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Course Mathematical Methods of Physics COURSE MATHEMATICAL METHODS OF PHYSICS. R.J.Kooman University of Leiden spring 2007. 1 TABLE OF CONTENTS. Chapter I. Ordinary linear di®erential equations. 1.1. Linear ¯rst order di®erential equations. 1.2. Linear di®erential equations with constant coe±cients. 1.3. The determinant of Wronski. 1.4. Frobenius' method: power series solutions. Reduction of the order. 1.5. Singular points. 1.6. The hypergeometric di®erential equation. 1.7. The confluent hypergeometric di®erential equation. 1.8. The adjoint di®erential operator. 1.9. Integral solutions of di®erential equations. 1.10. Asymptotic expansions; the method of steepest descent. Chapter II. Hilbert spaces, Fourier series and operators. 2.1. Banach spaces and Hilbert spaces. 2.2. Orthogonal sets and Fourier series. 2.3. Classical Fourier series. 2.4. Bounded operators. 2.5. Compact operators. 2.6. The spectral theorem for compact normal operators. 2.7. Distributions. Chapter III. Integral equations. 3.1 Volterra integral equations of the second kind. 3.2 Fredholm integral equations of the second kind. Integral equations with separated kernel. 3.3. Solution by an integral transform. Chapter IV. Sturm-Liouville theory. 4.1 Unbounded operators. 4.2 Sturm-Liouville systems. 4.3 Green's functions for Sturm-Liouville operators. 4.4 Asymptotic behaviour of the solutions of Sturm-Liouville problems. 4.5 Application: separation of variables and PDE's Chapter V. Partial di®erential equations. 5.1 General concepts. 5.2 Quasilinear PDE's of ¯rst order. Characteristics. 5.3 Linear PDE's of second order. Classi¯cation. 5.4 The di®usion equation. 5.5 The elliptic case: the equation of Laplace; harmonic functions. 5.6 The equation of Helmholtz. 5.7. The hyperbolic case: the wave equation in one and several dimensions. Chapter VI. Tensor algebra. 6.1. The dual of a vector space. 6.2. Tensors and tensor products. Tensor product of vector spaces. 6.3. Symmetric and antisymmetric tensors; the wedge product. 6.4. Cartesian tensors. 6.5. Application: isotropic elastic bodies. 6.6. The Hodge star operator. 2 Chapter VII. Tensor analysis and di®erential geometry. 7.1. Tensor analysis in Euclidian space. Tangent and cotangent space. The metric tensor. 7.2. The covariant derivative. Parallel displacement. 7.3. Divergence, curl and Laplacian in arbitrary coordinates. 7.4. Di®erentiable manifolds. 7.5. Integration of p-forms. 7.6. The exterior derivative; Poincar¶e'slemma and Stokes' theorem. 7.7. The Lie derivative. Interior product of a p-form and a vector. Time derivative of integrals. Divergence and flux of a vector ¯eld. 7.8. Riemannian and pseudo-Riemannian manifolds. Isometries and Killing vector ¯elds. 7.9. Connections and geodesics. 7.10. The Riemann curvature tensor. Geodesic deviation. 7.11. General relativity. 7.12. Lorentz vectors and tensors. 7.13. The Hodge star operator; the Maxwell equations. Chapter VIII. Groups and representations. 8.1. Groups: general concepts and de¯nitions (subgroups, homomorphisms, quotient groups, direct product). 8.2. Representations of ¯nite groups. Characters. Tensor product of representations. Representation of a subgroup. 8.3. Physical applications: dipole moments, degeneracy of energy states, normal modes, vibrational modes of a water molecule. Chapter IX. Lie groups and Lie algebras. 9.1. Matrix groups. In¯nitesimal transformations. 9.2. Lie groups. The Lie algebra of a Lie group. The exponential map. 9.3. The structure of Lie algebras. The adjoint representation. The Killing form. Compact Lie algebras. 9.4. Representation of compact Lie groups. 9.5. Representation of Lie algebras. Casimir operators. SO(3) and SO(3; 1). Chapter X. Calculus of variations. 10.1 The functional derivative. 10.2 The Euler Lagrange equation. 10.3. Lagrange multiplicators. 10.4. The case of free boundary conditions. 10.5. Geodesics. 10.6. Eigenvalue problems. 10.7. Noether's theorem. 3 MATHEMATICAL METHODS OF PHYSICS. PROBLEM SET. spring 2007. R.J.Kooman, University of Leiden. 4 Chapter I: Ordinary linear di®erential equations. 