Complex Conjugate of matrix

Question

I have written the following code

\[Psi] = {\[Alpha] Cos[B t/2] + 
I \[Beta] Sin[B t/2] E^(-I \[Phi]), \[Alpha] I Sin[B t/2] E^(
 I \[Phi]) + \[Beta] Cos[B t/2]};
\[Psi]1 =  Assuming[{B, t, \[Phi] } \[Element] Reals , Conjugate[\[Psi]]]
R = KroneckerProduct[\[Psi], \[Psi]1]
v = Eigenvectors[R]
Normalize /@ v // FullSimplify

but the matrix [Psi]1 takes the complex conjugate of the whole element and not just that of i, a and b, which, in turn, makes the normalized eigenvectors somewhat complicated to work with. What can i change to my code? The ideal would be to get

$$ \psi1=\begin{pmatrix} \alpha^{*}\cos(Bt/2)-i\beta^{*}\sin(Bt/2)e^{i\varphi} & -ia^{*}\sin(Bt/2)e^{-i\varphi}+\beta^{*}\cos(Bt/2) \end{pmatrix} $$

Answer 1

ComplexExpand assumes that all variables are real except those mentioned as second argument.

Try:

ψ = {α Cos[B t/2] + 
    I β Sin[B t/2] E^(-I ϕ), α I Sin[
      B t/2] E^(I ϕ) + β Cos[B t/2]};

ComplexExpand[Conjugate[ψ], {α, b }] // Simplify

This gives: