Quantum Physics 2005 - Rensselaer Polytechnic Institute

Quantum Physics 2005Notes-1Course Information, Overview,The Need for Quantum MechanicsNotes 1Quantum Physics F20051

Contact Information:Peter Persans, SC1C10, x2934email: persap @ rpi.eduPhysics Office: SC 1C25 x6310 (mailbox here)Office hours: Thurs 2-4; M12-2 and by emailappointment. Please visit me!Home contact: 786-1524 (7-10 pm!)Notes 1Quantum Physics F20052

Topic Overview123456Notes 1Why and when do we use quantum mechanics?Probability waves and the Quantum MechanicalState function; Wave packets and particlesObservables and OperatorsThe Schrodinger Equation and some specialproblems (square well, step barrier, harmonicoscillator, tunneling)More formal use of operatorsThe single electron atom/ angular momentumQuantum Physics F20053

Intellectual overview The study of quantum physics includesseveral key parts:– Learning about experimental observations ofquantum phenomena.– Understanding the meaning and consequences ofa probabilistic description of physical systems– Understanding the consequences of uncertainty– Learning about the behavior of waves andapplying these ideas to state functions– Solving the Schrodinger equation and/or carryingout the appropriate mathematical manipulations tosolve a problem.Notes 1Quantum Physics F20054

The course 2 lecture/studios per week T/F 12-2 Reading quiz and exercises every class 15% of grade Homework 35% 2 regular exams 30% Final 20%Notes 1Quantum Physics F20055

Textbook Required: Understanding Quantum Physics byMichael Morrison Recommended:Fundamentals of Physics by Halliday, Resnick Handbook of Mathematical Formulas by Spiegel(Schaums Outline Series) References:Introduction to the Structure of Matter, Brehm andMullenQuantum Physics, Eisberg and ResnickNotes 1Quantum Physics F20056

Academic Integrity Collaboration on homework and in-classexercises is encouraged. Copying isdiscouraged. Collaboration on quizzes and examinations isforbidden and will result in zero for that testand a letter to the Dean of Students. Formula sheets will be supplied for exams.Use of any other materials results in zero forthe exam and a letter to the DoS.Notes 1Quantum Physics F20057

The class really starts now.Notes 1Quantum Physics F20058

Classical Mechanics A particle is an indivisible point of mass. A system is a collection of particles withdefined forces acting on them. A trajectory is the position and momentum(r(t), p(t))of a particle as a function of time. If we know the trajectory and forces on aparticle at a given time, we can calculate thetrajectory at a later time. By integratingthrough time, we can determine the trajectoryof a particle at all times.rd 2rrm 2 !"V (r , t );dtNotes 1rdrrp (t ) mdtQuantum Physics F20059

Some compelling experiments:The particlelike behavior ofelectromagnetic radiation Simple experiments that tell us that light hasboth wavelike and particle-like behavior–––––Notes 1The photoelectric effect (particle)Double slit interference (wave)X-ray diffraction (wave)The Compton effect (particle)Photon counting experiments (particle)Quantum Physics F200510

The photoelectric effectKraneKraneKrane In a photoelectric experiment, we measure thevoltage necessary to stop an electron ejected froma surface by incident light of known wavelength.Notes 1Quantum Physics F200511

Interpretation of the photoelectriceffect experimentEinstein introduced the idea that light carries energy inquantized bundles - photons.The energy in a quantum of light is related to the frequencyof the electromagnetic wave that characterizes the light.The scaling constant can be found from the slope ofeVstop vs wave frequency, # . It is found that:eVstop h# ! % material E photon ! %where h 6.626 x10-34 joule-secFor light waves in vacuum, c # 3 x108 m/s,so we can also write:eVstop hc ! % materialA convenient (non-SI) substitution is hc 1240 eV-nm.Notes 1Quantum Physics F200512

from Millikan’s 1916 paperNotes 1Quantum Physics F200513

R. A. Millikan, Phys Rev 7, 355, (1916)Notes 1Quantum Physics F200514

Compton scattering(from Krane) The Compton effect involves scatteringof electromagnetic radiation fromelectrons. The scattered x-ray has a shiftedwavelength (energy) that depends onscattered directionNotes 1Quantum Physics F200515

