Pre-quantum mechanics Modern Physics Historical “problems” were resolved by modern treatments which lead to the development of quantum mechanics need special relativity EM radiation is transmitted by massless photons which have energy and momentum. Mathematically use wave functions (wavelength, frequency, amplitude, phases) to describe “particles” with non-zero mass have E and P and use wave functions to describe SAME

Blackbody Radiation Late 19th Century: try to derive Wien and Stefan-Boltzman Laws and shape of observed light spectra used Statistical Mechanics (we’ll do later) to determine relative probability for any wavelength l need::number of states (“nodes”) for any l - energy of any state - probability versus energy the number of states = number of standing waves = N(l)dl = 8pV/l4 dl with V = volume Classical (that is wrong) assigned each node the same energy E = kt and same relative probability this gives energy density u(l) = 8p/l4*kT u  infinity as wavelength  0 u wavelength

Blackbody Radiation II Modern, Planck, correct: E = hn = hc/ l Energy and frequency are the same. Didn’t quite realize photons were a particle From stat. Mech -- higher energy nodes/states should have smaller probability try 1: Prob = exp(-hn/kt) - wrong try 2: Prob(E) = 1/(exp(hn/kt) - 1) did work will do this later. Planck’s reasoning was obscure but did get correct answer…..Bose had more complete understanding of statistics Gives u(l) = 8p/ l4 * hc/ l * 1/(exp(hc/lkt) - 1) Agrees with experimental observations u Higher Temp wavelength

Photoelectric effect f Ee f Eg Photon absorbed by electron in a solid (usually metal or semiconductor as “easier” to free the electron) . Momentum conserved by lattice if Eg > f electron emitted with Ee = Eg - f (f is work function) Example = 4.5 eV. What is largest wavelength (that is smallest energy) which will produce a photoelectron? Eg = f or l = hc/f = 1240 eV*nm/4.5 eV = 270 nm f Ee in solid Conduction band Ee f Eg

Compton Effect g g g e e e e g + e  g’+ e’ electron is quasifree. What are outgoing energies and angles? Conservation of E and p  photon is a particle Einitial = Efinal or E + me = E’ + Ee’ x: py = p(e’)cosf + p( g’) cosq y: 0 = p(e’)sinf - p (g’)sinq 4 unknowns (2 angles, 2 energies) and 3 eqns. Can relate any 2 quantities 1/Eg’ - 1/Eg = (1-cosq)/me c2 Feymann diag g g g q f e e e e

Compton Effect 1/Eg’ - 1/Eg = (1-cosq)/mec2 g’ Z g e if .66 MeV gamma rays are Comptoned scattered by 60 degrees, what are the outgoing energies of the photon and electron? 1/Eg’ - 1/Eg = (1-cosq)/mec2 1/Egamma’ = 1/.66 MeV + (1-0.5)/.511 or Egamma’ = 0.4 MeV and Te = kinetic energy = .66-.40 = .26 MeV g’ Z g q f e

Brehmstrahlung + X-ray Production e+Z  g’+ e’+Z electron is accelerated in atomic electric field and emits a photon. Conservation of E and p. atom has momentum but Eatom =p2/2/Matom. And so can ignore E of atom. Einitial = Efinal or Egamma = E(e)- E(e’) Ee’ will depend on angle  spectrum E+Ze+Z+ g Brem e +g  e + g Compton e+Z+g  e+Z photoelectric Z+g  Z+e+e pair prod energy e e e z e g g Z brehm g

Particle antiparticle Pair Production A photon can convert its energy to a particle antiparticle pair. To conserve E and p another particle (atom, electron) has to be involved. Pair is “usually” electron+positron and Ephoton = Ee+Epos > 2me > 1 MeV and atom conserves momentum g + Z  e+ + e- + Z can then annihilate electron-positron pair. Need 2 photons to conserve momentum e+ + e-  g + g ALSO: Mu-mu pairs p+cosmic MWB Z Particle antiparticle Usually electron positron g

Electron Cross section Brehmstrahlung becomes more important with higher energy or higher Z from Rev. of Particle Properties

