Sheet 06.6: Curvilinear Integration, Matrices I

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1 Fakultät für Physik R: Rechenmethoden für Physiker, WiSe 5/6 Doent: Jan von Delft Übungen: Benedikt Bruognolo, Dennis Schimmel, Frauke Schwar, Lukas Weidinger Sheet 6.6: Curvilinear Integration, Matrices I Posted: Friday,..5 Due: Friday, 7..5, 3: Central Tutorial:..5 [](E/M/A) means: problem counts points and is easy/medium hard/advanced Example Problem : Exponential integrals of the form dx x n e ax [] Points: (a)[](m); (b)[](m) Calculate the integral I n (a) = dx x n e ax (with a R, a >, n N) using two different methods: (a) repeated partial integration, and (b) repeated differentiation: (a) Calculate I, I and I by using partial integration where necessary. Then use partial integration to show that I n (a) = n a I n (a) for all n. Use this relation iteratively to determine I n (a) as a function of a and n. (b) Show that taking n derivatives of I (a) with respect to a yields I n (a) = ( ) n d n I (a). Then da n calculate these derivatives for a few small values of n. From the emerging pattern, deduce the general formula for I n (a). Example Problem : Elliptical polar coordinates: area of an ellipse [] Points: (a)[](m); (b)[](e) (a) Let f : R R be a function that depends on the coordinates x and y only in the combination (x/a) + (y/b). Show that a two-dimensional area integral of f over R can be written as ˆ I = dxdy f ( (x/a) + (y/b) ) ˆ = πab dµ µ f(µ), R by transforming from cartesian coordinates to elliptical polar coordinates, defined as follows: x = µa cos φ, y = µb sin φ, µ = (x/a) + (y/b), φ = arctan(ay/bx). Hint: For a = b =, they correspond to polar coordinates. For a b, the local basis is not orthogonal! (b) Using a suitable function f, calculate the area of an ellipse with semi-axes a and b, with a and b defined by (x/a) + (y/b). Example Problem 3: Jacobian determinant for cylindrical coordinates [] Points: [](E)

2 Calculate the Jacobian determinant coordinates. (x,x,x 3 ) (ρ,φ,) for the transformation from Cartesian to cylindrical Example Problem : Volume and moment of inertia of a conical frustum [] Points: (a)[](e); (b)[](e) The moment of inertia of a rigid body with respect to a rotation axis is defined as I = V dv ρ (r)d (r), where ρ (r) is the density at the point r, and d (r) the perpendicular distance from r to the rotation axis. Let F = {r R 3 H H, x + y a} be a homogeneous conical frustum centered on the -axis. Calculate, using cylindrical coordinates, (a) its volume V F (a), and (b) its moment of inertia I F (a) with respect to the axis, as functions of the dimensionless, positive scale factor a, the length parameter H, and the mass m of the frustum. [Check your results: V F (3) = πh 3, I F () = 93π 7 MH.] e H H Example Problem 5: Volume of a buoy [] Points: (a)[](e); (b)[](m) Consider a buoy, with its tip at the origin, bounded from above by a sphere centered the origin, with x + y + R, and from below by a cone with tip at the origin, with a (x + y ). (a) Show that the half angle at the tip of the cone is given by θ = arctan(/a). θ R (b) Use spherical coordinates to calculate the volume V (R, a) of the buoy as a function of R and a. [Check your results: V (, 3) = (6π/3)( 3/).] Example Problem 6: Wave functions of the two-dimensional harmonic oscillator [] Points: (a)[,5](e); (b)[,5](e); (c)[3](m) The quantum mechanical treatment of a two-dimensional harmonic oscillator leads to so-called wave functions, Ψ nm : R C, r Ψ nm (r), with n N, m Z, m = n, n +,..., n, n, which have a factoried form when written in terms of polar coordinates, Ψ nm (r) = R n m (ρ)z m (φ), with Z m (φ) = π e imφ. The wave functions satisfy the following orthogonality relation : O mm nn ˆR da Ψ nm (r)ψ n m (r) = δ nn δ mm. Verify these for n =, and, where the radial wave functions have the form: R (ρ) = e ρ /, R (ρ) = ρe ρ /, R (ρ) = ρ e ρ /, R (ρ) = [ρ ]e ρ /. Proceed as follows. Due to the product form of the wave function Ψ, each area integral separates into two factors that can be calculated separately, Onn mm = P m m mm nn P, where P is a radial integral and P an angular integral.

