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Quantum Physics 2005 Notes-1 Course Information, Overview, The Need for Quantum Mechanics Notes 1 Quantum Physics F2005 1 Contact Information: Peter Persans, SC1C10, x2934 email: persap @ rpi.edu Physics Office: SC 1C25 x6310 (mailbox here) Office hours: Thurs 2-4; M12-2 and by email appointment. Please visit me! Home contact: 786-1524 (7-10 pm!) Notes 1 Quantum Physics F2005 2 Topic Overview 1 2 3 4 5 6 Notes 1 Why and when do we use quantum mechanics? Probability waves and the Quantum Mechanical State function; Wave packets and particles Observables and Operators The Schrodinger Equation and some special problems (square well, step barrier, harmonic oscillator, tunneling) More formal use of operators The single electron atom/ angular momentum Quantum Physics F2005 3 Intellectual overview • The study of quantum physics includes several key parts: – Learning about experimental observations of quantum phenomena. – Understanding the meaning and consequences of a probabilistic description of physical systems – Understanding the consequences of uncertainty – Learning about the behavior of waves and applying these ideas to state functions – Solving the Schrodinger equation and/or carrying out the appropriate mathematical manipulations to solve a problem. Notes 1 Quantum Physics F2005 4 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 1 Quantum Physics F2005 5 Textbook • Required: Understanding Quantum Physics by Michael 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 and Mullen Quantum Physics, Eisberg and Resnick Notes 1 Quantum Physics F2005 6 Academic Integrity • Collaboration on homework and in-class exercises is encouraged. Copying is discouraged. • Collaboration on quizzes and examinations is forbidden and will result in zero for that test and a letter to the Dean of Students. • Formula sheets will be supplied for exams. Use of any other materials results in zero for the exam and a letter to the DoS. Notes 1 Quantum Physics F2005 7 The class really starts now. Notes 1 Quantum Physics F2005 8 Classical Mechanics • A particle is an indivisible point of mass. • A system is a collection of particles with defined 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 a particle at a given time, we can calculate the trajectory at a later time. By integrating through time, we can determine the trajectory of a particle at all times. r d 2r r m 2 = !"V (r , t ); dt Notes 1 r dr r p (t ) = m dt Quantum Physics F2005 9 Some compelling experiments: The particlelike behavior of electromagnetic radiation • Simple experiments that tell us that light has both wavelike and particle-like behavior – – – – – Notes 1 The photoelectric effect (particle) Double slit interference (wave) X-ray diffraction (wave) The Compton effect (particle) Photon counting experiments (particle) Quantum Physics F2005 10 The photoelectric effect Krane Krane Krane • In a photoelectric experiment, we measure the voltage necessary to stop an electron ejected from a surface by incident light of known wavelength. Notes 1 Quantum Physics F2005 11 Interpretation of the photoelectric effect experiment Einstein introduced the idea that light carries energy in quantized bundles - photons. The energy in a quantum of light is related to the frequency of the electromagnetic wave that characterizes the light. The scaling constant can be found from the slope of eVstop vs wave frequency, # . It is found that: eVstop = h# ! % material = E photon ! % where h = 6.626 x10-34 joule-sec For light waves in vacuum, c = $# = 3 x108 m/s, so we can also write: eVstop = hc $ ! % material A convenient (non-SI) substitution is hc = 1240 eV-nm. Notes 1 Quantum Physics F2005 12 from Millikan’s 1916 paper Notes 1 Quantum Physics F2005 13 R. A. Millikan, Phys Rev 7, 355, (1916) Notes 1 Quantum Physics F2005 14 Compton scattering (from Krane) • The Compton effect involves scattering of electromagnetic radiation from electrons. • The scattered x-ray has a shifted wavelength (energy) that depends on scattered direction Notes 1 Quantum Physics F2005 15 