PHYS 216 Final, Fall 2013

Physics 216, Final Exam December 16, 2013, 2 hours Solve ten problems out of twelve 1. Check Stokes’ theorem using the function −→ V = ay�x + bx�y (a and b are constants) and the circular path of radius R, centered at the origin in the x − y plane. 2. Show that: (a) If A is a diagonal matrix with all diagonal elements different, and B is a matrix com- muting with A (AB = BA) then B is diagonal. (b) The trace of a product of a symmetric and antisymmetric matrix is zero. 3. Find the eigenvalues and eigenvectors of the matrix A =   1 0 0 0 0 1 0 1 0   . 4. Use the exponential Fourier series to evaluate the expansion of the delta function δ (ϕ1 − ϕ2) . Use the result to verify that 1 2π ∞ � m=−∞ π � −π f (ϕ1) eim(ϕ1−ϕ2)dϕ1 = f (ϕ2) 5. A rectangular pulse is described f (x) = � 1, |x| < a 0, |x| > a . Find the exponential Fourier transform F (t) then use Parseval’s theorem to evaluate the integral ∞ � −∞ sin2 t t2 dt. 6. Let {Qn, n = 0, · · · , ∞} be an orthogonal set of Sturm-Liouville polynomials with respect to the weight function w (x) ∞ � −∞ w (x) Qn (x) Qm (x) dx = 0, n ̸= m where Qn (x) satisfies the differential equation 1 w d dx � wα (x) dQn dx � + λnQn = 0. Show that � Q′ n = dQn dx , n = 1, · · · , ∞ � satisfy the property ∞ � −∞ w (x) α (x) Q′ n (x) Q′ m (x) dx = 0, n ̸= m Hint: Integrate by parts only on one of the dQn dx and use the defining equation for Qn. 7. Using only the generating function e x 2(t− 1 t) = ∞ � n=−∞ Jn (x) tn show that Jn (−x) = (−1)n Jn (x) . 1 8. The quantum mechanical angular momentum operator is given by −→ L = i−→r × −→ ∇. Show that L± = Lx ± iLy = ±e±iϕ � ∂ ∂θ ± i cot θ ∂ ∂ϕ � 9. Consider the Hermite differential operator 2H = − d2 dx2 + x2 and the raising and lowering operators a± = 1 √ 2 � x ∓ d dx � . Show that [a−, a+] = 1, and [N, a±] = ±a± where N = a+a−. Express H in terms of N. Let ψn (x) be the eigenstate satisfying Nψn (x) = nψn (x) . Deduce that ψn (x) is an eigenstate of H with eigenvalue � n + 1 2 � , i.e. Hψn (x) = � n + 1 2 � ψn (x). 10. Find the Laurent series expansion of the function f (z) = 1 (z−1)(z−2) in the annulus 1 < |z| < 2 by direct expansion. (Hint: first express f (z) in terms of 1 z−1 and 1 z−2 and use power series expansion in region of validity). Verify the coefficients a0 and a−1 using the formula an = 1 2πi � C f(z) zn+1 dz where f (z) = ∞ � n=−∞ anzn. 11. Find the residue at z = −i of the function 1 z4−1. Find the integral � C 1 z4−1dz where C is a circle of radius 1 2 centered at z = −i. 12. Using contour integration evaluate the definite integral 2π � 0 1 3+2 cos θdθ. 2