Quantum Mechanics Lecture Notes

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Quantum Mechanics Lecture Notes J. W. Van Orden Department of Physics Old Dominion University August 21, 2007Contents 1 Introduction: The Old Quantum Theory 1 1.1 Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Thermodynamics and Statistical Physics . . . . . . . . . . . . . . . . 4 1.2.1 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 The Photoelectric E ect . . . . . . . . . . . . . . . . . . . . . 6 1.3 Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Cathode Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 The Bohr-Sommerfeld Model . . . . . . . . . . . . . . . . . . 11 1.4 Wave Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Electron Di raction . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Mathematical Background to Quantum Mechanics 19 2.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 The Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.3 Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.4 The Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . 25 2.2 The Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 The Schr odinger Equation 39 3.1 Wave Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 The Schr odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 The Interpretation of the Wave Function . . . . . . . . . . . . . . . . 47 3.5 Coordinate Space and Momentum Space Representations . . . . . . . 49 3.6 Di erential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7 The Heisenberg Uncertainty Relation . . . . . . . . . . . . . . . . . . 53 3.8 Review of Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . 56 3.9 Generalization of the Schr odinger Equation . . . . . . . . . . . . . . . 57 4 The Time-Independent Wave Function 61 i5 Solutions to the One-Dimensional Schr odinger Equation 67 5.1 The One-dimensional In nite Square Well Potential . . . . . . . . . . 71 5.2 Bound States in a Finite Square Well Potential . . . . . . . . . . . . 80 5.2.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 The One-Dimensional Harmonic Oscillator . . . . . . . . . . . . . . . 89 6 Scattering in One Dimension 97 6.1 The Free-Particle Schr odinger Equation . . . . . . . . . . . . . . . . . 97 6.2 The Free Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 The Step Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7 The Solution of the Schr odinger Equation in Three Dimensions 111 7.1 The Schr odinger Equation with a Central Potential . . . . . . . . . . 113 7.1.1 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 116 7.2 The Radial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.3 Solution of the Radial Equation for a Free Particle . . . . . . . . . . 118 7.4 The Finite Spherical Well . . . . . . . . . . . . . . . . . . . . . . . . 120 7.4.1 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.4.2 The In nite Spherical Well . . . . . . . . . . . . . . . . . . . . 123 7.5 The Coulomb Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.5.1 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.5.2 Radial Wave Functions . . . . . . . . . . . . . . . . . . . . . . 128 8 Formal Foundations for Quantum Mechanics 131 8.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.3 Heisenberg Representation . . . . . . . . . . . . . . . . . . . . . . . . 138 8.4 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9 Symmetries, Constants of Motion and Angular Momentum in Quan- tum Mechanics 143 9.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.2 Time Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 9.3.1 Rotations of a Vector . . . . . . . . . . . . . . . . . . . . . . . 146 9.3.2 Rotations of Wave Functions . . . . . . . . . . . . . . . . . . . 149 9.4 The Angular Momentum Operators in Spherical Coordinates . . . . . 152 9.5 Matrix Representations of the Angular Momentum Operators . . . . 156 9.6 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.6.1 Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9.6.2 The Intrinsic Magnetic Moment of Spin-1/2 Particles . . . . . 164 9.6.3 Paramagnetic Resonance . . . . . . . . . . . . . . . . . . . . . 166 ii10 Addition of Angular Momenta 171 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.2 Addition of Two Angular Momenta . . . . . . . . . . . . . . . . . . . 172 10.2.1 General Properties of Clebsch-Gordan Coecients . . . . . . . 178 10.2.2 Example: Addition of j = 1 and j = 1 . . . . . . . . . . . . 180 1 2 11 Rotation Matices and Spherical Tensor Operators 183 11.1 Matrix Representation of Rotations . . . . . . . . . . . . . . . . . . . 183 11.2 Irreducible Spherical Tensor Operators . . . . . . . . . . . . . . . . . 188 11.2.1 The Tensor Force . . . . . . . . . . . . . . . . . . . . . . . . . 191 11.3 The Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . 192 12 Time-Independent Perturbation Theory 195 12.1 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 12.2 Perturbation Theory for Non-Degenerate States . . . . . . . . . . . . 197 12.2.1 One-Dimensional Harmonic Oscillator in an Electric Field . . 202 12.3 Degenerate State Perturbation Theory . . . . . . . . . . . . . . . . . 206 12.4 Leading Order Corrections to the Hydrogen Spectrum . . . . . . . . . 