Compton effect We use energy and momentum conservation laws: Energy hv ' = h# ! K electron : K = kinetic energy Momentum x component: h $ = y component: 0 = h $' h $' cos & + ' mv cos ( sin & + ' mv sin ( h $ '! $ = )$ = (1 ! cos & ) mc m = mass of electron; ' =relativistic correction to electron momentum mv Notes 1 Quantum Physics F2005 16 Conclusions from the Compton Effect X-ray quanta of wavelength $ have: Kinetic energy: K = Momentum: p = Notes 1 Quantum Physics F2005 hc $ h $ 17 Double slit interference The intensity maxima in a double slit wave interference experiment occur at: d sin ( = n$ where d is the distance between the slits. (The width of the overall pattern depends on the width of the slits.) Notes 1 Quantum Physics F2005 18 X-ray diffraction • In x-ray diffraction, x-ray waves diffracted from electrons on one atom interfere with waves diffracted from nearby atoms. • Such interference is most pronounced when atoms are arranged in a crystalline lattice. Bragg diffraction maxima are observed when: 2d sin ( = n$ d = the distance between adjacent planes of atoms in the crystal. Notes 1 Quantum Physics F2005 19 Laue X-ray Diffraction Pattern • A Laue diffraction pattern is observed when x-rays of many wavelengths are incident on a crystal and diffraction can therefore occur from many planes simultaneously. …pretty from Krane Notes 1 Quantum Physics F2005 20 The bottom line on light • In many experiments, light behaves like a wave (c=phase velocity, #=frequency, $=wavelength). • In many other experiments, light behaves like a quantum particle (photon) with properties: energy: E photon = h# = momentum: p = hc $ h $ and thus : E = pc Notes 1 Quantum Physics F2005 21 Some compelling experiments: The wavelike behavior of particles • Experiments that tell us that electrons have wave-like properties – electron diffraction from crystals (waves) • Other particles – proton diffraction from nuclei – neutron diffraction from crystals Notes 1 Quantum Physics F2005 22 Electron diffraction from crystals • electron diffraction patterns from single crystal (above) and polycrystals (left) [from Krane] Notes 1 Quantum Physics F2005 23 Proton diffraction from nuclei from Krane Notes 1 Quantum Physics F2005 24 Neutron diffraction from Krane Diffraction of fast neutrons from Al, Cu, and Pb nuclei. [from French, after A Bratenahl, Phys Rev 77, 597 (1950)] Notes 1 Quantum Physics F2005 25 Electron double slit interference • Electron interference from passing through a double slit from Rohlf Notes 1 Quantum Physics F2005 26 Helium diffraction from LiF crystal from French after Estermann and Stern, Z Phys 61, 95 (1930) Notes 1 Quantum Physics F2005 27 Alpha scattering from niobium nuclei Angular distribution of 40 MeV alpha particles scattered from niobium nuclei. [from French after G. Igo et al., Phys Rev 101, 1508 (1956)] Notes 1 Quantum Physics F2005 28 The bottom line on particles • In many experiments, electrons, protons, neutrons, and heavier things act like particles with mass, kinetic energy, and momentum: 1 2 p2 p = mv and K = mv = for non-relativistic particles 2 2m • In many other experiments, electrons, protons, neutrons, and heavier things act like waves with : h h $= = p 2mK Notes 1 Quantum Physics F2005 29 The De Broglie hypothesis p= h $ for everything Notes 1 Quantum Physics F2005 30 Some other well-known experiments and observations • optical emission spectra of atoms are quantized • the emission spectrum of a hot object (blackbody radiation) cannot be explained with classical theories Notes 1 Quantum Physics F2005 31 Emission spectrum of atoms from a random astronomy website For hydrogen: 1 1 h# = !13.6eV 2 ! 