Photon Cross Section vs E

Rutherford Scattering off Nuclei (briefly in Chap 11) partially SKIP First modern. Gave charge distribution in atoms. Needed l << atomic size. For 1 MeV alpha, p=87 MeV, l=h/p = 10-12cm kinematics: if Mtarget >> Malpha then little energy transfer but large possible angle change. Ex: what is the maximum kinetic energy of Au A=Z+N=197 after collision with T=8Mev alpha? Ptarget = Pin+Precoil ~ 2Pin (at 180 degrees) Ktarget = (2Pin)2/2/Mtarget = 4*2*Ma*Ka/2/Mau = 4*4/197*8MeV=.7MeV recoil A target a in

Rutherford Scattering II Assume nucleus has infinite mass. Conserve Ea = Ta +2eZe/(4per) conserve angular momentum La = mvr = mv(at infinity)b E+R does arithmetic gives cot(q/2) = 2b/D where D= zZe*e/(4pe Ka) is the classical distance of closest approach for b=0 don’t “pick” b but have all ranges 0<b<atom size all alphas need to go somewhere and the cross section is related to the area ds = 2pbdb (plus some trigonometry) gives ds/dW =D2/16/sin4q/2 b=impact parameter a q b Z

Rutherford Scattering III already went over kinematics Rutherford scattering can either be off a heavier object (nuclei)  change in angle but little energy loss  “multiple scattering” or off light target (electrons) where can transfer energy but little angular change (energy loss due to ionization, also produces “delta rays” which are just more energetic electrons).

Particles as Waves EM waves (Maxwell Eqs) are composed of individual (massless) particles - photons - with E=hf and p = h/l and E = pc observed that electrons scattered off of crystals had a diffraction pattern. Readily understood if “matter” particles (with mass) have the same relation between wavelength and momentum as photons Bragg condition gives constructive interference 1924 DeBroglie hypothesis: “particles” (those with mass as photon also a particle…) have wavelength l = h/p What is wavelength of K = 5 MeV proton ? Non-rel p=sqrt(2mK) = sqrt(2*938*5)=97 MeV/c l=hc/pc = 1240 ev*nm/97 MeV = 13 Fermi p=50 GeV/c (electron or proton) gives .025 fm (size of proton: 1 F)

Bohr Model I From discrete atomic spectrum, realized something was quantized. And the bound electron was not continuously radiating (as classical physics) Bohr model is wrong but gives about right energy levels and approximate atomic radii. easier than trying to solve Schrodinger Equation…. Quantized angular momentum (sort or right, sort of wrong) L= mvr = n*hbar n=1,2,3... (no n=0) kinetic and potential Energy related by K = |V|/2 (virial theorem) gives radius is quantized a0 is the Bohr radius = .053 nm = ~atomic size

Bohr Model II En = K + V = E0/n2 where E0 = -13.6 eV for H E0 = -m/2*(e*e/4pehbar)2 = -m/2 a2 where a is the fine structure constant (measure of the strength of the EM force relative to hbar*c= 197 ev nm) Bohr model quantizes energy and radius and 1D angular momentum. Reality quantizes energy, and 2D angular momentum (one component and absolute magnitude) for transitions

p m 60 MeV 1.6 keV bb quarks q=1/3 2.5 GeV .9 keV Bohr Model III E0 = -m/2*(e*e/4pehbar)2 = -m/2 a2 (H) easily extend Bohr model. He+ atom, Z=2 and En = 4*(-13.6 eV)/n2 (have (zZ)2 for 2 charges) reduced mass. 2 partlces (a and b) m =ma*mb/(ma+mb) if other masses En = m/(me )*E0(zZ/n)2 Atom mass E(n=1) e p .9995me -13.6 eV m p 94 MeV 2.6 keV p m 60 MeV 1.6 keV bb quarks q=1/3 2.5 GeV .9 keV

Developing Wave Equations Need wave equation and wave function for particles. Schrodinger, Klein-Gordon, Dirac not derived. Instead forms were guessed at, then solved, and found where applicable So Dirac equation applicable for spin 1/2 relativistic particles Start from 1924 DeBroglie hypothesis: “particles” (those with mass as photon also a particle…) have wavelength l = h/p

vel=<x(t2)>-<x(t1)> Wave Functions Particle wave functions are similar to amplitudes for EM waves…gives interference (which was used to discover wave properties of electrons) probability to observe =|wave amplitude|2=|y(x,t)|2 particles are now described by wave packets if y = A+B then |y|2 = |A|2 + |B|2 + AB* + A*B giving interference. Also leads to indistinguishibility of identical particles t1 t2 merge vel=<x(t2)>-<x(t1)> (t2-t1) Can’t tell apart

