Problem Sets

Problem Set 1 (due Sep. 10, Mon)

Chapter 1, Prob 1a, 1b.
Chapter 1, Prob 2b, 2d, 2e.

Problem Set 2 ( due Sep 17, Mon)

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 and express $T(x,t)$ with it.
Consider the wave equation

$\dfrac{\partial^{2}\phi(x,t)}{\partial t^{2}}-\dfrac{\partial^{2}\phi (x,t)}{\partial x^{2}}=\rho(x,t),$ $-\infty$ < $x$ < $\infty,$ $t$ > $0,$

where $\rho(x,t)$ is a given source function. Let the initial conditions be

$\phi(x,0)=f(x)$, $\phi_{t}(x,0)=g(x)$

Find the Green function $G(x-x^{\prime},t-t^{\prime})$ for the wave equation above and express $\phi(x,t)$ with it.

Problem Set 3 ( due Sep 24, Mon)

Solve $(\nabla^{2}-m^{2})u(\overrightarrow{x})=\rho(\overrightarrow{x)}$ in the 3-dimensional space with the boundary condition of $u(\overrightarrow{x})$ vanishing at the infinity. Express your solution witht the Green function for this problem.

Problem Set 4 ( due Oct 1, Mon)

Consider the temperature $T(x,t)$ in an infinite chopstick with a heat conductivity coefficient

$c(t)=\dfrac{1}{1+t}.$ The equation satisfied by $T(x,t)$ is

$\dfrac{\partial T}{\partial t}=$ $\dfrac{1}{1+t}\dfrac{\partial^{2}T}{\partial x^{2}}.$

The boundary conditions are

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

and the initial condition is

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

Express the solution of this problem with a Green function.
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.
Problem 4, Chapter 7.

Problem Set 5 ( due Oct 10, Wed)

Consider the problem of an electron scattered by a potential. The electron wavefunction $\Psi(x)$\ satisfies

$(\dfrac{d^{2}}{dx^{2}}+k^{2})\Psi(x)=-gV(x)\Psi(x).$

In the above, $k^{2}$ is the energy of the electron and $gV(x)$ is the potential for the electron. We shall assume that $V(x)$ vanishes rapidly as $x\rightarrow\pm\infty.$ If an electron wave $e^{ikx}$ is incident from $x=-\infty$, the asymptotic conditions for $\Psi(x)$ are

$\Psi(x)\simeq e^{ikx}+R$ $e^{-ikx},$ $x\rightarrow-\infty,$

$\simeq T$ $e^{ikx},$ $ x\rightarrow \infty$

where $R$ and $T$ are constants (they are called the reflection coefficient and the transmission coefficient, respectively.)
1. Use the Green's function method to show that the electron wavefunction satisfies the integral equation
  
  $\Psi(x)=e^{ikx}-\int_{-\infty}^{\infty}\dfrac{e^{ik\left\vert x-x^{\prime}\right\vert }}{2ik}gV(x^{\prime})\Psi(x^{\prime})dx^{\prime}$.
  
  Show that
  
  $R=-\int_{-\infty}^{\infty}\dfrac{e^{ikx^{\prime}}}{2ik}gV(x^{\prime})\Psi(x^{\prime})dx^{\prime}$
  
  and
  
  $T=1-\int_{-\infty}^{\infty}\dfrac{e^{-ikx^{\prime}}}{2ik}gV(x^{\prime})\Psi(x^{\prime})dx^{\prime}$
2. Prove that
  
  $\left\vert T\right\vert ^{2}+\left\vert R\right\vert ^{2}=1.$
  
  Hint: Make use of the fact that the Wronskian $(\Psi^{\ast}\dfrac{d\Psi}{dx}-\Psi\dfrac{d\Psi^{\ast}}{dx})$ is a constant.
3. Find the lowest-order approximations for $R$ and $T$ when $g$ is small and $V(x)=e^{-\left\vert x\right\vert }$.

Problem Set 6 ( due Oct 15, Mon)

Consider

$\dfrac{d^{2}\Psi}{dx^{2}}+\lambda^{2}[k^{2}-V(x)]\Psi=0,$ $\lambda >>1,$ $-\infty$ < $ x $ < $ \infty,$ $V(\pm\infty)=0,$

in which there are two turning points: $-a$ and $0.$ Let

$k^{2}-V(x) $ < $ 0,$ $-a$ < $x$ < $0$,

and let

$k^{2}-V(x)>0,$ for $x>0$ as well as for $x$ <$-a.$

Let an electron of momentum $k$ traveling from $x=-\infty$ incident upon the potential $V(x)$.
1. Explain why we have to start with the WKB solution from $x>0,$ and that this solution can be chosen to be
  
  $\Psi(x)\simeq T\dfrac{\exp i[\int_{0}^{x}p(x^{\prime})dx^{\prime}]} {\sqrt{p(x)}}.$ $ x>0.$
  
  where $T$ is a constant.
2. Show that the continuation of this solution to the neighborhood of the origin is a multiple of
  
  $e^{-i\pi/4}[$Bi$(-\rho)+i$Ai$(-\rho)]$.
  
