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Prof. Dr. S. Dietrich Dr. M. Gross ([email protected]) MSc. G. Zarubin Advanced Quantum Theory winter semester 2016/17 Problem sheet no. 2 (http://www.is.mpg.de/dietrich/teaching/AQT) 31 October 2016 4. Compton effect An incident photon with wave vector k0 is scattered from a particle with rest mass m0 and initial momentum p0 = 0. The scattered photon has the wave vector k. Based on the conservation of the relativistic energy and momentum, determine the shift λ − λ0 of the wave length of the scattered photon as a function of the scattering angle θ between k and k0 . (Remark: Special relativity yields the relation E 2 = c2 p2 + m2 c4 between the energy E and the momentum p of an object of rest mass m.) 5. Bohr model Consider the motion according to classical mechanics of an electron of charge −e in the electrostatic field of a proton of charge +e. (a) Determine the effective potential for the motion of the electron around the nucleus as a function of the radial distance. (b) According to the Bohr model, electrons move on circular orbits and the angular momentum L can assume the values L = n~, n ∈ {1, 2, . . .}. Determine the possible energies En , orbital radii rn , and velocity ratios vn /c, where c is the speed of light. (c) Bohr assumed that only radiation of frequency ω(n1 , n2 ) = En1 − En2 , ~ n1 > n2 (1) can be emitted. Compare ω(n + 1, n) for n → ∞ with the orbital frequency which results from classical mechanics using the results of part (b). (d) For velocities v c, the power radiated by an accelerated charge is given by the classical expression e2 P = v̇2 (2) 6π0 c3 in SI units. For the orbit corresponding to energy E1 , estimate the time during which the electron (considered to be a classical particle) would crash into the nucleus. please turn over 1 6. Fourier transform Given a function f : R → C, its Fourier transform f¯, provided it exists, is defined by Z ∞ dx ¯ √ f (x) exp(−iqx). f (q) := (3) 2π −∞ The following relations hold: Z ∞ dq √ f¯(q) exp(iqx) f (x) = 2π −∞ Z ∞ Z ∞ 2 dx |f (x)| = dq |f¯(q)|2 −∞ Consider the Gaussian (inverse Fourier transform) (4) (Parseval’s theorem). (5) −∞ x2 f (x) = √ exp − 2 . 2σ 2πσ 2 Calculate the Fourier transform f¯, plot f (x) and f¯(q), and check Eqns. (4) and (5). 1 2 (6)