Wave Functions Describe particles with wave functions y(x) = S ansin(knx) Fourier series (for example) Fourier transforms go from x-space to k-space where k=wave number= 2p/l. Or p=hbar*k and Fourier transforms go from x-space to p-space position space and l/k/momentum space are conjugate the spatial function implies “something” about the function in momentum space

Wave Functions (time) If a wave is moving in the x-direction (or -x) with wave number k can have kx-wt = constant gives motion of wave packet the sin/cos often used for a bound state while the exponential for a right or left traveling wave

Wave Functions (time) Can redo Transform from wave number space (momentum space) to position space normalization factors 2p float around in Fourier transforms the A(k) are the amplitudes and their squares give the relative probability to have wavenumber k (think of Fourier series) could be A(k,t) though mostly not in our book as different k have different velocities, such a wave packet will disperse in time. See sect. 2-2. Not really 460 concern…..

Heisenberg Uncertainty Relationships Momentum and position are conjugate. The uncertainty on one (a “measurement”) is related to the uncertainty on the other. Can’t determine both at once with 0 errors p = hbar k electrons confined to nucleus. What is maximum kinetic energy? Dx = 10 fm Dpx = hbarc/(2c Dx) = 197 MeV*fm/(2c*10 fm) = 10 MeV/c while <px> = 0 Ee=sqrt(p*p+m*m) =sqrt(10*10+.5*.5) = 10 MeV electron can’t be confined (levels~1 MeV) proton Kp = .05 MeV….can be confined

Heisenberg Uncertainty Relationships Time and frequency are also conjugate. As E=hf leads to another “uncertainty” relation atom in an excited state with lifetime t = 10-8 s |y(t)|2 = e-t/t as probability decreases y(t) = e-t/2teiMt (see later that M = Mass/energy) Dt ~ t DE = hDn Dn > 1/(4p10-8) > 8*106 s-1 Dn is called the “width” or and can be used to determine ths mass of quickly decaying particles if stable system no interactions/transitions/decays

Schrodinger Wave Equation Schrodinger equation is the first (and easiest) works for non-relativistic spin-less particles (spin added ad-hoc) guess at form: conserve energy, well-behaved, predictive, consistent with l=h/p free particle waves

Schrodinger Wave Equation kinetic + potential = “total” energy K + U = E with operator form for momentum and K gives (Hamiltonian) Giving 1D time-dependent SE For 3D:

Operators (in Ch 5) Operators transform one function to another. Some operators have eigenvalues and eigenfunctions Only some functions are eigenfunctions. Only some values are eigenvalues In x-space or t-space let p or E be represented by the operator whose eigenvalues are p or E

Continuous function look at “matrix” elements Operators Hermitian operators have real eigenvalues and can be diagonalized by a unitary transformation easy to see/prove for matrices Continuous function look at “matrix” elements

Operators By parts Example 1 O = d/dx Usually need function to be well-behaved at boundary (in this case infinity).

Commuting Operators mostly skip Some operators commute, some don’t (Abelian and non-Abelian) if commute [O,P]=0 then can both be diagonalize (have same eigenfunction) conjugate quantities (e.g. position and momentum) can’t be both diagonalized (same as Heisenberg uncertainty) (sometimes)

Interpret wave function as probability amplitude for being in interval dx

No forces. V=0 solve Schr. Eq Example No forces. V=0 solve Schr. Eq Find average values

Momentum vs. Position space Mostly SKIP Can solve SE (find eigenvalues and functions, make linear series) in either position or momentum space Fourier transforms allow you to go back and forth - pick whichever is easiest

Momentum vs. Position space example SKIP Expectation value of momentum in momentum space integrate by parts and flip integrals

Probability Current SKIP Define probability density and probability current. Good for V real gives conservation of “probability” (think of a number of particles, charge). Probability can move to a different x V imaginary gives P decreasing with time (absorption model)

Probability and Current Definitions SKIP With V real Use S.E. to substitute for substitute into integral and evaluate The wave function must go to 0 at infinity and so this is equal 0

Probability Current Example SKIP Supposition of 2 plane waves (right-going and left-going)