Spatial component of the wave function

Question

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 $x=L$"?

Answer 1

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 $$ L = \mbox{half of the spatial period of the wave}$$