I Introduction 1. 3 Units In this book we will usually use SI units, although eV will frequently be used. In ddition to these units, it is often convenient to devise units of energy that are tai- lored to a particular problem. For example, the eV, while convenient for the Bohr atom, is so much smaller than nuclear quantum levels that the Mev, one million eV is usually used. Another commonly used unit of"energy"is the MHz. Technically, this is a unit of frequency, but, in accord with Equation 1. 1, it may be regarded as a unit of energy if it is understood that it is actually the energy divided by Planck's constant. Table C 1 contains a listing of some of these contrived, but nonetheless. very useful, units and their relationships to the eV Problems involving atomic or molecular calculations can be facilitated using ye nother system of units, atomic units, abbreviated a.u. These units can lessen the calculational burden because many of the common atomic parameters are set equal to unity. After obtaining answers in a u. it is a relatively simple matter to convert back to more familiar units In atomic units, by definition, the electronic charge e, the mass of the electron and h are all set equal to unity. The unit of length is chosen to be the bohr radius ao=[(4 6o)h2/(mee 2). Thus, in atomic uni e=1=h The conversion between a u. and si units can be effected with the aid of Table 1.2 unit of velocity is simply the velocity of the electron in the first Bohr orbit while the unit of time is the period of the electron in the first Bohr orbit divided by 2T. The unit of energy is twice the ground-state Bohr energy or 27.2 eV. The extra factor of 2 is merely a convenience. Interestingly, from the definition of the fine structure constant, Equation 1.35, the speed of light is simply a. That is, in a u. the speed of light is 137 a.u. of length/a u. of time(se Problem 6) Table 1.2 Atomic units (au) Value (sI Mass 910×10-31k Charge 160×10-19c Angular momentum 106×10-34Js Velocity ao/o= l/ac 242×10-17 e2/(4r∈o 4.36×10-18 Electric field (4x∈04)=15.14×10Vm Bohi /(2me)=1 9.274×10-24J/T
24 1 Introduction 1.3 Units In this book we will usually use SI units, although eV will frequently be used. In addition to these units, it is often convenient to devise units of energy that are tailored to a particular problem. For example, the eV, while convenient for the Bohr atom, is so much smaller than nuclear quantum levels that the MeV, one million eV, is usually used. Another commonly used unit of “energy” is the MHz. Technically, this is a unit of frequency, but, in accord with Equation 1.1, it may be regarded as a unit of energy if it is understood that it is actually the energy divided by Planck’s constant. Table C.1 contains a listing of some of these contrived, but nonetheless, very useful, units and their relationships to the eV. Problems involving atomic or molecular calculations can be facilitated using yet another system of units, atomic units, abbreviated a.u. These units can lessen the calculational burden because many of the common atomic parameters are set equal to unity. After obtaining answers in a.u. it is a relatively simple matter to convert back to more familiar units. In atomic units, by definition, the electronic charge e, the mass of the electron me, and are all set equal to unity. The unit of length is chosen to be the Bohr radius a0 = (4π 0) 2 / mee2 . Thus, in atomic units e = 1 = = me = 1 4π 0 = 1 (1.60) The conversion between a.u. and SI units can be effected with the aid of Table 1.2. Notice that the unit of velocity is simply the velocity of the electron in the first Bohr orbit while the unit of time is the period of the electron in the first Bohr orbit divided by 2π. The unit of energy is twice the ground-state Bohr energy, or 27.2 eV. The extra factor of 2 is merely a convenience. Interestingly, from the definition of the fine structure constant, Equation 1.35, the speed of light is simply c = α−1. That is, in a.u. the speed of light is 137 a.u. of length/a.u. of time (see Problem 6). Table 1.2 Atomic units (a.u.) Quantity a.u. Value (SI) Mass me = 1 9.10 × 10−31 kg Charge e = 1 1.60 × 10−19 C Angular momentum = 1 1.06 × 10−34 Js Length a0 = 1 5.29 × 10−11 m Velocity v0 = αc 2.20 × 106 m/s Time a0/v0 = 1/αc 2.42 × 10−17 s Energy e2/ (4π 0a0) = 1 4.36 × 10−18 J Electric field e/ 4π 0a2 0 = 1 5.14 × 1011 V/m Bohr magneton e/ (2me) = 1/2 9.274 × 10−24 J/T
Problems In practice two of these quantities are used more than the others, length and nergy. Both have names in a.u. although they are seldom used. The unit of length n a.u. is the bohr and the unit of energy is the hartree, so named for D. R. Hartree who proposed the unit in 1926. Usually, however, most physicists simply say"one au. of length”or“ one a u. of energy. 1.4 Retrospecti A variety of experimental observations during the late nineteenth and early twen- tieth centuries showed that classical physics was inadequate for describing many phenomena. These observations led to the formulation of quantum phy know it today, the subject of this book. Only a few of these early studies that led to the development of contemporary quantum physics have been discussed in his chapter because the chapter is intended primarily as background for the re- mainder of the book. Emphasis was placed on key points that required resolution in the new quantum physics including the particlelike behavior of light and the wavelike behavior of particles. This wave-particle duality, which Bohr elaborated upon in his principle of complementarity, is dealt with in quantum mechanics by incorporating both the Planck relation E= hv and the de broglie wavelength a = h/p in the mathematical formulation. It is the inclusion of these quanti- ties that accounts for the quantized nature of subatomic energy levels. It remains merely"to deduce an equation of motion that adequately describes quantum me- chanical systems. After that we are done--except for the calculational details. In essence, this is all of quantum physics, at least in a first formulation which we now 1.5 References I. N. Bohr, "On the quantum theory of line-spectra, in"Sources of Quantum Mechanics, edited 7) 1. Light of wavelength A illuminates a metal surface and photoelectrons having leroy one which emits light of wavelength 2/2 and the photoelectrons are observed to have a maximum kinetic energy of 4.28 V. What is the work function of the metal? find a table of work functions and decide which metal it is of mercury atoms that is excited by electrons in the Fran xperiment decays back to the state from which it was excited by emitting light, what will be the wavelength of that light?