3 (a) Find general expressions for P and P as integrals over R or Z functions, respectively. (b) Compute the angular integral P mm for arbitrary values of m and m. (c) Now compute those radial integrals that arise in combination with P, namely P, P, P, P and P. Hint: The Euler identity, e iπk = if k Z, is useful for evaluating the angular integral, and dx x n e x = n! for the radial integrals. Background information: The functions Ψ nm (r) are the eigenfunctions of a quantum mechanical particle in a two-dimensional harmonic potential, V (r) r, where n and m are quantum numbers that specify a particular eigenstate. A particle in this state is found with probability Ψ nm (r) da within the area element da at position r. The total probability for being found anywhere in R equals, hence the normaliation integral yields Onn mm = for every eigenfunction Ψ nm (r). The fact that the area integral of two eigenfunctions vanishes if their quantum numbers are not equal, reflects the fact that the eigenfunctions form an orthonormal basis in the space of square-integrable complex functions on R. Example Problem 7: Matrix multiplication [] Points: [](E) Compute all possible products of two of the following matrices: ( ) ( 3 ) 3 P =, Q = 5, R = 6 ( 3 6 ). Example Problem 8: Spin matrices [3] Points: (a)[,5](e); (b)[,5](e); (c)[](e) The following matrices are used for the description for quantum mechanical particles with spin : S x = (a) Compute S = S x + S y + S. ( ), S y = ( ) i, S i = ( ). (b) Compute the commutators [S x, S y ], [S y, S ] and [S, S x ], and express each result through one of the matrices given above. Hint: [A, B] = AB BA. (c) The results from (b) can be compactly summaried in an equation of the form [S i, S j ] = a ijk S k for {i, j, k} {x, y, } (with summation over k). Find the tensor a ijk. Example Problem 9: Matrix multiplication [] Points: (a)[](m); (b)[](m) Let A and B be N N matrices with matrix elements a i j = A j δ i m and b i j = B i δ i j. Remark: Since the indices i and j are specified on the left, they are not summed over on the right even though the index i appears twice in b i j on the right. (a) For N = 3 and m =, write these matrices explicitly in the usual matrix representation and calculate the matrix product AB explicitly. 3

4 (b) Calculate the product AB for arbitrary N N and m N. [Total Points for Example Problems: ] Homework Problem : Gaussian integrals [5] Points: (a)[](m); (b)[](m); (c)[](m); (d)[](m); (e)[](m) (a) Show that the two-dimensional Gaussian integral I = dxdy +y ) e (x has the value I = π. Hint: Use polar coordinates; the radial integral can be solved by substitution. (b) Now calculate the one-dimensional Gaussian integral I (a) = dx e ax (here a R, a > ). Hint: I = [I ()]. Explain why! and below, Determine the value of the x n Gaussian integral, I n (a) = dx xn e ax (with n N), using two different methods: (c) repeated partial integration, and (d) repeated differentiation: (c) Calculate I, I and I by using partial integration where required. Then use partial integration to show that I n (a) = n I a n (a) holds for all n. Use this relation iteratively to determine I n (a) as a function of a and n. (d) Show that taking n derivatives of I (a) with respect to a times yields I n (a) = ( ) n d n I (a) da n. Then calculate these derivatives for a few small values of n. From the emerging pattern, deduce the general formula for I n (a). (e) Calculate the one-dimensional Gaussian integral with a linear term in the exponent: I (a, b) = dx e ax +bx (with a, b R, a > ). Hint: Complete the square: Write the exponent in the form ax + bx = a(x C) + D and then substitute x = x C. Homework Problem : Volume calculations with elliptical polar coordinates [] Points: (a)[](m); (b)[](m); (c)[](a,bonus) In the following, use elliptical coordinates in two dimensions, defined as x = µa cos φ, y = µb sin φ, with a, b R, a > b >. Calculate the volume V (a, b, c) of the following objects T, E and C, as a function of the length parameters a, b and c. (a) T is a tent with an elliptical base with semi-axes a and b. The height of its roof is described by the height function h T (x, y) = c [ c (x/a) (y/b) ]. b (b) E is an ellipsoid with semi-axes a, b and c, defined by (x/a) +(y/b) + (/c). (c) C is a cone with height c and an elliptical base with semi-axes a and b. All cross sections parallel to the base are elliptical, too. Hint: Augment the elliptical coordinates by another coordinate, (analagously to passing from polar to cylindrical coordinates). a x c a x b y y