213 12.4.1 The Stark E ect . . . . . . . . . . . . . . . . . . . . . . . . . 218 13 Feynman Path Integrals 223 13.1 The Propagator for a Free Particle . . . . . . . . . . . . . . . . . . . 228 14 Time-Dependent Perturbation Theory 231 14.1 The Interaction Representation . . . . . . . . . . . . . . . . . . . . . 233 14.2 Transition Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . 236 14.3 Time-Independent Interactions . . . . . . . . . . . . . . . . . . . . . . 237 14.4 Gaussian Time Dependence . . . . . . . . . . . . . . . . . . . . . . . 239 14.5 Harmonic Time Dependence . . . . . . . . . . . . . . . . . . . . . . . 240 14.6 Electromagnetic Transitions . . . . . . . . . . . . . . . . . . . . . . . 241 14.6.1 Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . 241 14.6.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 15 Many-body Systems and Spin Statistics 249 15.1 The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . 250 15.2 The Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . 252 15.2.1 The Permutation Operator . . . . . . . . . . . . . . . . . . . . 257 15.2.2 n Particles in a Potential Well . . . . . . . . . . . . . . . . . . 259 15.2.3 When is symmetrization necessary? . . . . . . . . . . . . . . . 260 15.3 The Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 15.3.1 The Bulk Modulus of a Conductor . . . . . . . . . . . . . . . 264 15.4 The Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 15.4.1 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 iii15.4.2 A Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . 266 15.4.3 The Nucleon-Nucleon Potential . . . . . . . . . . . . . . . . . 271 16 Scattering Theory 273 16.1 The Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . 274 16.2 The Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 16.3 Partial Waves and Phase Shifts . . . . . . . . . . . . . . . . . . . . . 278 16.4 The Low-Energy Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 282 16.5 Scattering from a Spherical Well . . . . . . . . . . . . . . . . . . . . . 283 ivChapter 1 Introduction: The Old Quantum Theory Quantum Mechanics is the physics of matter at scales much smaller than we are able to observe of feel. As a result, we have no direct experience of this domain of physics and therefore no intuition of how such microscopic systems behave. The behavior of quantum systems is very di erent for the macroscopic systems of Classical Mechanics. For this reason, we will begin by considering the many historical motivations for quantum mechanics before we proceed to develop the mathematical formalism in which quantum mechanics is expressed. The starting point is to review some aspects of classical physics and then to show how evidence accumulated for atomic systems that could not be explained in the context of classical physics. 1.1 Classical Physics What we now call \Classical Physics" is the result of the scienti c revolution of the sixteenth and seventeenth centuries that culminated in Newtonian mechanics. The core of this physics is Newton's laws describing the motion of particles of matter. The particles are subject to forces and Newton's Second LawF =ma can then be used to describe the motion of the particle in terms of a second-order di erential equation. By specifying the position and velocity of the particle at some initial time, the motion of the particle is determined at all subsequent time. There is nothing in principle in classical physics that prevents the initial conditions from being determined to arbi- trary accuracy. This property that allows all subsequent motion to be predicted from Newton's Laws, the force laws and the initial conditions is called \classical causality" and classical physics is said to be deterministic since the motion is determined by the initial conditions. The major direction of physics after Newton was to incorporate as much physical phenomena as possible into the framework of Newtonian physics. Thus Newton's laws were applied to the motions of extended objects, to the motion of uids and elastic 1bodies and to link mechanics to thermodynamics by means of the global conservation of energy. One application, which will be of particular interest in this course, was the descrip- tion of wave motion in uids and elastic materials. The waves are the manifestation of the collective motion of a macroscopically continuous medium. Since the waves are seen as deviations of some quantity such as the height of the surface of water from some average value, the wave is characterized by the amplitude of this deviation and the sign of the deviation at any point and time may be either positive or negative. Since the waves are solutions to linear di erential equations, waves can be added by simply adding the deviation of the wave from equilibrium at every point at any given time. The fact that this deviation may be either positive or negative leads to the wave motion being either cancelled or enhanced at di erent points which produces the typical wave phenomena of interference and di raction. The average energy den- sity carried by a mechanical wave is proportional to the square of the amplitude of the wave, and independent of the