Problem Sets

Problem Set 1 (due Sep. 9, Mon)

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 one-dimensional rod and $\rho(x,t)$ is a given source function. Let the initial temperture be

$T(x,0)=f(x)$.

Find the Green function $G(x-x^{\prime},t-t^{\prime})$ 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?

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

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

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?

Evaluate the following integrals:
1. $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
2. $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
The Green function for the Schrodinger equation of one spatial dimension satisfies

($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $G(x-x^{\prime},t-t^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime}),$

where $G(x-x^{\prime},t-t^{\prime})$ vanishes for $t$ < $t^{\prime}.$ Find $G(x-x^{\prime},t-t^{\prime})$ by expressing the Dirac delta functions with their Fourier integrals, i.e.

$\delta(t-t^{\prime})=\int_{-\infty}^{\infty}e^{-ik_{0}(t-t^{\prime})} dk_{0}/(2\pi)$

etc. and carrying out the integration over $k_{0}$.

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. $\int_{-\infty}^{\infty}e^{\lambda(x^{2}-x^{4})}dx.$
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 $\lambda$.

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 $e^{-x^{4}}).$

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?

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?

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

$y(-1)=-1,y(1)=3.$

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

$y(-1)=-1,y(1)=3.$