1a. Use Frobenius' method to ¯nd two linearly independent solutions of the di®erential equation y00(z)+ y(z) = 0. b. Show that the power series converge for all z 2 C. c. Why are the two solutions you found linearly independent? 2a. Find two linearly independent solutions of the Airy equation y00(z) + zy(z) = 0. b. Show that the power series converge for all z 2 C. 3. Show that y(z) is a solution of the Hermite equation y00(z) ¡ 2zy0(z) + ¸y(z) = 0 if and only if 2 w(z) = y(z)e¡z =2 is a solution of the equation w00(z) + (¸ + 1 ¡ z2)w(z) = 0. 4a. Solve the di®erential equation using the substitution z = es: 1 1 4y00(z) + y0(z) ¡ y(z) = 0: z z2 b. Give the singular points in C [ f1g. Which singular points are regular? 5a. Give two linearly independent solutions of Laguerre's equation about z = 0: zy00(z) + (1 ¡ z)y0(z) + ¸y(z) = 0: b. For which values of ¸ is there a polynomial solution? c. Show that the power series converge for all z 2 C. d. Is z = 1 an ordinary or a singular point of the DE? If singular, is it regular or irregular? 6a. Show that substitution of x = cos θ transforms Chebyshev's equation (1 ¡ x2)y00(x) ¡ xy0(x) + ¸y(x) = 0 into the constant coe±cient equation w00 + ¸w = 0. b. Show that for ¸ = N 2 the DE has a polynomial solution of degree N. c. Show that the general solution of the Chebyshev equation for ¸ = N 2 is y(x) = A cos(N arccos x)+ B sin(N arccos x). For which A; B is the solution a polynomial? 5 7a. Give two linearly independent power series solutions about z = 0 of the DE 2z n(n + 1) y00(z) ¡ y0(z) ¡ y(z) = 0 1 ¡ z2 (1 ¡ z2)z2 where n 2 Z¸0. b. For what z 2 C do the power series converge? c. Give the singular points of the DE in C [ f1g. Are they regular or not? 8. Give the singular points of the Bessel equation x2y00(x) + xy0(x) + (x2 ¡ º2)y(x) = 0 and ¯nd out if they are regular or not. 9. Consider the DE x2y00(x) ¡ 3xy0(x) + 4y(x) = 0: a. Use Frobenius' method to ¯nd a solution of the DE. Call it y1(x) b. Give a solution y2 that is linearly independent from y1. Use reduction of the order. 10. Prove the following properties of the Bessel functions Jº (using the power series representation): d a. (xº J (x)) = xº J (x): dx º º¡1 d b. (x¡º J (x)) = ¡x¡º J (x): dx º º+1 n c. J¡n(x) = (¡1) Jn(x) for n 2 Z. 11. Let y00(z) + P (z)y0(z) + Q(z)y(z) = 0 be a di®erential equation with three regular singular points in z = 0; 1 en z = 1 (and no other singular points). p(z) q(z) a. Show that P (z) = , Q(z) = where p(z); q(z) are polynomials of degree 1 and z(z ¡ 1) z2(z ¡ 1)2 2, respectively. b. Show that the are numbers ®; ¯ 2 C such that u(z) = z®(z ¡ 1)¯y(z) is a solution of the hyperge- ometric di®erential equation z(1 ¡ z)u00(z) + (b ¡ (a + c + 1)z)u0(z) ¡ acu(z) = 0: (y) c. Show that (y) has, besides F (a; c; b; z), also a solution z1¡bF (a ¡ b + 1; c ¡ b + 1; 2 ¡ b; z). Why are these solutions linearly independent if b 6= 1? What happens if b = 1? Z ¼=2 dθ ¼ 1 1 d. Show that the elliptic function of the ¯rst kind K(z) = p is equal to F ( ; ; 1; z2). 2 0 1 ¡ z2 sin θ 2 2 2 r r 2 sin x 2 cos x 12a. Show that J (x) = p en J (x) = p . 1=2 ¼ x ¡1=2 ¼ x 6 b. Write J3=2(x) and J¡3=2(x) in terms of x; sin x and cos x. (Use problem 10). c. Show that for n = 0; 1; 2;::: there exist polynomials Pn and Qn or degrees n and n ¡ 1 resp. such that ¡n¡1=2 ¡n¡1=2 Jn+1=2(x) = x (Pn(x) cos x+Qn(x) sin x);J¡n¡1=2(x) = x (Pn(x) sin x¡Qn(x) cos x): r ¼ The spherical Bessel functions m = 0; 1; 2;::: are de¯ned as jm(x) = Jm+1=2(x) and r 2x ¼ n (x) = J (x). m 2x ¡m¡1=2 d. Show that jm(x) and nm(x) are solutions of the DE x2y00(x) + 2xy0(x) + (x2 ¡ m(m + 1))y(x) = 0: Z ¼ ½ ¡ ¢ 1 1 ¢ n for n 2 Z, n even 13a. Show that cosn θdθ = 2n n=2 2¼ ¡¼ 0 for n odd. Z ¼ 1 iz cos θ b. Prove that e dθ = J0(z). 2¼ ¡¼ (Hint: give a power series for the integrand and use a.) p 14. (zeroes of the Bessel function.) Let ym(x) = xJm(x) for m 2 R. a. Show that µ ¶ 1=4 ¡ m2 y00 (x) + 1 + y (x): m x2 m b. Show that y1=2(x) = a sin x for some a > 0. c. Use (b) and theorem 1.5 to prove that Jm(x) has for jmj · 1=2 in¯nitely many positive (and also in¯nitely many negative) zeroes. d. Use problem 10 to show that Jm(x) has in¯nitely many zeroes for all real values of m.
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