frequency. At the beginning of the nineteenth century, mechanics, thermodynamics, electro- magnetic phenomena and optics were not yet united in any meaningful way. Perhaps the most glaring problem was the question of the nature of light. The long and heated argument as to whether light was corpuscular of a wave was nally settled in the nineteenth century by demonstrating in such experiments as Young's two slit experiment that light could interfere and di ract and was therefore a wave. During the course of this century, the understanding electromagnetic phenom- ena developed rapidly culminating in Maxwell's equations for electromagnetic elds. The great triumph of Maxwell's equations was the prediction of wave solutions to Maxwell's equations that led to the uni cation of electrodynamics and optics. The Maxwell's equations were also veri ed by the discovery of radio waves by Hertz. There were still obstacles to the uni cation of electromagnetism with mechanics. The na- ture of electromagnetic currents was not understood until the very last part of the century and there appeared to be no supporting medium for electromagnetic waves as was the case with all mechanical waves. An attempt to deal with the latter problem was to propose the existence of a all-pervading medium called the ether that was the medium that supported electromagnetic waves. It was the unsatisfactory nature of this hypothesis that led Einstein to develop the special theory of relativity in 1905. The situation with thermodynamics was much more satisfactory. The discovery that heat was a form of energy that could be created from mechanical energy or work and that could be in turn used to produce mechanical energy led to the concept of global conservation of energy. That is, energy can be transformed from one type to another but cannot be created or destroyed. During the nineteenth century the laws and logical structure of thermodynamics were codi ed and applied to a variety of phenomena. In addition, through the kinetic theory of gasses and the development of statistical mechanics by Boltzman and Gibbs it was shown that thermodynamics could be described by the average motions of complicated systems of very large num- 2bers of particles composing either uids or solids. Since the exact motion of such large collections of particles could not be determined, statistical methods are used to describe average properties of macroscopic systems. As a result thermodynamics and statistical mechanics are not deterministic. It should be emphasized that this is because the exact knowledge of the microscopic state of the system is impractical but not impossible in principle. It is also important for an understanding of the motivation behind the develop- ment of quantum mechanics to note that the nineteenth century is also the time when chemistry became a quantitative science. It was noted that the speci c grav- ities of various elements were approximately integer multiples of that of hydrogen. Experiments by Faraday on electrolysis indicated that the change in electric charges of various ions in this process indicated that charge appeared as multiples of some xed elementary charge. It was discovered that the light spectrum given of by vari- ous materials when heated by a ame or an electric arc showed discrete lines rather than a continuous distribution of wavelengths. It was shown that these lines were characteristic of each element and that, therefore, the spectra of materials could be used to identify the presence of known elements and to nd new ones. This was used Kirchho in 1859 to show that the absorbtion spectra of the sun indicated the existence of sodium in the stellar atmosphere. Finally the periodic table of elements (Mendeleev, 1869) showed that the chemical properties of various elements display a regular pattern as a function of the atomic number. As the century progressed, more elements were discovered (some of which did not actually exist) and an ever increasing collection of of improved spectroscopic information about these elements was amassed. All of this led to the belief on the part of chemists and physicists that the atomic description of matter was at least a useful tool if not a reality. The reality of atoms and molecules as chemically fundamental constituents of matter was not demonstrated until Einstein's paper of 1905 on Brownian motion, where the erratic motion of small particles suspended in uids was described as the result of the col- lective result of large numbers of collisions between the molecules of the uid and the small particles. One result of this period of great progress in the uni cation of classical physics and in the development of chemistry was that a great amount of new measurements were accumulated that were not fully incorporated into the structure of classical physics and which ultimately proved this structure to be inadequate. We will now examine some of these problems and see how they led to the development of what is now called the old quantum theory. 