2 n n f i Notes 1 Quantum Physics F2005 32 Blackbody radiation from Eisberg and Resnick Notes 1 Quantum Physics F2005 33 A review of wave superposition and interference Many of the neat observations of quantum physics can be understood in terms of the addition (superposition) of harmonic waves of different frequency and phase. In the next several pages I review some of the basic relations and phenomena that are useful in understanding wave phenomena. Notes 1 Quantum Physics F2005 34 Harmonic waves y=A sin(k(x±vt)+*) or y=Asin(kx±+t+*) s in ( x ) 1 0 .5 0 -4 -2 0 2 4 -0 .5 -1 x $ = wavelength; k = 2, $ 2, = 2,# T = period ; + = T Notes 1 Quantum Physics F2005 35 Phase and phase velocity y=Asin(kx±+t) (kx±+t)= phase when change in phase=2, = repeat Phase velocity = speed with which point of constant phase moves in space. vp=+/k=$# Tra ve ling wa ve 1 0.5 0 -4 -2 0 2 4 -0.5 -1 d is ta nc e (m ) Notes 1 Quantum Physics F2005 36 Complex Representation of Travelling Waves Doing arithmetic for waves is frequently easier using 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 1 Quantum Physics F2005 ] 37 Adding waves: same wavelength and direction, 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 : 2 0 2 01 2 02 . = . +. + 2. 01. 02 cos(&1 ! &2 ) and: . 01 sin &1 +. 02 sin &2 tan / = . 01 cos &1 +. 02 cos &2 Notes 1 Quantum Physics F2005 38 Adding like-waves, in words • Amplitude – When two waves are in phase, the resultant amplitude is just the sum of the amplitudes. – When two waves are 1800 out of phase, the resultant is the difference between the two. • Phase – The resultant phase is always between the two component phases. (Halfway when they are equal; closer to the larger wave when they are not.) (see adding_waves.mws) Notes 1 Quantum Physics F2005 39 Another useful example of added waves: diffraction from a slit a s path length difference Let’s take the field at the view screen from an element of length on the slit ds as Esds Ignore effects of distance except in the path length. Notes 1 Quantum Physics F2005 40 Single slit diffraction d+ = Re(. s ei ( kz ( s ) !+t ) ds ) z ( s ) = z0 + s sin ( . (( ) = . s e a/2 i ( kz0 !+t ) iks sin ( e ds ∫ !a / 2 = . se i ( kz0 !+t ) a sin 0 where ak 0 = sin ( 2 2 0 sin 2 2 . = (. s a) 2 0 First zero at 0 =, , so sin ( = 2, $ = ak a 0 Notes 1 Quantum Physics F2005 41 Adding waves: traveling in opposite directions Equal 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 pattern both in time and space separately =Standing wave Notes 1 Quantum Physics F2005 42 Adding waves: different wavelengths . 1 = . 01 cos(k1 x ! +1t ) . 2 = . 01 cos(k2 x ! +2t ) 4 3 2 1 0 1 101 201 -1 -2 -3 -4 5% k diff BEATS 1 . = 2. 01 cos ( k1 + k2 ) x ! (+1 + +2 )t 2 1 × cos ( k1 ! k2 ) x ! (+1 ! +2 ) t 2 )+ )k x! t = . 01 cos kx ! + t cos 2 2 • The resultant wave has a quickly varying part that waves at the average wavelength of the two components. • It also has an envelope part that varies at the difference between the component wavelengths. Notes 1 Quantum Physics F2005 43 Adding waves: group velocity • Note that when we add two waves of differing + and k to one another, the envelope travels with a different speed: monochromatic wave 1: v phase1 = monochromatic wave 2: v phase 2 = beat envelope: vgroup +1 k1 +2 k2 )+ = )k see group_velocity.mws Notes 1 Quantum Physics F2005 44 Adding many waves to make a pulse • In order to make a wave pulse of finite width, we have to add many waves of differing wavelengths in different amounts. • The mathematical approach to finding out how much of each wavelength we need is the Fourier transform: 1 1 f ( x) = 1 ∫ A( k ) cos kxdk + ∫ B(k ) sin kxdk , 0 0 where : 1 A( k ) = ∫ f ( x) cos kxdx !1 1 B( k ) = ∫ f ( x) sin kxdx !1 Notes 1 Quantum Physics F2005 45 The Fourier transform of a Gaussian pulse • We can think of a Gaussian pulse as a localized pulse, whose position we know to a certain accuracy )x=22x. f ( x) = Notes 1 1 2 x 2, Quantum Physics F2005 e ! x 2 / 22 x2 46 Finding the transform I will drop overall multiplicative constants because I am interested in the shape of A(k) 1 A( k ) 3 ∫ e ! x 2 / 22 x2 e ! ikx 1 2 dx = ∫ e ! ax e ! ikx dx !1 !1 where a = 1/ 22 x2 (solving by completing the square:) 1 1 2 A( k ) 3 ∫ e ! ax e ! ikx dx = ∫ e !1 letting 0 = x a ! ! x a! ik 2 a 2 !k 2 / 4a dx !1 ik 2 a , !k 2 / 4a , ! k 22 x2 / 2 1 !k 2 / 4a 1 ! 