Problem Sets
Problem Set 1 (due Sep. 9, Mon)
 Chapter 1, Prob 1a, 1b;
 Chapter 1, Prob 2b, 2d, 2e.
Problem Set 2 ( due Sep 16, Mon)

Consider the Laplace equation which holds in a two dimensional strip:
$(\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}})u(x,y)=0,$ $\infty$ < $x$ < $\infty,$ $0$ < $y$ < $a,$
The function $u(x,y)$ satisfies the boundary conditions
$u(x,0)=f(x),$ $u(x,a)=0,$ $\infty$ < $x$ < $\infty,$
($f(x)$ is a given function) as well as
$u(\pm\infty,y)=0.$
Find $u(x,y).$

Consider the equation
($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $\Psi(x,t)=\rho(x,t),$
where $\rho(x,t)$ is a given function and where $\Psi(x,t)$ vanishes at the infinity of space, i.e,
$\Psi(\pm\infty,t)=0.$
$\Psi(x,t)$ is required to satisfy the initial condition
$\Psi(x,0)=f(x),$
Find $\Psi(x,t).$
Problem Set 3 (due Sep 25, Wed)

The wavefunction of the electron $\Psi(x,t)$ satisfies the Schrodinger equation
($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $\Psi(x,t)=U(x,t)\Psi(x,t),$
where $U(x,t)$ is a given function (physically, it is the potential which acts on the electron). The wavefunction $\Psi(x,t)$ satisfies the same boundary conditions and initial condition as those given in prob 2.
Use the result in prob 2 to convert the Schridinger equation into an integral equation.
Find the perturbtion series for $\Psi(x,t).$

Consider the heat equation
$\dfrac{\partial T(x,t)}{\partial t}\dfrac{\partial^{2}T(x,t)}{\partial x^{2}}=\rho(x,t),$ $\infty$ < $x$ < $\infty,$ $t$ > $0,$
where $T(x,t)$ is the temperature of a onedimensional rod and $\rho(x,t)$ is a given source function. Let the initial temperture be
$T(x,0)=f(x)$.
Find the Green function $G(xx^{\prime},tt^{\prime})$ for the heat equation above in a closed form (not as an integral) and express $T(x,t)$ with it.

The Green function for the wave equation is given by the integral
$G(x,t)=\int_{\infty}^{\infty}\dfrac{dk}{(2\pi)}e^{ikx}\cos kt.$
Evaluate this integral. What does this result mean physically?
Problem Set 4 (due Sep 30, 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 5 (due Oct 16, Wed)

Evaluate the integral
$F_{\epsilon}(x)\equiv\int_{\infty}^{\infty}e^{ikx}e^{\epsilon k^{2}} \dfrac{dk}{2\pi}.$
Show that in the limit $\epsilon\rightarrow0,$ $F_{\epsilon}(x)$ becomes the Dirac delta function.

Consider the Schrodinger equation
$[\dfrac{d^{2}}{dx^{2}}+\lambda^{2}(Eg\left\vert x\right\vert )]\Psi(x)=0,$ $\infty$ < $x$ < $\infty,$
where
$\lambda^{2}=m/(2\pi^{2}h^{2}),$
$h$=$6.626x10^{27}erg\sec,$ and $g$ is a constant. Use the WKB method to determine, approximately, the energy eigenvalues$.$

Find the approximate energy eigenvalues for the radial Schrodinger equation
$[\dfrac{d^{2}}{dr^{2}}+\lambda^{2}(E+\dfrac{e^{2}}{r})]\Psi(r)=0$, $\infty>r>0.$
In the above, $e$ is the electric charge. Also, $\dfrac{e^{2}}{r}$ is the attractive Coulomb potential the the positively charged hydrogen nucleus
provides the negatively charged electron. Note that the energy $E$ is negative and that the boundary condition at $r=0$ is
$\Psi(0)=0$. The exact result is $E=\dfrac{e^{4}\lambda^{2}}{4(n+1)^{2}},n=0,1,2\cdot\cdot\cdot.$ Which of the eigenvalues you find are good approximations of the exact result?
Problem Set 6 (due Oct 21, Mon)