Compton effectWe use energy and momentum conservation laws:Energyhv ' h# ! K electron : K kinetic energyMomentumx component:h y component: 0 h 'h 'cos & ' mv cos (sin & ' mv sin (h '! ) (1 ! cos & )mcm mass of electron;' relativistic correction to electron momentum mvNotes 1Quantum Physics F200516

Conclusions from the Compton EffectX-ray quanta of wavelength have:Kinetic energy: K Momentum: p Notes 1Quantum Physics F2005hc h 17

Double slit interferenceThe intensity maxima in a double slit waveinterference experiment occur at:d sin ( n where d is the distance between the slits.(The width of the overall pattern depends onthe width of the slits.)Notes 1Quantum Physics F200518

X-ray diffraction In x-ray diffraction, x-ray waves diffracted fromelectrons on one atom interfere with waves diffractedfrom nearby atoms. Such interference is most pronounced when atomsare arranged in a crystalline lattice.Bragg diffraction maximaare observed when:2d sin ( n d the distance between adjacentplanes of atoms in the crystal.Notes 1Quantum Physics F200519

Laue X-ray Diffraction Pattern A Laue diffractionpattern is observedwhen x-rays of manywavelengths areincident on a crystaland diffraction cantherefore occur frommany planessimultaneously. prettyfrom KraneNotes 1Quantum Physics F200520

The bottom line on light In many experiments, light behaves like awave (c phase velocity, # frequency, wavelength). In many other experiments, light behaves likea quantum particle (photon) with properties:energy: E photon h# momentum: p hc h and thus : E pcNotes 1Quantum Physics F200521

Some compelling experiments:The wavelike behavior of particles Experiments that tell us that electrons havewave-like properties– electron diffraction from crystals (waves) Other particles– proton diffraction from nuclei– neutron diffraction from crystalsNotes 1Quantum Physics F200522

Electron diffraction from crystals electron diffraction patternsfrom single crystal (above)and polycrystals (left) [fromKrane]Notes 1Quantum Physics F200523

Proton diffraction from nucleifrom KraneNotes 1Quantum Physics F200524

Neutron diffractionfrom KraneDiffraction of fast neutrons from Al, Cu, and Pb nuclei.[from French, after A Bratenahl, Phys Rev 77, 597(1950)]Notes 1Quantum Physics F200525

Electron double slit interference Electron interference from passing through adouble slitfrom RohlfNotes 1Quantum Physics F200526

Helium diffraction from LiF crystalfrom French after Estermann and Stern, Z Phys 61, 95 (1930)Notes 1Quantum Physics F200527

Alpha scattering from niobium nucleiAngular distribution of 40 MeV alpha particles scatteredfrom niobium nuclei.[from French after G. Igo et al., Phys Rev 101, 1508(1956)]Notes 1Quantum Physics F200528

The bottom line on particles In many experiments, electrons, protons,neutrons, and heavier things act like particleswith mass, kinetic energy, and momentum:1 2 p2p mv and K mv for non-relativistic particles22m In many other experiments, electrons,protons, neutrons, and heavier things act likewaves with :hh p2mKNotes 1Quantum Physics F200529

The De Broglie hypothesisp h for everythingNotes 1Quantum Physics F200530

Some other well-known experimentsand observations optical emission spectra of atoms arequantized the emission spectrum of a hot object(blackbody radiation) cannot be explainedwith classical theoriesNotes 1Quantum Physics F200531

Emission spectrum of atomsfrom a random astronomy websiteFor hydrogen: 11 h# !13.6eV 2 ! 2 n nfi Notes 1Quantum Physics F200532

Blackbody radiationfrom Eisberg and ResnickNotes 1Quantum Physics F200533

A review of wave superposition andinterferenceMany of the neat observations of quantumphysics can be understood in terms of theaddition (superposition) of harmonic waves ofdifferent frequency and phase.In the next several pages I review some of thebasic relations and phenomena that are usefulin understanding wave phenomena.Notes 1Quantum Physics F200534

Harmonic wavesy A sin(k(x vt) *) or y Asin(kx t *)s in ( x )10 .50-4-2024-0 .5-1x wavelength; k 2, 2, 2,#T period ; TNotes 1Quantum Physics F200535

Phase and phase velocityy Asin(kx t)(kx t) phasewhen change in phase 2, repeatPhase velocity speed with which point ofconstant phase moves in space.vp /k #Tra ve ling wa ve10.50-4-2024-0.5-1d is ta nc e (m )Notes 1Quantum Physics F200536

