Consider a particle in a one$-$dimensional, infinitely deep box where $V(x)=0,$ for $0x$a and $V(x)=$ otherwise.

a$)$ Sketch the wave functions for the first three allowed energy levels.

b$)$ How does the number of nodes change with the quantum number $n?($Note: A node is a point where $(x)=0$ and is not at a boundary.$)$

c$)$ Where is the maximum of the probability density $|{}_n(x){|}^2$ at $n=1?$

d$)$ Given the definition of the wave vector $k=\frac{2}{}$ and allowed values of $k,$ what is the wavelength $$ of the wave function ${}_n(x)$ at $n=10?$