Phys 401 Spring 2021 Lecture #1 Summary 25 January, 2021
Phys 401Spring 2021Lecture #1 Summary25 January, 2021Welcome to Phys 401! It should be an exciting and action-packed semester as we delveinto the fascinating topic of Quantum Mechanics. Please do Homework 0 to review the math skillsthat you will need for this class. Note that we will move at a faster pace, and work at a higherlevel, than we did in Phys 371. I will skip many mathematical steps during the lectures (e.g.integration by parts, solving standard equations, performing algebraic simplifications, etc.), andyou should go back and supply these missing steps when you review your notes.We began by reviewing key ideas from Phys 371 βModern Physicsβ. The first big step inquantum mechanics came about from trying to understand the radiation emitted by blackbodyradiators. These are objects at some temperature ππ that emit radiation over a broad range ofwavelengths, and their emission properties are independent of material. It was found empiricallythat the total radiated power of such an object is given by the Stefan-Boltzmann law: π π ππππππππππ ππππ 4 , where π π ππππππππππ is the total radiated power per unit area of the object, having units ofStefan constant is 5.6703 10 8ππππ 2 πΎπΎ4ππππ 2. The, and ππ is the absolute temperature in Kelvin. Theradiated power is spread out over a broad range of wavelengths as shown below. The wavelengthππππππππ of the peak radiated power is related to the temperature of the blackbody as ππππππππ 1/ππ.The Wein displacement law says that ππππππππ ππ 2.898 10 3 ππ πΎπΎ. This empirical result turnedout to be a key observation for the development of the photon theory of light.1
A blackbody radiator is realized by creating a box with interior walls at temperature ππ. Theenergy density of the radiation in the box is given by ππ(ππ), which has units of Energy/(VolumeWavelength), or π½π½/(ππ3 ππππ). This is an energy density, both in terms of per unit volume, andper unit wavelength. If there is a small hole in one wall of the box that lets some of the radiationππto escape, the radiated power at wavelength ππ is given by π π (ππ) ππ(ππ), where ππ is the speed of4light. Hence the radiated power gives direct insight in to the energy density of radiation in thebox. Blackbody radiators are commercially available. The sun acts as a blackbody radiationsource to good approximation, at least when its π π (ππ) spectrum is measured outside the Earthβsatmosphere.The classical explanation was that ππ(ππ) ππ(ππ)πππ΅π΅ ππ, where ππ(ππ)ππππ is the number ofelectromagnetic modes with wavelengths between ππ and ππ ππππ per unit volume inside the box.This quantity ππ(ππ) 8ππππ4is derived here. The factor of πππ΅π΅ ππ comes from the statistical mechanicsidea of βequipartition of energy,β which says that every degree of freedom of the system acquiresπππ΅π΅ ππ/2 of energy in equilibrium. This idea is very successful in the thermodynamics of ideal gases3(where it predicts the internal energy ππ πππππ΅π΅ ππ), but it is a disaster for light in a box. It predicts2the Rayleigh-Jeans law ππ(ππ) 8πππππ΅π΅ ππ/ππ4 , which suffers from the βultraviolet catastropheβ in thatit predicts an infinite energy density in the wavelength going to zero (ultraviolet) limit. Thiscontradicts the experimental situation, as shown in the figure above, where it is seen that ππ(ππ) isstrongly suppressed to zero at short wavelengths. Houston, we have a problem!To address this problem Planck made two (unjustified and revolutionary) assumptions:0) The atoms in the walls of the box have electrons, and when these electrons vibrate atfrequency ππ they emit electromagnetic waves at frequency ππ. Likewise light at frequencyππ can be absorbed by the atoms in the walls and start oscillating at frequency ππ. Inequilibrium there is an equal energy flux from the walls to radiation and from radiationback into the walls.1) The atoms in the walls of the box have discrete energy levels given by πΈπΈππ ππππ, whereππ 0, 1, 2, Hence the atoms can only interact with light of energy ππ, 2ππ, 3ππ, etc.2) The energy of light is directly related to the frequency of oscillation of the EM fields asππ βππ, where β is a fudge factor with units of π½π½/π»π»π»π» or π½π½ π π . Surprisingly, this energy isindependent of the intensity of the light.Planck then adopted an assumption from statistical mechanics about the likelihood of findingan atom in the walls of the box being excited to energy state πΈπΈ, assuming the walls are inequilibrium with the radiation field at temperature ππ. It is given by the Maxwell-Boltzmann factorat temperature ππ: ππ(πΈπΈ ) π΄π΄ππ πΈπΈ/πππ΅π΅ ππ , where π΄π΄ is a normalization factor. Here ππ(πΈπΈ ) is the fractionof atoms in the wall of the box that are excited to a state of energy πΈπΈ. Note the negative exponential2
