Problem Sets

Problem Set 1 (due Sep. 23, Wed)

  1. Chapter 1, Prob 2a,2e;
  2. Chapter 1, Find the general solution of

    (D-1)(D-2)(D-3)y=2e$^{\cos x}.$

Problem Set 2 ( due Sep 21, Mon)

  1. Consider the wave equation

    $\dfrac{\partial^{2}}{\partial x^{2}}u(x,t)=\dfrac{\partial^{2}}{\partial t^{2}}u(x,t),$

    where $-\infty < x < \infty$ and $t>0$

    The wavefunction $u(x,t)$ satisfies the boundary condition

    $u(\pm\infty,t)=0$

    and the initial conditions

    $u(x,0)=f(x),$ and $u_{t}(x,0)=g(x)$,

    where $f(x)$ and $g(x)\ $are given functions. Find $u(x,t).$

  2. Find the eigenfunctions and the corresponding eigenvalues of the Sturm-Liouville problem

    $\dfrac{d^{2}\phi(\theta)}{d\theta^{2}}=-\lambda\phi(\theta),$ $0 < \infty < 2\pi,$

    with the homogenious (periodic) boundary conditions

    $\phi(0)=\phi(2\pi),$ $\phi^{\prime}(0)=\phi^{\prime}(2\pi).$

Problem Set 3 (due Sep 28, Mon)

  1. Let $y$ satisfy the equation

    $y"+x^{2}y=0$

    and the initial conditions $y(x_{0})=1,$ $y^{\prime}(x_{0})=0$ where $x_{0}>0.$

    1. Find the WKB approximation of $y(x)$ for $x>x_{0}$. For what values of $x$ do you expect it be a good approximation?
    2. Use the computor to obtain the numerical values of $y(x)$ as a function of $x$ for $x_{0}=1,5,10.$ Compute also the numerical values of the WKB approximation of $y(x)$. Compare the computor result with the approximate result.
  2. Problem 4, Chapter 7.

Problem Set 4 (due Oct 5, Mon)

  1. Consider the Schrodinger equation

    $[\dfrac{d^{2}}{dx^{2}}+\lambda^{2}(E$--$g\left\vert x\right\vert )]\Psi(x)=-0,$ --$\infty$ < $x$ < $\infty.$

    In the above, $\lambda^{2}=m/(2\pi^{2}h^{2})$ with $m$ the mass of the electron$,$ $h$ is the Planck's constant ($6.626x10^{-27}erg$--$\sec$), and $g$ is a constant.

    Use the WKB method to determine the approximate energy eigenvalues$.$

Problem Set 5 (due Oct 13, Tue)

  1. Using (7.71) in the textbook show that $a$ in (7.67) is equal to (-1)$^{n}$. Show also that (7.68) is satisfied.

Problem Set 6 (due Oct 26, Mon)

  1. Find the location and the width of the boundary layer for the problem $\epsilon y^{\prime\prime}-(\cos x)y^{\prime}+(\sin x)y=0,$ $0$ < $x$ < $1$ with the boundary conditions

    $y(0)=1,y(1)=2.$

  2. Solve approximately $\epsilon y^{\prime\prime}+(3\sin x)y^{\prime}+(\cos x)y=0,0$ < $x$ < $1$ with the boundary conditions

    $y(0)=-1,$ $y(1)=5.$ What are the location and the width of the boundary layer?

Problem Set 7 (due Nov 2, Mon)

  1. Solve approximately $\epsilon y^{\prime\prime}+(3\sin x)y^{\prime}+(\cos x)y=0,0$ < $x$ < $1$ with the boundary conditions

    $y(0)=-1,$ $y(1)=5.$ What are the location and the width of the boundary layer?

  2. Solve approximately $\epsilon y^{\prime\prime}-(3\sin x)y^{\prime}+(\cos x)y=0,$ $-1$ < $x$ < $1$ with the boundary conditions

    $y(-1)=1,$ $y(1)=2.$

Problem Set 8 (due Nov 9, Mon)

  1. Prob 1. Evaluate the following integrals:

    1. $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
    2. $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
  2. Prob 2. Find the leading term for the following integrals for $\lambda>>1$:

    1. $\int_{0}^{\pi/2}e^{-\lambda\cos t}dt,$
    2. $\int_{-1}^{1}e^{\lambda t^{2}}dt,$
    3. $\int_{-\infty}^{\infty}e^{\lambda(x^{2}-x^{4})}dx.$

Problem Set 9 (due Nov 30, Mon)

  1. Find the asymptotic form for

    $I(k)=\int_{-\infty}^{\infty}e^{-ikx}e^{-x^{4}},$ $k>>1,$

    (The integral above is the Fourier transform of $e^{-x^{4}}).$