Quantum Physics: Useful Constants, Uncertainty Principle, and Fourier Transform Pairs, Assignments of Quantum Physics

Download Quantum Physics: Useful Constants, Uncertainty Principle, and Fourier Transform Pairs and more Assignments Quantum Physics in PDF only on Docsity! John Venables 5-1675, john.venables@asu.edu Spring 2008 Background Topics continued Module 1, Lectures 4 and 5 PHY 571: Quantum Physics Useful constants in the Bohr atom • Fine structure constant: α = e2/ c 1/137; the value is given by α−1 = 137.0359895(61), dimensionless • (v/c) = Zα/n, is the electron velocity relativistic or not? H-atom, no, inner cores of heavy atoms, yes • Ground state energy E = −½mc2.(Zα/n)2; mc2 = rest energy of electron = 511 keV. So 100 eV e− is not relativistic, 1 MeV is (LEED and HVEM energies) • Radius of lowest Bohr orbit: a0 = ( /mc).Zα; Zα is a number, ( /mc) is the Compton wavelength/2π. The Z=1 value of a0 = 5.2977249(24)*10−11 m, i.e.0.530 Å. This H-atom radius is known as the Bohr radius. Uncertainty Principle: examples and exercises • Draw both types of "wave" experiments, and identify the conjugate variables involved; think of other examples: Problem set #1 Gedanken (thought) expt • Visualize a "particle" as a wave-packet, and draw this, and/or explore examples using the web resources • Think how one may write down the function f(x) describing a wave-packet which contains a spread ∆x and ∆p, or equivalently a spread of wave vectors ∆k • Explore the general relation between f(x) and g(k) in 1- dimension: these are Fourier Transform pairs. Web pages explore projects and other applications Topic 7: Fourier Transform Pairs • Work through the single slit, sometimes call the "top hat" function for f(x). What is the diffraction amplitude g(k)? • Reminders of cos(kx) and sin(kx) in complex number form, expressed in terms of exp(±ikx), i = √−1 • Gaussian function f(x) = exp(−x2/2α) (Gasiorowicz) or f(x) = exp(−(x−x0)2/2a2) (Liboff), what is g(k)? Quantitative measures of ∆x, ∆k, hence ∆x.∆k = #* • Lorentzian lineshape g(ω) = N/(ω2 +a2), what is f(t)?, related to shape of spectral lines (N = normalization). See projects http://venables.asu.edu/quant/fourier.html Two-slit Interference and Convolution • Draw f(x) = two "top hats" width a, separated by distance b: write as a convolution • f(x) = [δ(x-b/2) +δ(x+b/2)] * ftop(x) • convolution = integral [flattice(x')].fshape(x-x')dx' • g(k) = product (glattice(k) . gshape(k)), i.e. if f(x) = f1(x) * f2(x), then g(k) = g1(k).g2(k) g1 = interference pattern g2 = diffraction envelope See Fourier handouts (Cowley), web supplement 2A and Student project