  Show that, continuing this solution to the region $-a$ < $x$ < $0,$ $\Psi(x)$ is approximately
  
  $\Psi(x)\simeq e^{-i\pi/4}T$ $\dfrac{\exp[\int_{x}^{0} \eta(x^{\prime})dx^{\prime}]}{\sqrt{\eta(x)}},$
3. Show that the expression above can be written as
  
  $\Psi(x)\simeq e^{-i\pi/4}T\exp[\int_{-a}^{0}\eta(x^{\prime})dx^{\prime}]\dfrac{\exp[-\int_{-a}^{x}\eta(x^{\prime})dx^{\prime}]}{\sqrt{\eta(x)}},$ $-a$ < $x$ < $0.$
4. Show that, by choosing
  
  $T=$ $\exp[-\int_{-a}^{0}\eta(x^{\prime})dx^{\prime}],$
  
  we have
  
  $\Psi(x)\simeq e^{-i\pi/4}\dfrac{\exp[-\int_{-a}^{x}\eta(x^{\prime})dx^{\prime}]}{\sqrt{\eta(x)}},$ $-a$ < $x$ < $0.$
5. Prove that the continuation of $\Psi$ to the left of the turning point $x=-a$ is
  
  $\Psi(x)\simeq$ $e^{-i\pi/4}\dfrac{2\sin[\int_{-a}^{x}p(x^{\prime})dx^{\prime}+\pi/4]}{\sqrt{p(x)}},$ $x$ <$-a.$
6. Express the sine function above as a superposition of two exponentials and explain why $T$ can be regarded as the transmission coefficient for quantum tunneling.

Problem Set 7 ( due Oct 22, Mon)

Prove (7.68) in the textbook by making use of Bohr's quantization rule (7.71) together with $a=(-1)^{n}$.
Find the WKB eigenvalues for

$\dfrac{d^{2}\Psi}{dx^{2}}+(k^{2}-x^{4})\Psi=0,$ $-\infty$ < $x$ < $\infty,$

with the homogeneous boundary condditions $\Psi(\pm\infty)=0.$
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 $\lambda$.

Problem Set 8 ( due Oct 29, Mon)

Show that $ \lim_{\Delta z\rightarrow0} \dfrac{\Delta f}{\Delta z}$ exists for all $\Delta z$ if its real part and the imaginary part satifies the Cauchy-Riemann equations.
Find the leading asymptotic form of the integral

$I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda(x-x^{5}/5)}dx,$$ \lambda>>1.$

Problem Set 9 ( due Nov 5, Mon)

Find the leading asymptotic form of the integral

$I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda x}e^{-x^{4}/4}dx,$ $ \lambda>>1.$

Compare this asymptotic form with the numerical value of the integral.
Compute the leading asymptotic term of the following integral:

$I(\lambda)=\int_{0}^{\infty}e^{-\lambda x}e^{-1/x^{3}}dx,$

where $\lambda$ is very large.
The Bessel function of order $\nu,$ denoted by $J_{\nu}(z),$ has the integral representation

$J_{\nu}(x)=\dfrac{(x/2)^{\nu}}{\sqrt{\pi}\Gamma(\nu+1/2)}\int_{-\pi/2}^{\pi/2}e^{ix\sin\phi}(\cos\phi)^{2\nu}d\phi.$

If $J_{\nu}(x_{0})=0,$ we call $x_{0}$ a zero of the Bessel function $J_{\nu}(x).$ Determine these zeroes of very large magnitude.

Problem Set 10 ( due Nov 12, Mon)

1. Determine the unshaded regions of $f(z)=e^{iz^{5}}$ in the infinity of which $f(z)$ vanishes.
2. By deforming the contour of integration, express $J\equiv \int_{-\infty}^{\infty}e^{ix^{5}}dx$ with a Gamma function.
  
  (Note: $\Gamma(1+\rho)=\int_{0}^{\infty}e^{-t}t^{\rho}dt.)$
3. Find the values of $\int_{-\infty}^{\infty}\cos x^{5}$ $dx$ and $\int_{-\infty}^{\infty}\sin x^{5}$ $dx.$
Find the leading asymptotic form of the integral

$I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda x}e^{ix^{5}/5}dx,$ $ \lambda>>1.$

Problem Set 11 ( due Nov 19, Mon)

Find both $y_{in}$ and $y_{out}$ for the equation

$\epsilon y^{"}-y^{\prime}+(1+x^{3})y=0,$ $0$ < $x$ < $1, $ $y(0)=1,$ $y(1)=3.$

It is assumed that $\epsilon$ << $1.$ 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 12 (due Nov 26, Mon)

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

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

Problem Set 13 (due Dec 3rd, Mon)

Solve the differential equation

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

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