dependence on the ratio of the energy of the atom πΈπΈ to the thermal energy πππ΅π΅ ππ. This means thatit will be very unlikely to find atoms occupying energy states with πΈπΈ πππ΅π΅ ππ, which in turn meansthat electromagnetic modes of the box at that energy (which corresponds to a short wavelength)will have a very low probability of occupation, which will fix the ultraviolet catastrophe. Theresulting Planck blackbody radiation formula is ππ(ππ) 8ππβππ/ππ5ππ βππ/πππππ΅π΅ ππ 1. This function has the propertythat in the ultraviolet limit (ππ 0) it goes to zero exponentially fast, thus avoiding the ultravioletcatastrophe. To fit the blackbody emission data for ππ(ππ) one has to choose a fudge factor of β 6.626 10 34 π½π½ π π , which is known today as Planckβs constant.The photoelectric effect involves ultraviolet light impinging on a clean metal surface andliberating photoelectrons. The parameters are the intensity πΌπΌ and frequency ππ of the light, and themaximum kinetic energy of the photoelectrons, and the metal used for the photo-cathode. Anumber of observations were made about this effect (see the experimental setup below):1) A positive anode (collector) voltage resulted in a steady photocurrent ππ.2) The magnitude of the photocurrent scales with the light intensity ππ πΌπΌ.3) There was NO lag between the introduction of the light and the onset of photocurrent.Even in situations where the light intensity was so low that classically it would takehours to transfer enough energy to the electrons to begin liberating them from the metal,the first photoelectrons would appear essentially instantaneously after even weak lightwas turned on.4) If the anode (collector) potential was made sufficiently negative the photocurrent wouldcease. This is called the stopping potential ππ0 (with ππ0 0).5) The stopping potential is a measure of the maximum kinetic energy of the liberated1electrons: ππππ0 πππ£π£ 2 2ππππππ6) It was found that ππ0 is independent of the light intensity. Classically you would expectthat higher light intensity would result in more energetic photoelectrons, but this is notobserved.7) It was found that changing the frequency of the light would change the stoppingpotential. The higher the frequency of the light, the larger the magnitude of thestopping potential, and a linear (rather than quadratic) relationship was observed.Observations 3, 6 and 7 are clearly at odds with the expectations of classical physics.3
Einsteinβs 1905 paper originated the concept of a photon. He was bothered by the fact thatMaxwellβs equations predicted that light energy would continuously decrease to arbitrarily smallamounts as a light sphere around a source expanded outward. He proposed that this energy dilutionstopped when the light energy got down to some minimum quantum of energy. He also expectedthat time-dependent phenomena involving electromagnetic waves (such as the absorption oremission of light) might show new phenomena not described by Maxwellβs equations.Einstein made three proposals, with which he could explain all the experimental results on thephotoelectric effect:1) Adopt the energy quantization idea for electromagnetic fields, as proposed by Planck,namely the energy of the βlight particlesβ is related to the frequency of the EM waves asπΈπΈππππππβπ‘π‘ βππ.2) The βquantum of lightβ aka βphotonβ collides with a single electron in the metal andtransfers all of its energy to the electron at once.3) The energy of the resulting photoelectron is πΈπΈππππ βππ ππ, where ππ is the work functionof the metal, and varies from one metal to the next, but is in the range of 2 to 5 eV. Thework function is the binding energy of the electrons in the metal and depends on details.With this proposal, Einstein explained all the observations and made the following prediction.In the limit of low frequency, the photon will not have enough energy to liberate the electronsbecause βππ ππ. This leads to the threshold frequency πππ‘π‘ ππ/β below which there is nophotocurrent ππ. If one plots the stopping potential ππ0 vs. light frequency ππ for ππ πππ‘π‘ it shouldobey the photoelectric equation: ππ0 (β/ππ)ππ ππ/ππ, where the slope of the straight line should4