Problems 25 In practice two of these quantities are used more than the others, length and energy. Both have names in a.u. although they are seldom used. The unit of length in a.u. is the bohr and the unit of energy is the hartree, so named for D. R. Hartree who proposed the unit in 1926. Usually, however, most physicists simply say “one a.u. of length” or “one a.u. of energy.” 1.4 Retrospective A variety of experimental observations during the late nineteenth and early twentieth centuries showed that classical physics was inadequate for describing many phenomena. These observations led to the formulation of quantum physics as we know it today, the subject of this book. Only a few of these early studies that led to the development of contemporary quantum physics have been discussed in this chapter because the chapter is intended primarily as background for the remainder of the book. Emphasis was placed on key points that required resolution in the new quantum physics including the particlelike behavior of light and the wavelike behavior of particles. This wave–particle duality, which Bohr elaborated upon in his principle of complementarity, is dealt with in quantum mechanics by incorporating both the Planck relation E = hν and the de Broglie wavelength λ = h/p in the mathematical formulation. It is the inclusion of these quantities that accounts for the quantized nature of subatomic energy levels. It remains “merely” to deduce an equation of motion that adequately describes quantum mechanical systems. After that we are done—except for the calculational details. In essence, this is all of quantum physics, at least in a first formulation which we now embark upon. 1.5 References 1. N. Bohr, “On the quantum theory of line-spectra,” in “Sources of Quantum Mechanics,” edited by B. L. van der Waerden (Dover, New York, 1967). Problems 1. Light of wavelength λ illuminates a metal surface and photoelectrons having maximum kinetic energy of 1eV are ejected. The light source is replaced by a one which emits light of wavelength λ/2 and the photoelectrons are observed to have a maximum kinetic energy of 4.28 eV. What is the work function of the metal? Find a table of work functions and decide which metal it is. 2. If the state of mercury atoms that is excited by electrons in the Franck–Hertz experiment decays back to the state from which it was excited by emitting light, what will be the wavelength of that light?
I Introduction 3. Calculate the followin (a)The wavelengths in nm of the first three lines of the Lyman series and the Balmer series (b) The series limit of the Lyman and Balmer series. The series limit is defined 4. It is possible to form a hydrogenlike atom with a proton and a negative u-meson having mass mu N 200me. Find the radius of the first Bohr orbit in terms of ao, the velocity of the u-meson in the first Bohr orbit in terms of the same quantity for hydrogen, and the ionization energy from the ground state in electron-volts Show that ao= h/(e ca) and that the speed of the electron in the nth Bohr orbit is Un =ac/n 6. Using the definition of the fine structure constant a,(Equation 1.35), and its known value, show that, in atomic units, the speed of light is 137 a.u. of length er a.u. of time 7. Compton scattering experiments can be performed using protons rather thai electrons (a) Find the Compton wavelength of the proton in terms of the Compton wave- length of the electron. (b)If the apparatus is such that An/A must be 0.03, what must be the wavelength of the incident photon? In what region of the electromagnetic spectrum are photons of this energy? 8. Show that fitting de broglie waves to the circumference of the bohr orbits leads to the postulate that Bohr never made, that is, angular momentum is quantized in units of h
26 1 Introduction 3. Calculate the following: (a) The wavelengths in nm of the first three lines of the Lyman series and the Balmer series. (b) The series limit of the Lyman and Balmer series. The series limit is defined as the shortest possible wavelength. 4. It is possible to form a hydrogenlike atom with a proton and a negativeμ-meson having mass mμ ≈ 200me. Find the radius of the first Bohr orbit in terms of a0, the velocity of the μ-meson in the first Bohr orbit in terms of the same quantity for hydrogen, and the ionization energy from the ground state in electron-volts. 5. Show that a0 = / (mecα) and that the speed of the electron in the nth Bohr orbit is vn = αc/n. 6. Using the definition of the fine structure constant α, (Equation 1.35), and its known value, show that, in atomic units, the speed of light is 137 a.u. of length per a.u. of time. 7. Compton scattering experiments can be performed using protons rather than electrons. (a) Find the Compton wavelength of the proton in terms of the Compton wavelength of the electron. (b) If the apparatus is such that λ/λ must be ∼ 0.03, what must be the wavelength of the incident photon? In what region of the electromagnetic spectrum are photons of this energy? 8. Show that fitting de Broglie waves to the circumference of the Bohr orbits leads to the postulate that Bohr never made, that is, angular momentum is quantized in units of .