Evaluate the following integrals:
 $\int_{\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x3i)(x5i)}.$
 $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$

The Green function for the Schrodinger equation of one spatial dimension satisfies
($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $G(xx^{\prime},tt^{\prime})=\delta(xx^{\prime})\delta(tt^{\prime}),$
where $G(xx^{\prime},tt^{\prime})$ vanishes for $t$ < $t^{\prime}.$ Find $G(xx^{\prime},tt^{\prime})$ by expressing the Dirac delta functions with their Fourier integrals, i.e.
$\delta(tt^{\prime})=\int_{\infty}^{\infty}e^{ik_{0}(tt^{\prime})} dk_{0}/(2\pi)$
etc. and carrying out the integration over $k_{0}$.
Problem Set 7 (due Oct 28, Mon)

Find the leading term for the following integrals for $\lambda>>1$:
 $\int_{0}^{\pi/2}e^{\lambda\cos t}dt,$
 $\int_{1}^{1}e^{\lambda t^{2}}dt,$
 $\int_{\infty}^{\infty}e^{\lambda(x^{2}x^{4})}dx.$
Use the computer to evaluate the numerical values of these integrals and compare them with the numerical values of their leading terms as a function of $\lambda$.
Problem Set 8 (due Nov 4, Mon)

As we know, the Gamma function is defined by
$\Gamma(1+\lambda)=\int_{0}^{\infty}e^{t}t^{\lambda}dt..$
Find the asymptotic form of $\Gamma(1+\lambda)$ when $\lambda>>1.$
Hint: you will need to change the variable of integration (why?)

The Laplace transform of the function $f(x),$ $0$ < $x$ < $\infty$, is defined as
$L(s)=\int_{0}^{\infty}dx$ $e^{sx}f(x).$
Find the asymptotic form of $L(s)$ when $s>>1.$ Verify your result with the examples of $f(x)=\sin x$ and $f(x)=\cos x.$

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 $e^{x^{4}}).$
Problem Set 9 (due Nov 13, Wed)

Find the leading asymptotic form of the integral
$I(\lambda)=\int_{\infty}^{\infty}e^{i\lambda t}e^{it^{3}/3}dt, \lambda>>1.$

Determine the unshaded regions of $f(z)=e^{iz^{5}}$ in the infinity of which $f(z)$ vanishes.

Find both $y_{in}$ and $y_{out}$ for the equation
$\epsilon y^{"}y^{\prime}+(1+x^{3})y=0$, $0$ < $x$ < $1$, $y(0)=1$, $y(1)=3$, $\epsilon$ << $1$.
Where is the boundary layer and what is its width?
Problem Set 10 (due Nov 18, Mon)

Find both $y_{in}$ and $y_{out}$ for the equation
$\epsilon y^{"}y^{\prime}+(1+x^{3})y=0,$ $0$ < $x$ < $1,$ $y(0)=1,$ $y(1)=3,$ $\epsilon<<1.$
Where is the boundary layer and what is its width?

Find both $y_{in}$ and $y_{out}$ for the equation
$\epsilon y^{"}+2y^{\prime}+(1+x^{3})y=0,$ $0$ < $x$ < $1,$ $y(0)=1,$ $y(1)=0.$
It is assumed that $\epsilon<<1.$ Where is the boundary layer?
Problem Set 11 (due Nov 25, Mon)

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

Solve approximately $\epsilon y^{\prime\prime}+(3\sin x)y^{\prime}+(\cos x)y=0,0$ < $x$ < $1,$ $1$ < $x$ < $1$, with the boundary conditions
$y(1)=1,y(1)=3.$
Problem Set 12 (due Dec 2, Mon)

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

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