For a particle in a $2D$ square box:

a$)$ write down the wavefunction $()$ for the $n_1=3$ and $n_2=2$ state. $[2]$

b$)$ what is the energy and degeneracy of this state? $[2]$

c$)$ now determine the energy of this state by applying the appropriate operator. Do you get the same value as in $($b$)?[5]$

d$)$ Draw a picture of ${}^2$ on a square grid and clearly identify the location of the maxima and the nodal lines in terms of $x$ and $y.[5]$