0 2 A( k ) 3 e e d e e 0 = = ∫ a a !1 a Notes 1 Quantum Physics F2005 47 Transform of a Gaussian pulse: The Heisenberg Uncertainty Principle We can rewrite this in the standard form of a Gaussian in k: A( k ) 3 e ! k 2 / 22 k2 2 k where 2 = 1/ 2 2 x The result then is that 2x2k=1 for a Gaussian pulse. You will find that the product of spatial and wavenumber widths is always equal to or greater than one. Since the deBroglie hypothesis relates wavelength to momentum, p=h/$ we thus conclude that 2x2p>=h/2,. This is a statement of the Heisenberg Uncertainty Principle. Notes 1 Quantum Physics F2005 48 The Heisenberg Uncertainty Principle • This principle states that you cannot know both the position and momentum of a particle simultaneously to arbitrary accuracy. – There are many approaches to this idea. Here are two. • The act of measuring position requires that the particle intact with a probe, which imparts momentum to the particle. • Representing the position of localized wave requires that many wavelengths (momenta) be added together. • The act of measuring position by forcing a particle to pass through an aperture causes the particle wave to diffract. Notes 1 Quantum Physics F2005 49 The Heisenberg Uncertainty Principle • Position and momentum are called conjugate variables and specify the trajectory of a classical particle. We have found that if one wants to specify the position of a Gaussian wave packet, then: )x)p = h • Similarly, angular frequency and time are conjugate variables in wave analysis. (They appear with one another in the phase of a harmonic wave.) )+)t = 1 • Since energy and frequency are related Planck constant we have, for a Gaussian packet: )E )t = h Notes 1 Quantum Physics F2005 50 The next stages • We have seen through experiment that particles behave like waves with wavelength relationship: p=h/$. • The next stage is to figure out the relationship between whatever waves and observable quantities like position, momentum, energy, mass… • The stage after that is to come up with a differential equation that describes the wavy thing and predicts its behavior. • There is still a lot more we can do before actually addressing the wave equation. Notes 1 Quantum Physics F2005 51 References • Krane, Modern Physics, (Wiley, 1996) • Eisberg and Resnick, Quantum Physics of Atoms…, (Wiley, 1985) • French and Taylor, An Introduction to Quantum Physics, (MIT, 1978) • Brehm and Mullin, Introduction to the Structure of Matter, (Wiley, 1989) • Rohlf, Modern Physics from / to 4, (Wiley, 1994) Notes 1 Quantum Physics F2005 52 Addendum – Energy and momentum The notation for energy, momentum, and wavelength in Morrison is somewhat confusing because he does not always clearly distinguish between total relativistic energy (which includes mass energy), and kinetic energy (which does not.) Here goes my version: 1 p2 2 1) Classical kinetic energy-momentum relation: T = m0 v = 2 2m0 2) Relativistic total energy-momentum relationship: E 2 = p 2 c 2 + m02 c 4 with E = T + m0 c 2 . When you want to find the wavelength from classical kinetic energy, use 1 and p = p2 h2 T= = 2m0 2m0 $ 2 ⇒ $= $= 2 m c 1+ 0 E Notes 1 2 = $ : h hc = 2m0T 2m0 c 2T When you want to find the wavelength for a relativistic particle use 2 and p = hc / E h h $ : hc / (T + m0 c 2 ) m c2 0 1+ (T + m0 c 2 ) 2 Quantum Physics F2005 53 Addendum – comparing classical and relativistic formulas for wavelength 1) Classical: $= 2) Relativistic: $ = h hc = 2m0T 2m0 c 2T hc / (T + m0 c 2 ) m c2 0 1! (T + m0 c 2 ) 2 hc = 2 2 2 2 (T + m c ) ! ( m c ) 0 0 To compare the two expressions, let's Taylor expand eq. 2 in $= hc 5 hc 2 T T m0 c 2 2 m0 c 2 + ! 1 1 2 m0 c 2 m c 0 They become the same at small T! Notes 1 = hc = 2 m0 c 2 T m c 2 + 1 ! 1 0 T : m0 c 2 hc 2m0 c 2T Quantum Physics F2005 54 Addendum – comparing classical and relativistic formulas for wavelength 1) Classical: $= h hc = 2m0T 2m0 c 2T hc 2) Relativistic: $ = 2 m0 c 2 T m c 2 + 1 ! 1 0 To find the difference between the two forms, let's keep all the terms in hc $= 2 m0 c 2 T m c 2 + 1 ! 1 0 $Re lativistic 5 $Classical Notes 1 hc 5 m0 c 2 T 2 T + 1 ! 1 +2 m0 c 2 m0 c 2 = T : m0 c 2 hc T 2Tm0 c 2 + 1 2 m c 2 0 T 5 $Classical 1 ! 2 T 4m0 c + 1 2 2 m c 0 1 Quantum Physics F2005 55