think that his work on relativity also deserves a prize. The citation for Einstein’s prize reads: “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” Milliken was also awarded a Nobel Prize, his in 1923, the citation for which reads: “for his work on the elementary charge of electricity and on the photoelectric effect.” C.E. Burkhardt, J.J. Leventhal, Foundations of Quantum Physics, 1 DOI: 10.1007/978-0-387-77652-1 1, C Springer Science+Business Media, LLC 2008

I Introduction In 1901 Max Karl Ernst Ludwig Planck published his revolutionary In equation form, it is E=nhv where E and v are the energy and frequency of an oscillator in the solid; n is a positive integer. The constant h=6.626x 10-34 J-s is Planck's constant. For this innovation Planck was awarded the Nobel Prize in 1918. the citation for which reads in recognition of the services he rendered to the advancement of Physics by his discovery of energy quant Equation 1.1, the Planck relation, is often written in terms of the angular fre- quency a= 2v and h= h/2. The symbol h is read"h-bar"and E=nho Einsteins explanation of the photoelectric effect rested on Plancks assumption that Equation 1. I also applied to light emitted by the oscillators. As a consequence, it was inferred that light(electromagnetic radiation) could be considered to be made up of bundles or"quanta"called photons, each having energy E and frequency v. Thus was born the concept of wave particle duality. That is, light exhibits both particle properties, quanta having energy E, and wave properties as represented by the frequency v. It is common to speak of light in terms of the wavelength a rather han the frequency, in which case Equation 1. I takes the form λ where c is the speed of light Now, what are the details of the photoelectric effect? The observations are best understood in terms of the experiments. A schematic diagram of the apparatus used by Lenard, and later Milliken, is shown in Fig. 1.la Light of a fixed frequency(monochromatic light)illuminates an elemental metal, the photocathode. Electrons are emitted from the photocathode, collected on the v=constant Fig 1.1 (a) Schematic iagram of the apparatus used n the photoelectric effect. The photocathode and anode are labeled PC and A resp light of frequency hv (b) Simulated data

2 1 Introduction In 1901 Max Karl Ernst Ludwig Planck published his revolutionary hypothesis. In equation form, it is E = nhν (1.1) where E and ν are the energy and frequency of an oscillator in the solid; n is a positive integer. The constant h = 6.626 × 10−34 J·s is Planck’s constant. For this innovation Planck was awarded the Nobel Prize in 1918, the citation for which reads “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta”. Equation 1.1, the Planck relation, is often written in terms of the angular fre￾quency ω = 2πν and = h/2π. The symbol is read “h-bar” and E = nω (1.2) Einstein’s explanation of the photoelectric effect rested on Planck’s assumption that Equation 1.1 also applied to light emitted by the oscillators. As a consequence, it was inferred that light (electromagnetic radiation) could be considered to be made up of bundles or “quanta” called photons, each having energy E and frequency ν. Thus was born the concept of wave particle duality. That is, light exhibits both particle properties, quanta having energy E, and wave properties as represented by the frequency ν. It is common to speak of light in terms of the wavelength λ rather than the frequency, in which case Equation 1.1 takes the form E = hc λ (1.3) where c is the speed of light. Now, what are the details of the photoelectric effect? The observations are best understood in terms of the experiments. A schematic diagram of the apparatus used by Lenard, and later Milliken, is shown in Fig. 1.1a. Light of a fixed frequency (monochromatic light) illuminates an elemental metal, the photocathode. Electrons are emitted from the photocathode, collected on the Fig. 1.1 (a) Schematic diagram of the apparatus used in the photoelectric effect. The photocathode and anode are labeled PC and A, respectively. Monochromatic light of frequency hν illuminates the photocathode. (b) Simulated data

