# Homework Due April 10 — MTH 496

Homework due April 10.

In these exercises you will consider the eigenstates of the Hamiltonian:

$\displaystyle H=\frac{1}{2}P^{2}+aS(Q)$

where ${a>0}$ and ${S}$ is the step function,

$\displaystyle S(x)=\begin{cases} 1, & \mbox{ if }x>0,\\ 0, & \mbox{ if }x<0. \end{cases}$

As always ${P=-i\frac{\partial}{\partial x}}$.

Exercise 1 Suppose ${f}$ is a real valued, twice differentiable function of ${x}$ and that ${f(x)}$ and ${f'(x)}$ both vanish faster than any polynomial as ${x\rightarrow\pm\infty}$. Prove that

$\displaystyle \int_{-\infty}^{\infty}f(x)Hf(x)dx\ge0.$

Exercise 2 Consider ${0.

1. Show that there is a unique solution to ${H\psi=E\psi}$ of the form

$\displaystyle \psi_{E}\left(x\right)=\begin{cases} e^{ikx}+re^{-ikx}, & \ \mbox{if }x<0,\\ te^{-\kappa x}, & \ \mbox{if }x>0, \end{cases}$

for suitable ${k,\kappa>0}$ and ${r,t\in\mathbb{C}.}$ Find ${k}$, ${\kappa}$, ${r}$ and ${t}$ in terms of ${a}$ and ${E}$.

2. Show that ${\left|r\right|^{2}=1}$.
3. Show that any bounded solution of ${H\psi=E\psi}$ is of the form ${\psi=A\psi_{E}}$ for some ${A\in\mathbb{C}}$.

Exercise 3 Consider ${E>a}$.

1. Show that there is a unique solution to ${H\psi=E\psi}$ of the form

$\displaystyle \psi_{E}^{R}\left(x\right)=\begin{cases} e^{ikx}+re^{-ikx}, & \ \mbox{if }x<0,\\ te^{ik'x}, & \ \mbox{if }x>0, \end{cases}$

for suitable ${k,k'>0}$ and ${r,t\in\mathbb{C}}$. Find ${k}$, ${k'}$, ${r}$ and ${t}$ in terms of ${a}$ and ${E}$.

2. Show that there is another solution to ${H\psi=E\psi}$ of the form

$\displaystyle \psi_{E}^{L}\left(x\right)=\begin{cases} te^{-ikx}, & \mbox{ if }x<0,\\ e^{-ik'x}+re^{ik'x}, & \mbox{ if }x>0, \end{cases}$

for suitable ${k,k'>0}$ and ${r,t\in\mathbb{C}}$. Find ${k}$, ${k'}$, ${r}$ and ${t}$ in terms of ${a}$ and ${E}$.

3. Show, for ${\psi_{E}^{R}}$ that

$\displaystyle k\left|r\right|^{2}+k'\left|t\right|^{2}=k,$

for ${\psi_{E}^{L}}$ that

$\displaystyle k'=k'\left|r\right|^{2}+k\left|t\right|^{2},$

and that ${\lim_{E\rightarrow\infty}r=0}$.

4. Prove that any solution to ${H\psi=E\psi}$ is of the form ${\psi=A\psi_{E}^{R}+B\psi_{E}^{L}}$ for some ${A,B\in\mathbb{C}}$.

Exercise 4 Given any wavefunction ${\psi}$ the function

$\displaystyle J_{\psi}(x)=\mathrm{Re}\left[\overline{\psi\left(x\right)}P\psi\left(x\right)\right]$

is interpreted as the momentum density. Prove that

1. For ${0, ${J_{\psi_{E}}\left(x\right)=0}$ for all ${x}$.
2. For ${E>a}$, the momentum density ${J_{\psi_{E}^{R}}\left(x\right)}$ is a positive constant and that

$\displaystyle J_{\psi_{E}^{R}}\left(x\right)=-J_{\psi_{E}^{L}}\left(x\right).$

CHALLENGE PROBLEM: Show that there are continuous functions ${m(E)}$ for ${0 and ${m^{R}\left(E\right),m^{L}\left(E\right)}$ for ${E>a}$ so that for suitable ${\psi}$, say ${C^{\infty}}$ and vanishing faster than any power as ${x\rightarrow\infty}$, we have

$\displaystyle \begin{array}{rcl} \psi(x) & = & \int_{0}^{a}\widetilde{\psi}\left(E\right)\psi_{E}\left(x\right)m(E)dE+\int_{a}^{\infty}\widetilde{\psi}^{R}\left(E\right)\psi_{E}^{R}\left(x\right)m^{R}(E)dE\\ & & +\int_{a}^{\infty}\widetilde{\psi}^{L}\left(E\right)\psi_{E}^{L}\left(x\right)m^{L}\left(E\right)dE \end{array}$

where

$\displaystyle \begin{array}{rcl} \widetilde{\psi}\left(E\right) & = & \int_{-\infty}^{\infty}\overline{\psi_{E}\left(x\right)}\psi(x)dx,\ \widetilde{\psi}^{R}\left(E\right)=\int_{-\infty}^{\infty}\overline{\psi_{E}^{R}\left(x\right)}\psi(x)dx,\\ & & \ \mbox{and }\ \widetilde{\psi}^{L}\left(E\right)=\int_{-\infty}^{\infty}\overline{\psi_{E}^{L}\left(x\right)}\psi(x)dx. \end{array}$

$\displaystyle$