have a universal value of β/ππ independent of the metal used in the cathode. This is in agreementwith experiments (see HW 1).Roentgen discovered in 1895 that X-rays are produced when cathode rays (electrons) areproduced with a very high potential ( 104 Volts) and directed in to a metal target. The resultingbremsstrahlung (braking radiation) is an electromagnetic wave. The wavelengths of the resultingelectromagnetic waves are less than 1 nm.X-ray emission has three properties:1) There is a continuous spectrum expected classically from βbraking radiation.β2) There are sharp emission lines that appear for sufficiently high accelerating voltages.These lines depend on the type of metal used as a target.3) The continuous spectrum has a sharp cutoff at short wavelengths, found empirically tobe described by the equation ππππππππ accelerating voltage of the electrons.1240ππ(ππ)ππππ ππ (Duane-Hunt Rule), where ππ is theEinstein pointed out that X-ray generation is the inverse of photo-electron emission, andused the photoelectric equation to derive ππππππππ βππππππ 1239.8ππ(ππ)ππππ ππ, in good agreement with theDuane-Hunt Rule. The observed sharp lines are a consequence of the discrete energy levels presentin atoms.Most solid materials are crystalline and are made up of atoms or molecules that areregularly spaced in a periodic array. When x-rays come in to such a crystal structure theyencounter a periodic potential that can give rise to sharp and intense diffracted beams. Thesebeams arise from reflections of electromagnetic waves from parallel planes of atoms that act aspartially reflecting mirrors, and occur when constructive interference occurs in reflection. Afteridentifying a set of parallel planes, one can calculate the constructive interference diffractioncondition by requiring that each wave that penetrates one layer deeper must traverse a distance5
corresponding to an integer number (ππ) of additional wavelengths before rejoining the beamreflected from the layer above. This gives rise to the condition that ππππ 2ππ sin ππ, where ππ 1, 2, 3, , ππ is the wavelength of the x-rays, ππ is the spacing of the parallel planes of atoms, and ππis the angle of incidence of the x-ray beam relative to the parallel planes. This is called Braggβslaw, and is a consequence of the wave nature of x-rays.We briefly reviewed Compton scattering with an emphasis on the energy and momentumof light particles (photons) involved in the scattering process. X-rays of wavelength ππ scatter fromstationary electrons, changing to a longer wavelength ππβ² and moving off in a direction at angle ππrelative to their incident direction. The Compton formula is: ππβ² ππ βππππ(1 cos ππ), where β isPlanckβs constant, ππ is the electron mass, and ππ is the speed of light in vacuum. The idea thatlight has an energy given by πΈπΈ βππ, where ππ is the frequency of oscillation of the electromagneticwaves, and light has a momentum given by ππ π¬π¬ππ ππππππππ βππ, is consistent with ideasππintroduced by Planck and Einstein to explain the blackbody radiation spectrum and the photoelectric effect, respectively. One derives the Compton formula simply by enforcing conservationof energy and momentum for the βphoton-electron particle-like collision.β This illustrates theparticle nature of light and further supports the concept of a photon. The fact that light sometimesacts like a wave and sometimes acts like a particle is called βwave-particle duality.βThe Nuclear AtomAtomic spectroscopy shows that atoms have many discrete spectral lines in emission andabsorption. Each type of atom has its own unique spectrum of lines, a kind of βfingerprint.β Itwas found that Hydrogen (and one-electron ions) have the simplest structure of spectral lines.Regularities in the spectral wavelengths were observed involving the inverse squares of integers.For example the Rayleigh-Ritz formula βpredictedβ the wavelengths of many spectral lines as1ππππππ π π 1ππ 2 1ππ 2 , where ππ and ππ are positive integers with ππ ππ. The factor π π is the Rydberg6
