Particle in a $3-$dimensional box

Consider a particle confined in a $3-$dimensional cubic box with an edge length of $L.$ The potential energy of the particle is given by

$V(x,y,z)=\{\begin{array}{ccc}0& if0xL\text{a}\text{n}\text{d}0yL\text{a}\text{n}\text{d}& 0zL\\ +& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

The Schr$$dinger equation for this system is given by

$-\frac{{}^2}{2m}(\frac{de{l}^2}{del{x}^2}+\frac{de{l}^2}{del{y}^2}+\frac{de{l}^2}{del{z}^2})(x,y,z)=E(x,y,z),$

where the wave function $(x,y,z)$ is a function of $x,y,$ and $z$ and ${}^{**}(x,y,z)(x,y,z)$ gives the probability density at a position $(x,y,z).($Hint: Read section $3-9$ of the textbook.$)$

$($a$)$ Write all the boundary conditions.

$($b$)$ The solution to the Schr$$dinger equation is given by

$]=[1,2,3,$cdots

Show that the wave functions $($energy eigenfunction$){}_{n_x,n_y,n_z}(x,y,z)$ are normalized.

$($c$)$ By substituting the wave functions ${}_{n_x,n_y,n_z}(x,y,z)$ into the LHS of the Schr$$dinger equation, find the eigenenergy expression $E_{n_x,n_y,n_z}.$

$($d$)$ What is the energy if the particle is in state $(n_x,n_y,n_z)=(1,2,1)?$ Are there any other eigenstates that give the same energy?

$($e$)$ What is the probability that the particle is found in the cubic region given by $0x\frac{L}{6},0y\frac{L}{6},$ and $0z\frac{L}{6}$ if the particle is in state $(n_x,n_y,n_z)=(1,2,1).$