Question

Ψ(x,t)
=Ax/a, if 0<=x<=b , =A(b-x)/(b-a), if a <=x<=b, =0, otherwise, whr a,b are constant...1.sketch
Ψ(x,0)
as a fuction of x

Answer 1

Hint:
I think that your function is defined as below:
\[\Large \psi \left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{A\frac{x}{a},\quad 0 \leqslant x \leqslant a} \\
\begin{gathered}
\hfill \\
A\frac{{b - x}}{{b - a}},\quad a \leqslant x \leqslant b \hfill \\
\hfill \\
0,\quad otherwise \hfill \\
\end{gathered}
\end{array}} \right.\]
In such way, the wave function is continuous at point \(x=a\)
Now, before to draw the graph of such function, we have to normalize it, with this condition:
\[\Large \int_{ - \infty }^{ + \infty } {{{\left| {\psi \left( x \right)} \right|}^2}} dx = 1\]
after a simple computation, we get:
\[\Large A = \sqrt {\frac{3}{b}} \]
therefore, we have to draw the graph of this function:
\[\Large \psi \left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{\sqrt {\frac{3}{b}} \left( {\frac{x}{a}} \right),\quad 0 \leqslant x \leqslant a} \\
\begin{gathered}
\hfill \\
\sqrt {\frac{3}{b}} \left( {\frac{{b - x}}{{b - a}}} \right),\quad a \leqslant x \leqslant b \hfill \\
\hfill \\
0,\quad otherwise \hfill \\
\end{gathered}
\end{array}} \right.\]