I-1CHAPTER I. INTRODUCTION TO QUANTUM THEORYA. Historical perspectiveThe topics that will be covered during approximately the first 70% of this course focus onthe microscopic properties of matter. The objective is to gain a rudimentary understanding of thebehavior of atoms and molecules, as well as the theoretical framework used in the description ofmicroscopic particles. The historical development of humankind's understanding of the physicalprinciples involved is briefly sketched below. (See Table I-1).The description of macroscopic properties utilizes the tools of classical mechanics andthermodynamics. Classical mechanics evolved during the 1600's, beginning with theastronomical observations of Tycho Brahe and their interpretation by Kepler. The additionalinformation provided by the mechanical experiments of Galileo enabled Newton (in 1687) todevise an elegant theory of mechanics. A similar revolution involving the unification of a large variety of experimental factsconcerning electricity and magnetism took place in the nineteenth century. All of thesephenomena, including the wave propagation theory of light, were brought together within oneconceptual framework by the electromagnetic theory of Maxwell in 1864.The beginning of the twentieth century brought an unprecedented series of experimentswhich unveiled new phenomena that defied explanation in terms of the universally accepted lawsof classical physics. Ultimately, these events led to one of the most stimulating periods ofdevelopment ever to occur in the physical sciences. As a result of this activity, intuitive ideasconcerning the nature of microscopic systems and space/time relationships underwent drasticrevision. These new concepts are embodied in what is now called "modern physics", which iscomprised of the quantum theory and the theory of relativity.One of the most important experiments leading to the development of quantum theorymeasured the spectral distribution of black body radiation. The theoretical explanation of this phenomenon introduced the crucial concept of quantization.I-2Table I-1. Major events in the evolution of physical conceptsClassical Mechanics: 1600's1600 - Tycho Brahe's astronomical observations - Interpretation by Kepler - Mechanical experiments of Galileo1687 - Newton's theory of mechanicsClassical electromagnetism: 1800's1800 - Experiments on electricity and magnetism1864 - Maxwell's electromagnetic theoryRelativity: 1900's1905 - Einstein's theory of relativityQuantum theory: 1900's1896 - Discovery of radioactivity by Becquerel1897 - Thompson's experiments with electrons1911 - Rutherford model of the atom1913 - Bohr model of the atom1924 - Wave properties of particles (de Broglie/Davisson & Germer) 1927 - Quantum mechanics (Schrödinger & Heisenberg)A(x,t) ' A0sin (kx !ωt)I-3B. Failures of Classical Physics1. Black Body RadiationWhen an object is heated, it emits electromagnetic radiation. A black body has the abilityto absorb and emit all radiation regardless of its frequency. It can be approximated by an ovenwith a small hole through which the radiation is sampled. The frequency distribution of blackbody radiation is shown in Fig. I-1. Notice that the radiation distribution has a maximum, ν,maxthat increases with temperature (e.g., ν 6 red 6 orange 6 yellow 6 white).maxIn the late 1800's, Rayleigh and Jeans examined the problem of black body radiation. Using classical concepts, they were able to obtain a relatively simple formula for the spectrum ofblack body radiation. They began by assuming the electromagnetic radiation was generated bythe vibrations of atomic oscillators in the walls of the oven, much as radio waves are generatedby the oscillation of an electric current in a dipole antenna. Under the steady state conditions ofthermal equilibrium, the radiation inside the oven must exist in the form of standing waves withnodes at the walls. Before delving into the details of the analysis by Rayleigh and Jeans, a shortdigression into the general properties of waves will be beneficial.When a wave propagates through an elastic medium (e.g., air, water), each particle of themedium vibrates in simple harmonic motion. Therefore, the displacements of the particles aregiven by a sine or cosine function of the space coordinates. The distance between any twosuccessive particles whose motion is in phase is defined to be the wavelength λ of the wave. Since the wave has moved a distance of one wavelength during the period T, the wave velocity isv = λ/T = λ ν , where ν is the frequency. Therefore.The general equation for a plane wave propagating to the right along the x-axis is .Frequency (1013 s-1)0 10 20 30 40 50U(ν) (10-17J-s/m3)024682000 K1750 Kvisible lightI-4Figure I-1. Frequency distribution of black body radiation at two temperatures.2A0sinα%β2cosα!β2A ' 2A0sin (kx!ωt) cos!dkx%dωt2I-5where A is the amplitude, ω = 2πν (angular frequency) and k = 2π/λ (propagation constant). oWhen two waves traverse the same region of space at the same time, the resultant amplitude willbe given by the sum of the two waves. This superposition principle leads to wave interference. Consider the sum of two waves having the same amplitude but differing by an amount dω inangular frequency and dk in propagation constant;A = A sin (kx!ωt) = A sin α1 0 0A = A sin [(k+dk)x ! (ω+dω)t] = A sin β .2 0 0 Their sum is A = A + A = A sin α + A sin β1 2 0 0 = (trig. identity)where α+β = (2k+dk)x!(2ω+dω)t . 2(kx!ωt)α !β = !dkx+dωt .Hence .This is a wave moving to the right which is characterized by beats caused by the second term(the modulating factor). An illustration of the interference of two waves and the resulting beatpattern is shown in Fig. I-2. By the appropriate choice of dω and dk, total constructive or totaldestructive interference can result.A special case of interest is a standing wave formed by two waves having the sameamplitudes, ω's, and k's traveling in opposite directions;A = A sin (kx!ωt) A = A sin (kx+ωt) .1 0 2 0 By performing the same exercise as above, the sum is found to beA = 2A sin kx cos ωt .0 This is the equation of a stationary wave, as may be seen by noting that at values of x for whichkx = nπ (where n is an integer), A = 0. Hence the wave has stationary nodes (see Fig. I-3).Amplitude-40-2002040Amplitude-40-20020X0.0 0.4 0.8 1.2 1.6 2.0Amplitude-80-40040I-6Figure I-2. A wave formed by the superposition of