PHYS 216 Exam 2, Fall 2016

Physics 216,Quiz 2 November 29, 2016, 90 minutes Do 8 problems out of 10. 1. A function f (x) = � −x, −π < x < 0 x, 0 < x < π . Is this function even or odd? Find the Fourier series. 2. The Fourier series of the function f (x) of period L = 2 which is equal to x on the interval (−1, 1) is given by f (x) = − 2 π ∞ � n=1 (−1)n 1 n sin nπx Use this expansion and Parseval theorem to show that ∞ � n=1 1 n2 = π2 6 3. Find the Fourier transform F (ω) of the function f (t) = e−a|t| where a > 0. 4. A rectangular pulse is represented by f (t) = � 1, |t| < a 0, |t| > a . Find Fourier transform and use the convolution theorem ∞ � −∞ f (t) f∗ (t) dt = ∞ � −∞ F (ω) F ∗ (ω) dω to evaluate the integral ∞ � 0 sin2 ω ω2 dω. 5. Consider the Sturm-Liouville operator £y = (1 − x2) d2y dx2 − x dy dx + n2y = 0. Find the weight function w (x) for this operator to be self-adjoint. 6. Use the recursion relation of the Legendre polynomials (2n + 1) xPn (x) = (n + 1) Pn+1 (x) + nPn−1 (x) and the values P0 = 1, P1 = x to prove by induction that Pn (−x) = (−1)n Pn (x) . 7. Given a set of orthogonal functions {Qn (x)} satisfying the Sturm-Liouville equation £Qn = 1 w d dx � wdQn dx � + γQn = 0 (1) with weight functions w (x) so that � w (x) Q∗ n (x) Qm (x) dx = 0, if n ̸= m. Show that the functions {Q′ n (x)} are orthogonal functions with weight w (x) α (x) . Hint: Substitute in the integral � w (x) Q∗ n (x) £Qm (x) dx = 0. 8. Express the function f (θ, φ) = sin θ � sin2 θ 2 cos φ + i cos2 θ 2 sin φ � +sin2 θ 2 as a sum of spherical harmonics. Use the values of Y m l (θ, φ) given in Table 15.4 page 760 in Arfken. 1 9. Hermite polynomials are defined by g (x, t) = exp � −t2 + 2tx � = ∞ � n=0 tnHn (x) . Use this to prove that Hn (−x) = (−1)n Hn (x) . Multiply g (x, t) by e− x2 2 and integrate to show that ∞ � 0 e− x2 2 Hn (x) dx = 0 if n = 2m + 1. 10. Use the relation xLk n = (2n + k + 1) Lk n − (n + k) Lk n−1 − (n + 1) Lk n+1 to show that ∞ � 0 e−xxk+1Lk n (x) Lk n (x) dx = (n + k)! n! (2n + k + 1) . 2