31.2 Thermodynamics and Statistical Physics 1.2.1 Black Body Radiation One of the phenomena that had long been known to man was that when an object is heated to a suciently high temperature that it begins to glow and that the color of the glowing object is related to the temperature of the object. This was used for millennia by metal smiths to determine when metal was suciently hot to be easily worked with hammers. In 1859 Kirchho showed on the basis of thermodynamics that the energy per unit area per unit time (the energy current density) of light given o by an a completely absorbing body is a function only of the temperature of the object and the frequency of the light emitted by the object and not to any particular physical properties of the emitter. A completely absorbing object is referred to as a black body. In practice a black body was constructed as a closed box which was sealed to light and heated to a uniform temperature. A small hole was placed in the box to allow the measurement of the intensity of the light in the cavity as a function of frequency. The black body distribution problem was of interest not only for its intrinsic scienti c value, but also because black bodies could be used as means of calibrating various kinds of lamps used for scienti c and commercial purposes . The veri cation of Kirchho 's prediction and the actual measurement of the spectral density for black body radiation posed a serious technical problem at the time and it was not until the last part of the nineteenth century that suciently reliable experiments were available. In the mean time several theoretical contributions to this problem were obtained. In 1879, Stefan proposed that the total electromagnetic energy in the cavity is propor- tional to the fourth power of the absolute temperature. In 1884 Boltzmann provided a proof of this using thermodynamics and electrodynamics. He showed that c J(;T ) = (;T ) (1.1) 8 where J(;T ) is the energy current density of emitted radiation and (;T ) is the spectral density or the energy per unit volume per unit time of radiation in the cavity and c is the speed of light. From this Boltzmann derived the Stefan-Boltzmann law Z 4 E(T ) =V d(;T ) =aVT (1.2) where E(T ) is the energy of the radiation in the cavity for temperature T and V is the volume enclosed by the cavity. In 1893, Wien derived the Wien displacement law    3 (;T ) = f (1.3) T 43 which states that the spectral distribution is proportional the times some function of the ratio of  to T . However on the basis of just thermodynamics and electro- dynamics it is not possible to determine this function. Wien conjectured that the spectral distribution was of the form  3 T (;T ) =  e (1.4) where and were unknown constants to be determined by data. In 1893 Paschen presented data from the near infrared that was in excellent agreement with Wien's formula. In 1900, Rayleigh derived a new formula using statistical mechanics that is given by 2 8 (;T ) = kT: (1.5) 3 c where k is the Boltzmann constant. In fact Rayleigh did not actually determine the constants in this expression. These were correctly determined by Jeans in 1905 and the equation is now called the Rayleigh-Jeans Law. Note that this law will not satisfy 2 the Stefan-Boltzmann condition since increases and  and, therefore, has an in nite integral over . This was referred to as the ultraviolet catastrophe. Also in 1900, two groups in Berlin consisting of Lummer and Pringsheim, and Rubens and Kurlbaum obtained data at lower frequencies further into the infrared. This data, however, was not in good agreement with Wien's Law. This data was immediately shown to Planck and he was told that the data was linear in T at small temperatures. It is not clear whether he knew of Rayleigh's work at that time, however. Planck knew that Wien's Law worked well at high frequencies and that the spectral density had to be linear inT as low temperatures. Using this he quickly guessed at a formula that would interpolate between the two regions. This is Planck's Law 3 8h 1 (;T ) = (1.6) h 3 c kT e 1 which contained a new constanth (Planck's constant) that could be determined from the data. Indeed this law provides an excellent representation of the black body spectral distribution. Figure 1.1 shows the three laws for the distribution functions at T = 1000K. Planck was now in a situation which is not uncommon for theoretical physicists, he had a formula that t the data, but did not have a proof. He then proceeded to try to derive his Law. He modeled the black body as a set of charges that were attached to harmonic oscillators. The acceleration of these particles then produced radiation. The assumption was that these oscillators were in thermal equilibrium with the radiation in the cavity. In obtaining his proof, he did two things which were not consistent with classical physics, he used a counting law for determining the probability of various con gurations that was not consistent with classical statistical 5Figure 1.1: Spectral distributions for the Wien, Rayleigh-Jeans and Planck laws at T = 1000K. mechanics and he was required to assume that the oscillators in the walls of the cavity could only radiate at a speci c energy E =h: (1.7) He referred to these bundles of energy as \quanta." This is a very radical departure from classical mechanics because according to electrodynamics the energy of the radi- ation should be determined by the magnitude of the oscillations and be independent of the frequency. Planck assumed that there must be some unknown physics associ- ated with the production of radiation of the oscillators, but that the description of the radiation in the cavity should still be consistent with Maxwell's equations which had recently been veri ed by an number of experiments. This is historically the beginning of quantum mechanics. 