Spatial component of the wave function The spatial component of a wave function is given as
$$\sin\left ( \frac{n \pi x}{L} \right )$$
then for $n=1$,
we get 
$$\sin\left ( \frac{ \pi x}{L} \right )$$
and this produces one half cycle of a sine wave over the distance $x=0$ to $x=L$.
It has been a while since I touched wave mechanics.
Could someone explain to me how I can 'see' the part "this produces one half cycle of a sine wave over the distance $x=0$ to $x=L$"?

First note that for
$$\sin\left(\frac{\pi x}{L}\right)=\sin\left(\frac{2\pi x}{2L}\right)$$
We have
$$ x = \mbox{spatial variable for position}$$
$$ 2L = \mbox{spatial period of the wave}$$
This implies that 
$$ L = \mbox{half of the spatial period of the wave}$$
