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 \centerline{\large Math.\ 401, Sec.\ 500
\hfill    Spring, 2005}
                                                                               
\bigskip
                                          
              
\centerline{\bfseries\large Homework 5, due February 23}
\bigskip
                                                                               

\noindent The first few problems here are a continuation of 
Problem~2 of the
previous assignment, which concerned the equation
 $$\frac{d^2y}{dt^2} + [\omega^2 -V(t)]y =0. \eqno(1) $$


\begin{enumerate}


                                                                               
\item
\begin{enumerate}
\item  In the equation
$$\frac{d^2y}{dt^2} + \omega^2 V(t)y =0 \eqno(2)$$
  make the change of time variable
 $$\tau \equiv\omega \int_0^t \sqrt{V(\tilde t)} 
\,d\tilde t.$$
{\slshape Hint:\/}   As an intermediate step you should find that
 $$\frac{d^2}{dt^2} = \omega^2V(t) \frac{d^2}{d\tau^2} + 
\omega\,{V'(t) \over 2\sqrt{V(t)}}
\, \frac d{d\tau} \,.$$
 Show that the result is the equation of a harmonic oscillator
 subjected to small, slowly varying damping. (Define $\epsilon =
1/\omega$.)

\item Apply
the two-variable method to the damped oscillator
equation you found in (a).  (For help, see 
J. D. Cole,  {\slshape Perturbation Methods in Applied 
Mathematics},
Sec.\ 3.6.)
You should get the same WKB approximation 
$$\displaystyle V(t)^{-\frac14} e^{\pm i \omega
\int_0^t \sqrt{V(\tilde t)}\,d\tilde t}
\eqno({*}{*}{*})$$
that was obtained in
class by applying the two-variable method directly to~(2).
\end{enumerate}

 \item   
\begin{enumerate}
\item In (2) make the change of  variables
 $$ z\equiv V^\frac14 y, \qquad
s \equiv \int_0^t \sqrt{V(\tilde t)} \,d\tilde t.$$
 [{\slshape Hint:\/}  First carry out the change of dependent 
variable;
 then change the time variable in analogy with the previous 
problem.]
 Show that the result is a special case of~(1).

  \item
 Problem  2(a) shows that equations (1) and (2) are, at root,
equivalent.
 Therefore, the solution to Problem 2(b) of the previous assignment
  should somehow match the
first-order WKB solution (derived in notes and in Problem 1).
Check this, by renaming $V$ in (2) as $1-\epsilon^2 V$ and 
comparing the various solutions we've found.
\end{enumerate}

\goodbreak


 \item The formula for the WKB approximation suggests that
 a solution which starts out proportional to 
$\,\exp(+i\omega\int\sqrt V)$
 remains so for all $t$ (rather than acquiring a component
proportional to $\,\exp(-i\omega\int\sqrt V)$\,).
 This is actually true
 (through arbitrarily high finite order in $1/\omega$)
  if $V$ is a smooth function. However,
a discontinuity in $V$ or its derivative
can  cause ``mixing'' of
positive- and negative-frequency complex exponentials.
 To see this:  For each of the following functions $V$,
write down the WKB solution 
 that behaves as $\,\exp(+i\omega\int\sqrt V)$
 on $-\infty<t<0$.
 Match it to a WKB solution
(linear combination of the two solutions (${*}{*}{*}$))
 on $0<t<\infty$ by requiring that the
solution and its derivative be continuous at 0, at least to lowest
order.

\begin{enumerate}
 \item   \quad $\displaystyle V(t) = \left\{ \begin{array}{ll}
     1 &\mbox{for $t<0$,}\\ 
     2 &\mbox{for $t>0$.} 
\end{array}\right.$

 \item \quad $\displaystyle V(t) = \left\{ \begin{array}{ll}
    1 &\mbox{for $t<0$}, \\
     1+t &\mbox{for $t>0$}.
 \end{array}\right.$
\end{enumerate}

\noindent Comment on the difference between the two cases.  
(Relate 
the 
asymptotic dependence on~$\omega$ to the degree of smoothness 
(continuity and differentiability) of~$V$.)

\item Consider the (second-order, linear) partial differential 
equation
$$ i\epsilon \,\frac{\partial u}{\partial t} =
-\, \frac{\epsilon^2}{2m} \nabla^2 u + V({\mathbf x})u. 
\eqno(\mathrm S)$$
Here $\mathbf x$ is a variable in $\mathbf R^3$ and $\nabla$ is 
the gradient with respect to $\mathbf x$;
$u(t,{\mathbf x})$ thus is a complex-valued function of four 
real variables.
Make the ansatz
$$u(t,{\mathbf x}) = e^{iS(t,{\mathbf x})/\epsilon}\,
[A_0(t,{\mathbf x}) + \epsilon A_0(t,{\mathbf x}) + \cdots]$$
(where $S$ and $A_n$ are independent of $\epsilon$).
Show that the first two equations in the perturbation hierarchy 
are
$$  \frac{\partial S}{\partial t} + \frac1{2m}(\nabla S)^2 +V =0
 \eqno (\mathrm{HJ})$$
and
$$ \frac{\partial A_0}{\partial t} + \frac 1m \nabla A_0 \cdot
\nabla S + \frac1{2m} \, A_0\nabla^2 S =0.
 \eqno (\mathrm{T})$$
{\slshape Remarks:\/}
In quantum mechanics (S) is the {\slshape Schr\"odinger 
equation\/} describing the behavior of a particle of mass~$m$;
$\epsilon$ is {\slshape Planck's constant}, usually written 
as~$\hbar$.
Then (HJ) (a~first-order, nonlinear equation for~$S$) is called 
the 
{\slshape Hamilton--Jacobi equation},
and in advanced classical mechanics courses it is shown that 
solving it is equivalent to solving the classical (nonquantum) 
equation of motion of the particle.
Finally, (T) (a~first-order, linear equation for~$A_0$)
is called the {\slshape transport equation\/} because
it can be solved by integration along the possible classical 
trajectories of the particle.

\end{enumerate}
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