Complex Representation of TravellingWavesDoing arithmetic for waves is frequently easierusing the complex representation using:e i( cos( i sin (So that a harmonic wave is represented as. ( x, t ) A cos(kx ! t - )[. ( x, t ) Re Aei ( kx ! t - )Notes 1Quantum Physics F2005]37

Adding waves: same wavelength anddirection, different phase. 1 . 01 sin(kx t &1 ). 2 . 02 sin(kx t &2 )taking the form:. R . 1 . 2 . 0 sin(kx t / )we find :20201202. . . 2. 01. 02 cos(&1 ! &2 )and:. 01 sin &1 . 02 sin &2tan / . 01 cos &1 . 02 cos &2Notes 1Quantum Physics F200538

Adding like-waves, in words Amplitude– When two waves are in phase, the resultantamplitude is just the sum of the amplitudes.– When two waves are 1800 out of phase, theresultant is the difference between the two. Phase– The resultant phase is always between the twocomponent phases. (Halfway when they areequal; closer to the larger wave when they arenot.)(see adding waves.mws)Notes 1Quantum Physics F200539

Another useful example of added waves:diffraction from a slitaspath length differenceLet’s take the field at the view screen from an element of length onthe slit ds as EsdsIgnore effects of distance except in the path length.Notes 1Quantum Physics F200540

Single slit diffractiond Re(. s ei ( kz ( s ) ! t ) ds )z ( s ) z0 s sin (. (( ) . s ea/2i ( kz0 ! t )iks sin (eds !a / 2 . sei ( kz0 ! t )asin 0whereak0 sin (220sin22. (. s a)20First zero at 0 , , so sin ( 2, ak a0Notes 1Quantum Physics F200541

Adding waves:traveling in opposite directionsEqual amplitudes:. R . 0 (sin( kx ! t -1 ) sin( kx t - 2 ))-1 - 2) cos t. R 2. 0 sin(kx 2 The resultant wave does not appear to travel– it oscillates in place on a harmonic patternboth in time and space separately Standing waveNotes 1Quantum Physics F200542

Adding waves: different wavelengths. 1 . 01 cos(k1 x ! 1t ). 2 . 01 cos(k2 x ! 2t )432101101201-1-2-3-45% k diffBEATS1. 2. 01 cos ( k1 k2 ) x ! ( 1 2 )t 21 cos ( k1 ! k2 ) x ! ( 1 ! 2 ) t 2) )kx!t . 01 cos kx ! t cos 2 2 The resultant wave has a quickly varying part that waves at theaverage wavelength of the two components. It also has an envelope part that varies at the differencebetween the component wavelengths.Notes 1Quantum Physics F200543

Adding waves: group velocity Note that when we add two waves of differing and k to one another, the envelope travelswith a different speed:monochromatic wave 1: v phase1 monochromatic wave 2: v phase 2 beat envelope: vgroup 1k1 2k2) )ksee group velocity.mwsNotes 1Quantum Physics F200544

Adding many waves to make a pulse In order to make a wave pulse of finite width, wehave to add many waves of differing wavelengths indifferent amounts. The mathematical approach to finding out how muchof each wavelength we need is the Fouriertransform:1 1 f ( x) 1 A( k ) cos kxdk B(k ) sin kxdk , 0 0where :1A( k ) f ( x) cos kxdx!11B( k ) f ( x) sin kxdx!1Notes 1Quantum Physics F200545

The Fourier transform of a Gaussianpulse We can think of a Gaussian pulse as alocalized pulse, whose position we know to acertain accuracy )x 22x.f ( x) Notes 112 x 2,Quantum Physics F2005e! x 2 / 22 x246

Finding the transformI will drop overall multiplicative constantsbecause I am interested in the shape of A(k)1A( k ) 3 e! x 2 / 22 x2e! ikx12dx e ! ax e ! ikx dx!1!1where a 1/ 22 x2(solving by completing the square:)112A( k ) 3 e ! ax e ! ikx dx e!1letting 0 x a !! x a!ik2 a2!k 2 / 4adx!1ik2 a, !k 2 / 4a, ! k 22 x2 / 21 !k 2 / 4a 1 ! 0 2A( k ) 3eedee0 aa!1aNotes 1Quantum Physics F200547