1.2.2 The Photoelectric E ect One of Einstein's papers of 1905 considered Planck's derivation of his radiation law. Einstein was well aware of the errors and conjectures that were necessary to Planck's derivation. As a result Einstein believed that Plank's Law was consistent with exper- iment but not with existing theory while the Rayleigh-Jeans Law was consistent with existing theory but not with experiment. He then proceeded to use Boltzmann statis- tics to examine the radiation in the regime where Wien's Law is consistent with data 61 and derived this result. This proof was also de cient , but in the process he made the hypothesis that the light in the cavity was quantized, in contrast to Planck's as- sumption that it was the material oscillators that were quantized. The light-quantum hypothesis was of course not consistent with the classical physics of Maxwell's equa- tions where the energy of the wave is related to the square of the electric eld and not to the frequency. This led to the work that was to lead to Einstein's 1922 Nobel prize. The rst application of the light-quantum hypothesis was to the photoelectric e ect. This e ect was rst seen by Hertz in 1887 in connection with his experiments with electromagnetic radiation. He noticed that the light from one electric arc could e ect the magnitude of the current in a second arc. In 1888 Hallwachs showed that ultraviolet light falling on a conductor could give it a positive charge. Since this was before the discovery of the electron, the nature of this e ect was a mystery. In 1899, J. J. Thomson showed that the charges emitted in the photolectric e ect were electrons which he had identi ed in cathode rays in 1897. In 1902 Lenard examined the photoelectric e ect using a carbon arc light source which could be varied in intensity by a factor of 1000. He made the surprising discovery that the maximum energy of electrons given o in the photoelectric e ect was independent of the intensity of the light. This is in contradiction to classical electrodynamics where the energy provided by the light source depends only on the intensity. In addition, he determined that the energy of the electrons increased with the frequency of the light, again in contradiction to classical theory. In 1905 Einstein proposed that the photoelectric e ect could be understood in terms of the light-quantum hypothesis. If the light quanta have an energy of h then the maximum energy of the emitted electrons should follow the formula E =hP (1.8) max where P is the amount of energy required to remove an electron from the conductor and is called the work function. Experimental con rmation of this formula was pro- vided in 1916 by Millikan who showed that this formula was consistent with his data to within 0.5%. In spite of this stunning con rmation, the light-quantum hypothesis was viewed with considerable skepticism by the majority of physicists at the time. 1.3 Atomic Physics The other path that led to the establishment of quantum mechanics was through atomic physics. As we have already seen a considerable amount of information had been collected during the nineteenth century associated with the regularities seen in 1 Einstein would return to this problem several times during the career, but a completely satis- factory derivation of Planck's Law was not achieved until Dirac did so in 1927. 7the chemical properties of various elements and with the very large amount of data that had been obtained on the spectra of the elements. Any acceptable theory of the atom would necessarily need to account for these phenomena. It had been noted by Maxwell in 1875 that atoms must have many degrees of freedom in order to produce the complicated spectra that were being observed. This implied that the atoms must have some complicated structure since a rigid body with only six degrees of freedom would not be sucient to describe the data. The problem here is that until the last few years of the nineteenth century any clues as to the physical structure of the atom was missing. 