Chapter 2 Elementary Wave Mechanics 2.1 What is doing the Waving? In nonrelativistic quantum physics, particles are treated as points. That is, they have no finite dimensions(zero volume)so they cannot, for example, spin. We are therefore justified in asking"what is doing the waving?"The answer is that it is the probability of finding the particle in a particular region of space. Actually, it is the probability of finding the particle within a particular range of some physically measurable parameters such as linear momentum or angular momentum, but let onfine our attention to coordinates for now. The application of quantum physics to solve problems thus becomes one of solving the appropriate equation of motion for the function that represents this probability. The mechanics of doing this is called quantum mechanics or, archaically, wave mechanics. The intention of this chapter is to introduce this equation of motion and, using it, to better understand the answer to the question what is doing the waving 2.2 A Gedanken Experiment--Electron Diffraction Revisited It is reasonable to ask if we can imagine an experiment that will demonstrate the wave nature of the probability and, simultaneously, the pointlike "structure"of the particles Gedanken is the German word for thought, so a Gedanken experiment is not one that can actually be performed, but one that can be imagined and used to understand a particular phenomenon. Modern technology has, however, made it possible to perform experiments that were envisioned as Gedanken experiments dur ing the development of quantum mechanics. Because of the counterintuitive nature of quantum physics, many Gedanken experiments were imagined, especially in the early development of quantum physics. For the present purpose, we return to the electron diffraction experiment described in Section 1.2.3 and use it to perform ken experiment. Imagine the screen to be constructed of a material that phosphoresces when struck by an electron. Phosphorescent materials continue to emit light after be- ing energized and we assume, for the purpose of this experiment, that our screen C.E. Burkhardt, J.J. Leventhal, Foundations of Quantum Physics DOI: 10.1007/978-0-387-77652-1-2, O Springer Science+Business Media, LLC 2008
Chapter 2 Elementary Wave Mechanics 2.1 What is Doing the Waving? In nonrelativistic quantum physics, particles are treated as points. That is, they have no finite dimensions (zero volume) so they cannot, for example, spin. We are therefore justified in asking “what is doing the waving?” The answer is that it is the probability of finding the particle in a particular region of space. Actually, it is the probability of finding the particle within a particular range of some physically measurable parameters such as linear momentum or angular momentum, but let us confine our attention to coordinates for now. The application of quantum physics to solve problems thus becomes one of solving the appropriate equation of motion for the function that represents this probability. The mechanics of doing this is called quantum mechanics or, archaically, wave mechanics. The intention of this chapter is to introduce this equation of motion and, using it, to better understand the answer to the question what is doing the waving. 2.2 A Gedanken Experiment—Electron Diffraction Revisited It is reasonable to ask if we can imagine an experiment that will demonstrate the wave nature of the probability and, simultaneously, the pointlike “structure” of the particles. Gedanken is the German word for thought, so a Gedanken experiment is not one that can actually be performed, but one that can be imagined and used to understand a particular phenomenon. Modern technology has, however, made it possible to perform experiments that were envisioned as Gedanken experiments during the development of quantum mechanics. Because of the counterintuitive nature of quantum physics, many Gedanken experiments were imagined, especially in the early development of quantum physics. For the present purpose, we return to the electron diffraction experiment described in Section 1.2.3 and use it to perform a Gedanken experiment. Imagine the screen to be constructed of a material that phosphoresces when struck by an electron. Phosphorescent materials continue to emit light after being energized and we assume, for the purpose of this experiment, that our screen C.E. Burkhardt, J.J. Leventhal, Foundations of Quantum Physics, 27 DOI: 10.1007/978-0-387-77652-1 2, C Springer Science+Business Media, LLC 2008