5 [Check your answers: if a = /π, b =, c = 3, then (a) V T = 3, (b) V E = 8, (c) V C =.] Homework Problem 3: Jacobian determinant for spherical coordinates [] Points: [](E) Calculate the Jacobian determinant (x,x,x 3 ) (r,θ,φ) for the transformation from Cartesian to spherical coordinates. Homework Problem : Volume and moment of inertia: cylindrical coordinates [3] Points: (a)[](m); (b)[](m); (c)[](a,bonus) Consider the homogeneous rigid bodies C, P and B specified below, each with density ρ. Use cylindrical coordinates to compute for each its volume V (a) and moment of inertia I(a) = ρ V dv d with respect to the symmetry axis, as functions of the dimensionless, positive scale factor a, the length parameter R, and the mass of the body, M. (a) C is a hollow cylinder with inner radius R, outer radius ar, and height R. [Check your results: V C () = 6πR 3, I C () = 5 6 MR.] (b) P is a paraboloid with height h = ar and curvature /R, defined by P = {r R 3 h, (x + y )/R } [Check your results: V P () = πr 3, I P () = 3 MR.] (c) B is the bowl obtained by taking a sphere, S = {r R 3 x + y + ( ar) a R }, with radius ar, centered on the point P : (,, ar) T, and cutting from it a cone, C = {r R 3 (x + y ) (a ), a }, symmetric about the axis, with apex at the origin. [Check your results: V B ( 3) = 6 9 πr3, I B ( 3 ) = 5 MR. What do you get for a =? Why?] x Hint: Firtst find, for given, the radial integration boundaries, ρ () ρ ρ (), then the integration boundaries, m. What do you find for m, the maximal value of? h P x Homework Problem 5: Volume integral over quarter sphere [] Points: [](M) Use spherical coordinates to calculate the volume integral F (R) = dv f(r) of the function Q f(r) = xy on the quadrant Q, defined by x + y + R and x, y. Sketch Q. [Check your result: F () = 6.] 5 Homework Problem 6: Wave functions of the hydrogen atom [] Points: [](M); (b)[](m); (c)[](m,bonus) Show that the volume integral P nlm = R 3 dv Ψ nlm (r) for the following functions Ψ nlm (r) = R nl (r)y m l (θ, φ), with spherical coordinates r = r(r, θ, φ), yields P nlm = : (a) Ψ (r) = R (r)y (θ, φ), R (r) = re r/, Y (θ, φ) = ( 3 π) / cos θ (b) Ψ 3 (r) = R 3 (r)y (θ, φ), R 3 (r) = r e r/3 8 3, Y (θ, φ) = ( 5 / 6π) (3 cos θ ) (c) Show that the so-called overlap integral O = R dv Ψ 3 3 (r)ψ (r) yields ero. 5

6 Hint: In = dx xn e x = n!. Background information: The Ψnlm (r) are quantum mechanical eigenfunctions of the hydrogen atom, where n, l and m are quantum numbers which specify the quantum state of the system. A particle in this state is found with probability Ψnm (r) dv within the volume element dv at position r. The total probability for being found anywhere in 3 equals, hence Pnlm = holds for every eigenfunction Ψnm (r). The figures each show a surface on which Ψnlm has a constant value. The eigenfunctions form an orthonormal basis in the space of square-integrable complex functions on 3, hence the volume integral of two eigenfunctions vanishes if their quantum numbers are not equal. R R Homework Problem 7: Matrix multiplication [] Points: [](M) Compute all possible products of two of the following matrices: P = 3 3 7, Q =, R = 6 6. Homework Problem 8: Spin matrices [] Points: (a)[,5](e); (b)[,5](e) The following matrices are used for the description for quantum mechanical particles with spin : i Sy = i i, S =. Sx =, i (a) Compute S = Sx + Sy + S. (b) Comute the commutators [Sx, Sy ], [Sy, S ] and [S, Sx ], and express each result through one of the matrices given above. Hint: [A, B] = AB BA. Homework Problem 9: Matrix multiplication [] Points: (a)[.5](e); (b)[.5](e) Let A and B be N N matrices with matrix elements aij = Ai δ in + j and bij = Bi δ ij. Remark: Since the indices i and j are specified on the left, they are not summed over on the right even though the index i appears twice in bij on the right. (a) For N = 3 and m =, write these matrices explicitly in the usual matrix representation and calculate the matrix product AB explicitly. (b) Calculate the product AB for arbitrary N N and m N. [Total Points for Homework Problems: ] 6