1.3.1 Cathode Rays One of the rst advances in this area involved the study of cathode rays. Cathode rays are seen as luminous discharges when current ows through partially evacuated tubes. This phenomenon had been known since the eighteenth century and demonstrations of it had been a popular entertainment. However, since pressures in these tubes could only be lowered by a small amount compared to atmospheric pressure, there was a sucient amount of gas in the tube that a great many secondary e ects were present, so it was dicult to study the cathode rays themselves. About a third of the way through the nineteenth century it became possible to produce tubes with much higher vacuums and to start to consider the primary e ect. The nature of these rays was a topic of some dispute. Some physicists (mostly English) believed that the rays were due to the motions of charged particles while others (mostly German) believed that the rays were actually due to ows or disturbances of the ether. Hertz showed in 1891 that cathode rays could pass through thin metal foils. His student Lenard then produced tubes with thin metal windows which would allow the cathode rays to be extracted from the tube. Using these he showed in 1894 that the cathode rays could not be molecular and that they could be bent in an external electric eld. Also in 1894 Thomson showed that the cathode rays moved with a velocity substantially smaller than the speed of light. In 1895 Perrin placed a small metal cup in a cathode ray tube to collect the rays and showed that the cathode rays carried a negative charge as had been indicated by the direction of deviation of the rays external elds. In 1897 Wiechert, Kaufmann and Thomson each performed experiments with the de ection of cathode rays in magnetic (or in the case of Thomson electric and magnetic) elds. By measuring the de ection of the cathode rays it was possible to determine the ratio of the charge to the mass, e=m. In all cases it was shown that the this ratio was on the order of 2000 times that of singly ionized hydrogen. This could of course be either due to a large charge or a small mass. Both Wiechert and Thomson speculated that the cause was the small mass of the particles constituting the cathode rays. In 1899 Thomson was able to measure the charge of the constituents of the cathode rays separately using the newly invented Wilson cloud chamber. He 8showed that this was roughly the same as the charge of ionized hydrogen determined from electrolysis. This then proved that the mass of the cathode ray particle was indeed much smaller than the hydrogen mass. Thus, the electron was born as the rst subatomic particle. Also in 1899, as was previously noted, Thomson determined that the particles emitted by the photoelectric e ect were also electrons and thus that ionization was the result of removing electrons from atoms. That is the atom was no longer immutable and could be broken down into constituent parts. This led to Thomson's model of the atom. This model assumed that the atom was composed of electrons moving in the electric eld of some positive background charge which was assumed to uniformly distributed over the volume of the atom. Initially Thomson proposed that there were as many as a thousand electrons in the atom, but later came to believe that the number of electrons was on the order of the atomic number of the atom. This model is often referred to as the \plum pudding" model of the atom. 1.3.2 Radioactive Decay In 1895 Roentgen discovered X-rays while experimenting with cathode ray tubes. While this has no direct impact on the development of quantum mechanics, it stimu- lated a great deal of experimental activity. One of those who was stimulated to look into the problem was Becquerel. Since the source of the X-rays appeared to come from the luminous spot on the wall of the tube struck by the cathode ray, Becquerel hypothesized that X-rays were associated with orescence. To test this hypothesis, in 1896 he began studying whether a phosphorescent uranium salt could expose pho- tographic plates wrapped in thick black paper. He would expose the salt to sun light to cause them to uoresce and then would place it on top of the plate. He found that it did indeed expose the photographic plate. At one point during the experiment the weather turned cloudy and he was unable to subject the salt to sunlight so he placed it along with the plate in a closed cupboard. After several days, he developed the plate and discovered that it had been exposed. Therefore the exposure of the plate was not the result of the orescence after all. He also discovered that any uranium salt, even those that were not phosphorescent, also exposed the plates. The radia- tion was, therefore, a property of uranium. He found that the uranium continued to radiate energy continuously with no apparent diminution over a considerable period of time. The hunt was now on for more sources of Becquerel rays. In 1897 both the Curies and Rutherford became engaged in the problem. The Curies soon began to discover a variety of known elements such as thorium which gave o the rays and also new radioactive elements such as radium and radon. They also proceeded to give themselves radiation poisoning. To this day their laboratory books are suciently radioactive that they are kept in a lead lined vault. Rutherford showed in 1898 that uranium gave o two di erent types of rays which 9he called - and -rays. The -rays were much more penetrating than the -rays. The -rays were soon identi ed as electrons. It was suspected that the -rays were related to helium since this element seemed to appear in the gases given o by radioactive materials, but it was not until 1908 that Rutherford made a completely convincing case that the -rays were doubly-ionized helium atoms or helium nuclei. For the purposes of our introduction to quantum mechanics this has two important consequences, one which adds