Assuming[a[Element]Reals, Conjugate[a]]//Simplify

Conjugate[a]

ClearAll["Global`*"]
{a, h, m} > 0;
p [Element] Reals;
n [Element] Integers;
En = (n Pi h)^2/(2 m a^2);
[CapitalPsi][x_, t_] := Sqrt[2/a] Sin[(n Pi x)/a] E^((-I En t)/h);
[Phi][p_,
   t_] := (2 Pi h)^(-1/2) Integrate[
    E^((-I p x)/h) [CapitalPsi][x, t], {x, 0, a}];
[CapitalPsi][x, t] // Simplify
[Phi][p, t] // Simplify
Conjugate[[Phi][p, t]]*[Phi][p, t] // ComplexExpand