constant and has a value of π π π π π»π» 1.096776 1071ππfor Hydrogen. Heavier single-electronions have slightly larger values of π π . Where does this regularity in the emission spectrum comefrom?Rutherford scattering (studied carefully in Phys410 Classical Physics) experiments consistof energetic Ξ±-particles (2 protons plus 2 neutrons) being scattered from materials like Au in theform of a thin sheet. It was found that a large number of the Ξ±-particles were scattered straightback, and the angular distribution of the scattering implies that much of the mass of the atom isconcentrated in a single positively charged entity. This entity is the nucleus of course, andRutherford showed that it has a dimension on the scale of 1 fm (10 15 m). The electrons aredistributed more or less uniformly in a cloud outside of the nucleus. This is the origin of thenuclear model of the atom.Neils Bohr developed a planetary model of the Hydrogen atom. One can imagine anelectron in an orbit around the proton in analogy with the earth around the sun. However, inclassical physics charges that accelerate are known to radiate. Thus it was not clear why anelectron in an orbit around the proton would not radiate continuously and spiral into the nucleus.Bohr made three bold postulates:1) Electrons orbit the nucleus in circular orbits called βstationary statesβ and do not radiatewhile in such states.2) Atoms radiate when electrons make transitions between stationary states of differentenergy.3) The angular momentum of electrons in stationary states is quantized in units of β β/2ππ.(Here, Planckβs constant appears in an entirely new context, very different from blackbodyradiation!)Bohr then did a βsemi-classicalβ calculation of the Hydrogen atom structure, assuming that theelectron and proton are attracted to each other by the Coulomb interaction. He arrived at thefollowing results:Using Newtonβs second law of motion for the bound state of a positively charged nucleus anda single negatively charged electron, he found the speed, angular momentum, and the radius of theelectron orbit, and the total energy of the Hydrogen atom. The speed of the electron in its circularππππ 2orbit of radius ππ: π£π£ 4ππππ0 ππππ, where the nucleus has charge ππππ, ππ0 is the permittivity of freespace, and ππ is the (reduced) mass of the electron.Now use the third postulate and assume a non-relativistic situation: The angular momentum isquantized as πΏπΏ β ππβ πππ£π£β ππππππ ππβ, with ππ 1, 2, 3, This leads to the quantized radiiof the Bohr orbits: ππππ ππ 2ππ0ππ, where ππ0 4ππππ0 β2ππππ 27 0.529 β« is called the Bohr radius. Note that
the Bohr radius expression is made up of only fundamental constants of nature. Hence the structureof the Hydrogen atom is universal and time invariant, as far as we know.The total energy of the hydrogen atom πΈπΈ ππ ππ (ππ is kinetic energy of the electron β theproton is assumed to be stationary, ππ is the Coulomb potential energy between the electron andππ 2proton) is found to be πΈπΈππ πΈπΈ0 ππ2 , with πΈπΈ0 ππ 2 /4ππππ0βππ 1137.036ππππ 2 ππ 2 /4ππππ0 2 (βππ)22 ππππ 22πΌπΌ 2 13.6 ππππ, where πΌπΌ is a famous dimensionless constant called the fine structure constant. ForHydrogenic atoms and single-electron ions one has πΈπΈππ 13.6 ππππ ππ 2ππ 2, with ππ 1, 2, 3, Why isthe energy negative? Negative energy represents a bound state of the electron and proton. Wedefine the zero of energy to be an electron and proton separated to infinity, with no kinetic energy.A negative energy comes about because the Coulomb attraction of the two particles, plus theirkinetic energy, corresponds to a lower energy state than having the particles at rest separated toinfinity. Also take note that Bohr predicts an infinite number of bound states of a proton andelectron (labelled by the set of all positive integers)!With these energy levels for the βstationary statesβ Bohr could now predict the wavelengthsof light emitted by a hydrogen atom as it made transitions between stationary states. He foundthat1ππππππ πΈπΈ0 ππ 2βππ 1ππππ2 1ππππ2 , where ππ and ππ refer to initial and final states labeled by positive integersππππ and ππππ , respectively. This explains the discreteness of the spectral lines as well as the regularitynoted in the Hydrogen emission spectrum involving the difference of the inverse squares ofintegers (the Rayleigh-Ritz formula). He thus made a prediction that the Rydberg constant isπΈπΈrelated to fundamental constants of nature as π π βππ0 measured value.