2 Elementary Wave Mechanics phosphoresces indefinitely. Now, let us lower the intensity of the electron beam so we can easily see each electron as it strikes the screen, lights it up, and leaves a signature of its presence in the form of a persistent pinpoint of light. The first elec tron strikes somewhere, we cannot predict where with certainty. From the known diffraction pattern we know where it is most likely to strike. Perhaps it is a contrary electron and strikes in a region in which the diffraction pattern has low intensity, perhaps not. Bear in mind that it is a single event. Wherever it strikes, it leaves its signature. A second electron arrives. It too leaves its signature again we do not know where it will land, only where it is most likely to land. After perhaps 100 electrons have struck the screen we have a pattern, but it may not look like the known diffraction pattern because 100 is not, statistically speaking, a very large number. When, however, a large number of electrons have struck the screen it is lit up with the known diffraction pattern. This pattern is composed of many points of light representing the point electrons, but the pattern represents the diffractic pattern characteristic of wave motion The important point to remember is that the particles are not magically turning into slithering sausages as they make their way through the narrow slit. They main- are required to demonstrate the wavelike properties of matter umber of them that tain their identity as point particles. It is, perhaps, Avogadro 2.3 The Wave Function Paramount to obtaining the probability distribution is the wave function, y (r, t). We use the capital Greek letter to designate the wave function when the time is included and, for now, we work with only one-dimension, x. We point out that the wave function need not be written in terms of any coordinates. It could be in terms of another variable(called an observable in quantum mechanics), but we will consider only coordinates and time for now. Now, by postulate, y(x, t) contains all the information that the uncertainty principle permits us to know about the particle. ng an asterisk to signify the complex conjugate, the probability that the particle will be found in the interval dx at time t is given by Y(x, t)y(x, ndx=ly(x, tl-dx provided y(r, t)has been normalized so that 业*(x,)业(x,t)dx=1 (2.2) Normalization assures that the total probability cannot exceed unity. The complex onjugate is required because y (x, t) may very well be a complex function. On he other hand, the probability must be real so the absolute value in Equation 2.1 assures us that the probability will be real. We see then that, while y(x, t)does not give physical information, its absolute square does. The quantity Iy(x, t)I-is
28 2 Elementary Wave Mechanics phosphoresces indefinitely. Now, let us lower the intensity of the electron beam so we can easily see each electron as it strikes the screen, lights it up, and leaves a signature of its presence in the form of a persistent pinpoint of light. The first electron strikes somewhere, we cannot predict where with certainty. From the known diffraction pattern we know where it is most likely to strike. Perhaps it is a contrary electron and strikes in a region in which the diffraction pattern has low intensity, perhaps not. Bear in mind that it is a single event. Wherever it strikes, it leaves its signature. A second electron arrives. It too leaves its signature. Again, we do not know where it will land, only where it is most likely to land. After perhaps 100 electrons have struck the screen we have a pattern, but it may not look like the known diffraction pattern because 100 is not, statistically speaking, a very large number. When, however, a large number of electrons have struck the screen it is lit up with the known diffraction pattern. This pattern is composed of many points of light representing the point electrons, but the pattern represents the diffraction pattern characteristic of wave motion. The important point to remember is that the particles are not magically turning into slithering sausages as they make their way through the narrow slit. They maintain their identity as point particles. It is, perhaps, Avogadro’s number of them that are required to demonstrate the wavelike properties of matter. 2.3 The Wave Function Paramount to obtaining the probability distribution is the wave function, (x,t). We use the capital Greek letter to designate the wave function when the time is included and, for now, we work with only one-dimension, x. We point out that the wave function need not be written in terms of any coordinates. It could be in terms of another variable (called an observable in quantum mechanics), but we will consider only coordinates and time for now. Now, by postulate, (x,t) contains all the information that the uncertainty principle permits us to know about the particle. Using an asterisk to signify the complex conjugate, the probability that the particle will be found in the interval dx at time t is given by ∗ (x,t) (x,t) dx = | (x,t)| 2 dx (2.1) provided (x,t) has been normalized so that ∞ −∞ ∗ (x,t) (x,t) dx = 1 (2.2) Normalization assures that the total probability cannot exceed unity. The complex conjugate is required because (x,t) may very well be a complex function. On the other hand, the probability must be real so the absolute value in Equation 2.1 assures us that the probability will be real. We see then that, while (x,t) does not give physical information, its absolute square does. The quantity | (x,t)| 2 is