Homework Due April 10 — MTH 496

Homework due April 10.

A PDF version of these exercises is available here.

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<E<a}.

  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<E<a}, {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<E<a} 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

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One thought on “Homework Due April 10 — MTH 496

  1. Pingback: Due date for HW 3 now Friday April 12 | Jeffrey Schenker

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