ππππ 2 πΌπΌ22βππ 1.098 1071ππ, close to theThe Bohr model is a great accomplishment in understanding the basic structure of theHydrogen atom and its interaction with light. However, it fails to explain the properties of morecomplicated atoms with two or more electrons. Nevertheless there were three pieces ofinformation that suggested more complex atoms also had quantized energy levels:1) All atoms, no matter how many electrons they have, show emission spectra made upof discrete lines.2) The peaks in x-ray emission from metal targets are signs that the innermost electronbound states in heavy atoms (like Cu, Mo, W) have discrete energy levels. Thesedeeply bound electrons with ππ 1 ππππ 2 essentially behave like hydrogen atoms withlarge ππ (like 29, 42, 74). Thus the binding energy is enhanced by a factor of ππ 2 , whichcan be thousands of electron volts. When high-voltage electrons collide with theseatoms in an x-ray target they can knock out these βinner shellβ electrons. The relaxationof the atom to fill that electron vacancy produces characteristic emission lines that show8
up in the x-ray emission spectrum. This was systematically explored by Moseley, whoused the Bohr model combined with x-ray emission peaks to predict the existence of 3new elements (Tc, Pm and Re).3) Inelastic electron scattering from Hg atoms (ππ 80) showed that the Hg atom had anexcited state 4.9 eV higher than the ground state. This is the Franck-Hertz experiment.This experiment also showed that ultraviolet light with a wavelength corresponding toan energy difference of 4.9 eV was emitted by the Hg gas only when the acceleratingpotential of the electrons exceeded 4.9 eV, making a direct connection between thediscrete energy levels measured by inelastic electron scattering and by light emission.At this point it became clear that some new thinking was required to understand thestructure of atoms. The three Bohr postulates are largely at odds with classical thinking, makingthe whole exercise somewhat dubious. It required a revolution in thinking about the nature ofmatter to make any further progress.So far we have seen that light sometimes acts like a wave (e.g. x-ray scattering from parallelplanes of atoms giving rise to bright diffraction peaks due to constructive interference β Braggscattering), and sometimes like a particle (e.g. an x-ray having a collision with a st
Spring 2021 . Lecture #1 Summary . 25 January, 2021 . Welcome to Phys 401! It should be an exciting and action-packed semester as we delve into th e fascinating topic of Quantum M echanics. Please do Homework 0 to review the math skills that you will need for this class. Note
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Credit is not given for both PHYS 102 and either PHYS 212 or PHYS 214. Prerequisite: PHYS 101. This course satisο¬es the General Education Criteria for: Nat Sci Tech - Phys Sciences . Credit or concurrent registration in PHYS 212. PHYS 246 Physics on the Silicon Prairie: An Introduction to Modern Computational Physics credit: 2 Hours. (https .
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
ENTM 20600 General Entomology & ENTM 20700 General Entomology Lab PHYS 17200 Modern Mechanics PHYS 21800 General Physics I PHYS 21900 General Physics II PHYS 22000 General Physics PHYS 22100 General Physics PHYS 24100 Electricity & Optics PHYS 27200 Electric & Magnetic Interactions
Letter to the Editor L541 Herrick D R 1976 J. Chem. Phys. 65 3529 Killingbeck J 1977 Rep. Prog. Phys. 40 963 Koch P M 1978 Phys. Rev. Lett. 41 99 Littman M G, Kash M M and Kleppner D 1978 Phys. Rev. Lett. 41 103 Ortolani F and Turchetti G 1978 J. Phys. B: Atom.Molec. Phys. 11 L207 Reinhardt W P 1976 Int. J. Quantum Chem. Symp. 10 359 Silverstone H J 1978 Phys. Rev.
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PHYS 0160 Introduction to Relativity, Waves and Quantum Physics 1 or PHYS 0060 Foundations of Electromagnetism and Modern Physics PHYS 0470 Electricity and Magnetism 1 PHYS 0500 Advanced Classical Mechanics 1 PHYS 1410 Quantum Mechanics A 1 PHYS 1530 Thermodynamics and Statistical Mechanics 1 S
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