1. I Early Experiments anode, and measured using an ammeter as shown in Fig. l.I. The photocathode and the anode are encased in a glass envelope from which the air has been evacuated. The potential difference between the photocathode and the anode is variable as shown and may be either positive or negative. Because the ejected electrons acquire kinetic nergy, the anode voltage VA, if sufficiently negative, can repel them and prevent them from being collected and detected. Several modes of data acquisition are employed, but one of the most striking is a plot of VA versus IA at fixed intensity of the light /. As seen in the hypothetical data in Fig. 1. Ib for three different intensities, the anode current saturates at sufficiently high values of VA, but the value of the stopping voltage VA =-Vs at which the electrons are turned around is independent of the intensity. This shows unequivo- cally that the electron kinetic energy is not determined by the intensity of the light. Moreover, experiments performed with different frequencies show that the value of Vs changes with both the frequency of the light and the material out of which the photocathode is constructed Einstein explained these data in terms of quanta of light called photons. These photons each carry an amount of energy in accord with Equation 1. 1. Thus, the kinetic energy imparted to each electron(having charge of magnitude e)depends upon the energy per photon, not 1, the number of photons per second falling upon the photocathode. Einstein wrote a simple relation between the photon energy hv, the electron kinetic energy K E, and the stopping voltage Vs KE=hv-evs (14) Equation 1. 4 tells us that the kinetic energy of the ejected electron is equal the photon energy hv minus the energy required to liberate the electron from the photo- thode. This amount of energy, called the work function W=evs, differs for each different photocathode material. Equation 1. 4 is usually written in the form KE=hv-w and is known as the Einstein relation It is not our goal here to study the photoelectric effect in detail. We wish to note that Einstein,s explanation clearly showed that light exhibited particle characteris- tics. While the wave properties of light had been known for centuries before the photoelectric effect, its explanation in terms of particles was revolutionary 1.1.2 The franck-Hertz Experiment The Franck-Hertz experiments provided early evidence of the quantization of atomic energy levels. They demonstrated that the amount of energy that could stored in an atom was not arbitrary. Rather, these energies come in discrete incre- ments. Moreover the increments were different for different atoms. For their work first reported in 1914, James Franck and Gustav Ludwig Hertz shared the 192

1.1 Early Experiments 3 anode, and measured using an ammeter as shown in Fig. 1.1. The photocathode and the anode are encased in a glass envelope from which the air has been evacuated. The potential difference between the photocathode and the anode is variable as shown and may be either positive or negative. Because the ejected electrons acquire kinetic energy, the anode voltage VA, if sufficiently negative, can repel them and prevent them from being collected and detected. Several modes of data acquisition are employed, but one of the most striking is a plot of VA versus IA at fixed intensity of the light I. As seen in the hypothetical data in Fig. 1.1b for three different intensities, the anode current saturates at sufficiently high values of VA, but the value of the stopping voltage VA = −VS at which the electrons are turned around is independent of the intensity. This shows unequivo￾cally that the electron kinetic energy is not determined by the intensity of the light. Moreover, experiments performed with different frequencies show that the value of VS changes with both the frequency of the light and the material out of which the photocathode is constructed. Einstein explained these data in terms of quanta of light called photons. These photons each carry an amount of energy in accord with Equation 1.1. Thus, the kinetic energy imparted to each electron (having charge of magnitude e) depends upon the energy per photon, not I, the number of photons per second falling upon the photocathode. Einstein wrote a simple relation between the photon energy hν, the electron kinetic energy K E, and the stopping voltage VS K E = hν − eVS (1.4) Equation 1.4 tells us that the kinetic energy of the ejected electron is equal the photon energy hν minus the energy required to liberate the electron from the photo￾cathode. This amount of energy, called the work function W = eVS, differs for each different photocathode material. Equation 1.4 is usually written in the form K E = hν − W (1.5) and is known as the Einstein relation. It is not our goal here to study the photoelectric effect in detail. We wish to note that Einstein’s explanation clearly showed that light exhibited particle characteris￾tics. While the wave properties of light had been known for centuries before the photoelectric effect, its explanation in terms of particles was revolutionary. 1.1.2 The Franck–Hertz Experiment The Franck–Hertz experiments provided early evidence of the quantization of atomic energy levels. They demonstrated that the amount of energy that could be stored in an atom was not arbitrary. Rather, these energies come in discrete incre￾ments. Moreover, the increments were different for different atoms. For their work, first reported in 1914, James Franck and Gustav Ludwig Hertz shared the 1925