another puzzle to the list of problems with classical physics and another which led to a greater understanding of the structure of the atom. The rst of these resulted from the publication by Rutherford and Soddy of the transformation theory. In this they proposed that radioactive decay causes one type of atom to be transformed into another kind. This was consistent with the pattern of radioactive elements that was being established. This is clearly a problem for those who thought that atoms were immutable, but for our purposes the real problem lies in the other part of this theory. It was proposed that the number of atoms that decayed in a given time period was proportional to the number of atoms present. Mathematically this is expressed as dN =N (1.9) dt where N(t) is the number of atoms of a given type at time t. This leads to the exponential decay law t N(t) =N(0)e : (1.10) The question that arises from this is: Why does one uranium atom decay now while another seemingly identical atom decays in 10,000 years? Clearly, from classical physics it should be expected that once the atom is created it should be possible to determine exactly when it will decay. You could argue that the atoms were created 10,000 years apart and were indeed decaying in the same way, but it can be shown that radioactive elements created by the decay of another radioactive element within a short period of time will also satisfy the same decay law. The apparent statistical nature of radioactive decays is a direct challenge to classical physics. The second aspect of radioactive decays that is important to the story of quantum mechanics is also associated with Rutherford. Because energetic -particles were given o by radioactive decays and these -particles could be columnated into a beam, the -particles could be used as a probe of the structure of the atom. In 1908 Rutherford and Geiger demonstrated that they could indeed be scattered from atoms. These experiments were very tedious and required that observers sit in the dark looking for light ashes given o by the scattered -particles when the hit a orescent screen. Each of these events had to be carefully counted and recorded. In 1909 Rutherford suggested to Geiger and a young undergraduate named Marsden, who was assisting him with the counting, that they should look for -particles that were scattered through more than 90 degrees. They found that 1 in 8000 of the -particles were indeed de ected by more than 90 degrees. This was extremely surprising since 10if the positive charge in the atom was distributed over the volume of the atom, many fewer -particles should be scattered at such large angles. This immediately led Rutherford to see that the positive charge in the atom should be concentrated in a very small part of the atom. This, in turn, resulted in the Rutherford model of the atom where the electrons orbit around a very small positive nucleus. This is clearly an improvement over the Thomson model since it explains the new scattering data, but as a classical model it does little to satisfy the criteria that the model deal with all of the previous data collected for atoms. There is nothing in either of these models that explains the regularities discovered in the atoms nor does it explain why the spectra should show discrete lines rather than a continuous spectrum which should be expected from a classical orbital model. In addition, neither of these two models is stable. Since any electron moving in a con ned space must accelerate, classical electrodynamics predicts that the electrons in these atomic models should be continually emitting light until they spiral into the center of the atom and come to a stop. 1.3.3 The Bohr-Sommerfeld Model Before proceeding to the Bohr model it is necessary to make a small digression. In 1885 Balmer considered four spectral lines in the spectrum of hydrogen measured by  Angstrum in 1868. These were referred to as H ,H ,H andH . He noted that the  ratios of the frequencies of the these states could be written as simple fractions and that these could be summarized by the expression   1 1  =R : (1.11) 2 4 n After reporting this work he was informed that there were additional data available and these t his formula with very good accuracy. Apparently, Bohr did not know of the Balmer formula until he was informed of it by a Danish colleague in 1913. Once he knew of the Balmer formula he saw a way to obtain an expression that would reproduce the formula for hydrogen-like atoms. To do this he assumed that there had to be stable solutions which he called stationary states the describe the ground and excited states of atoms. He then assumed that the spectra were due to light emitted when an electron moves from a higher energy state to a lower energy state. That is, h =E E (1.12) m n where E E . m n The model that he constructed assumed that the atom was described by an elec- tron in a circular orbit around a nucleus with positive charge Ze. Classically, these 11circular orbits will be stable if the Coulomb force on the electron produces the ap- 2 propriate centripetal acceleration. This is given by 2 2 Ze v =m (1.13) e 2 r r where m is the mass of the electron and v is its orbital