Problem Sets
Problem Set 1 (due Sep. 23, Wed)
- Chapter 1, Prob 2a,2e;
Chapter 1, Find the general solution of
(D-1)(D-2)(D-3)y=2e$^{\cos x}.$
Problem Set 2 ( due Sep 21, Mon)
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 $u(x,t).$
-
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)
-
Let $y$ satisfy the equation
$y"+x^{2}y=0$
and the initial conditions $y(x_{0})=1,$ $y^{\prime}(x_{0})=0$ where $x_{0}>0.$
- Find the WKB approximation of $y(x)$ for $x>x_{0}$. For what values of $x$ do you expect it be a good approximation?
- 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.
- Problem 4, Chapter 7.
Problem Set 4 (due Oct 5, Mon)
-
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)
- 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)
-
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
$y(0)=1,y(1)=2.$
-
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)
-
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?
-
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)
-
Prob 1. Evaluate the following integrals:
- $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
- $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
-
Prob 2. Find the leading term for the following integrals for $\lambda>>1$:
- $\int_{0}^{\pi/2}e^{-\lambda\cos t}dt,$
- $\int_{-1}^{1}e^{\lambda t^{2}}dt,$
- $\int_{-\infty}^{\infty}e^{\lambda(x^{2}-x^{4})}dx.$
Problem Set 9 (due Nov 30, Mon)
-
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 $e^{-x^{4}}).$