Part A : Bound States of the Finite Square Well, Transcendental Equations

(*For a given value of z0, plot the relevant functions for the finite square well bound state energy levels*)
z0:=8
Plot[{Tan[z],(Sqrt[(((z0/z))^2)-1])},{z,0,20},AxesLabel->{"z",""},PlotLabel->"Even States (n=1,3,5...)",PlotLegends->"Expressions"]
Plot[{-Cot[z],(Sqrt[(((z0/z))^2)-1])},{z,0,20},AxesLabel->{"z",""},PlotLabel->"Odd States (n=2,4,6...)",PlotLegends->"Expressions"]

(*You can use something like NSolve to find the roots of the transcendental equation, but using this function ("Reduce") we can specify the range over which we'd like to find all the solutions.*)
(*This cell uses the value of z0 entered at the top.*)
Reduce[Tan[z]==(Sqrt[(((z0/z))^2)-1]) && 0<z<z0,z]
Reduce[-Cot[z]==(Sqrt[(((z0/z))^2)-1]) && 0<z<z0,z]

Part D: Scattering States of the Finite Square Well

(*Use arbitrary units*)
V0:=1
a:=1
ℏ:=1
m:=5
T[En_]:=(1+((V0^2)/(4En(En+V0)))((Sin[((2a)/ℏ)(Sqrt[2m(En+V0)])])^2))^(-1)
Plot[T[En],{En,0,10},PlotRange->{0.9,1},PlotLabel->{"E (arbitrary units)","T"}]