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Developing Wave Equations

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 E+R Ch 3+5 and Griffiths Ch 1 try to show why choice is reasonable……..

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Developing Wave Equations

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  1. 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 • E+R Ch 3+5 and Griffiths Ch 1 try to show why choice is reasonable…….. • Start from 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 e or p gives .025 fm (size of p: 1 F) P460 - dev. wave eqn.

  2. 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 P460 - dev. wave eqn.

  3. 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 momentum space are conjugate • the spatial function implies “something” about the function in momentum space P460 - dev. wave eqn.

  4. 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 P460 - dev. wave eqn.

  5. 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/2teiM (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 P460 - dev. wave eqn.

  6. Bohr Model • From discrete atomic spectrum, realized something was quantized. And the bound electron was not continuously radiateing (as classical physics) • Bohr model is wrong but gives about right energy levels and approximate atomic radii. easier than trying to solve Scrod. Eg…. • 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 • a0 is the Bohr radius = .053 nm = ~atomic size P460 - dev. wave eqn.

  7. Bohr Model • En = K + V = E0/n2 where E0 = -13.6 eV for H . = -m/2*(e*e/4pehbar)2 • Bohr model quantizes energy and radius and 1D angular momentum. Reality has energy, and 2D angular momentum (one component and absolute magnitude) • for transitions • easily extend Bohr model. He+ atom, Z=2 and En = 4*(-13.6 eV)/n2 (have (zZ)2 for 2 charges) • reduced mass m =mamb/(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 P460 - dev. wave eqn.

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