speed. This implies that e 2 Ze 2 =m v : (1.14) e r This is, of course, true for any circular orbit and classically there are a continuous set of such orbits corresponding to any choice the radius r. To select the set of allowable stationary states, Bohr imposed the quantum condition that the kinetic energy for the stationary states is xed such that 1 n 2 m v = hf (1.15) e 2 2 where h is Planck's constant and f is the orbital frequency of the electron. The orbital frequency is in turn given by v f = : (1.16) 2r Substituting (1.16) into (1.15) leads to the expression h 1 nh  v =n = (1.17) 2m r m r e e h 34 where we have de ned the constant h  = = 1:05457  10 J s. We can now 2 substitute (1.17) into the stability condition (1.14) which yields 2 2 2 Ze n h  = : (1.18) 2 r m r e This can now be solved for the radius of the stationary state corresponding to the integer n to give 2 2 n h  r = : (1.19) n 2 m Ze e We can now now calculate the energy. First using (1.14) we can write 2 2 2 2 1 Ze 1Ze Ze 1Ze 2 E = m v = = : (1.20) e 2 r 2 r r 2 r 2 Here I am choosing to express the Coulomb force with constants appropriate for the esu system of electromagnetic units. This is somewhat simpler and the expressions can always be rewritten in terms of dimensionless quantities with values independent of the system of units. 12th The energy of the n stationary state can now be calculated using this and (1.19) giving 2 2 4 1 Ze Z e m e E = = : (1.21) n 2 2 2 n h 2 2 2h n 2 m Ze e We can now de ne a new dimensionless constant 2 e 1  = = : (1.22) hc  137 The energy can now be rewritten as 2 2 2 Z m c e E = : (1.23) n 2 2n The frequency of light that is emitted from the transition from a state n to a state m, where nm can now be written as   2 2 2 E E Z m c 1 1 n m e  = = : (1.24) nm 2 2 h 4h  m n If we identify 2 2 2 Z m c e R = (1.25) 4h  it is clear that the Balmer series corresponds to the special case where m = 2. It also useful to consider two alternate forms of (1.17). First we can rewrite this equation as m rv =nh : (1.26) e The left-hand side of this is just the angular momentum for a particle moving with uniform speed in a circle, so L =nh  (1.27) means that we can also state the quantization condition as the quantization of angular momentum. The second form is m v2r =nh: (1.28) e The left-hand side of this is just the momentum times the circumference of the circular orbit. This can be generalized as Z dlp =nh: (1.29) The left-hand side is called the action so this form of the condition means that the action is quantized. When Sommerfeld extended the Bohr model to allow for elliptical orbits and relativistic corrections it was the action form of the quantization condition that was used. 13Now lets return to the expression for the radius of the Bohr orbitals (1.19) for the case of hydrogen (Z = 1). This can be rewritten as h  2 2 r = n =a n (1.30) n 0 m c e 11 a = 5:292 10 m is called the Bohr radius and is the radius of the ground state 0 2 of the Bohr hydrogen atom. Although a is a small number, n grows very rapidly. 0 For the the radius to be 1mm, r 3 1:0 10 m  n = 4350: (1.31) = 11 5:292 10 m Bohr had now introduced a new theory for atoms, but all of the accumulated theory and observations show that classical physics works a macroscopic scales. It is, there- fore, necessary for the quantum theory to reproduce classical physics when the size of the object becomes on the macroscopic scale. This called the classical correspondence principle. Bohr stated this by observing that for large values of n his theory should reproduce the classical result. We can see how this occurs for the Bohr atom by noting that a classical electron moving in a circle with a positive charge at the center will radiate at the a frequency equal to the orbital frequency of the electron. That is nh  2 2 2 2 2 v nh  nh  m c m c m r e e e  =f = = = = = : (1.32) cl 2 2 4 2 2r 2r 2m r 2m 2hn  h  n e e Now consider the frequency for the Bohr model when the electron moves from a state n to a state n 1 using (1.24) for the case of hydrogen. This gives   2 2 2 2 m c 1 1 m c 2n 1 e e  = = : (1.33) n;n1 2 2 2 2 4h  (n 1) n 4h  n (n 1) In the limit where n becomes large this yields 2 2 m c e   (1.34) = n;n1 3 2hn  which agrees with the classical result. So the Bohr atom obeys the classical corre- spondence principle. As we will see, quantum mechanics in its current form is constructed such that it satis es the classical correspondence principle. The Bohr atom was revolutionary. For the rst time it was possible to reproduce spectroscopic data and a great number of advances were made in physics under its in uence. It did, however, have substantial problems. While it predicts spectra well for hydrogen and singly ionized helium, it does a poor job of reproducing the spectra of the neutral helium atom. Even when spectra a predicted, the model cannot account for the intensity of the spectral lines or for their widths. Bohr was able to make remarkable number of predictions with the model in conjunction with the correspondence principle, but in the end it did not provide a sucient basis to move forward with the study of quantum systems. 14

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