#$2.$ Consider the possible definition of the probability

density:

$P(x,t)=(x,t{)}^{**}(x,t)+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(x,t{)}^{**}(x,t)$

It is readily shown that this definition satisfies

conservation of probability, i$.$e$.$

$\frac{d}{dt}{\displaystyle \underset{-}{\overset{+}{}}}p(x,t)dx=0$

On the other hand, it will be shown in class that the

time$-$independent $($stationary state$)$ wavefunctions $-$

$(x,t)=(x)$

for a particle in a box may be written as

${}_n(x)=(\frac{2}{l}{)}^{\frac{1}{2}}sin\frac{nx}{l}$

where $l$ is the length of the ID box, $0xl$ and $n=1,2,3,$dots.

For $n=1$ show that the above definition of $P(x,t)$ does not satisfy

one or more of the conditions for a proper probability density.

$[$Note that $f(x)=0$ for $x<mn\; id="EditorElement\_11225243">0</mn>$ and $x>l.]$