# Complex Conjugating a Row Vector

The code for the problem is:

J = 1

newcoherentstate[α_, χ_,
m_] := ((ψ[α, χ])/(1 +
Abs[ψ[α, χ]]^2))^(J) *(ψ[α, \
χ])^(m)*Sqrt[Binomial[2 J, J + m]] ;

NewMatrix [α_, χ_] :=
Table[newcoherentstate[α, χ, m], {m, -J, J}];

NewMatrix1[α_, χ_] :=
ArrayReshape[NewMatrix[α, χ], {1, 2*J + 1}];

NewMatrixC[α_, χ_] :=
Conjugate[NewMatrix1[α, χ]];


Using matrix form, the result is:

What is making the Conjugate appear everywhere? I simply want the Conjugate function to take the complex conjugate of each term in the row vector.

What you observe is the fact that Mathematica assumes the most general case, since you did not specify any assumptions. If you know that e.g. \[alpha] is real-valued, you need to let Mathematica know this very important fact.

See this:

z = Exp[I*a];
Conjugate[z]


E^(-I Conjugate[a])

This is basically what you asked about in your question. IF a is real, then you need to provide this as Assumptions to e.g. Refine or Simplify (depending on your expression, Simplify might take more time than Refine due to its nature):

Simplify[Conjugate[z], Assumptions -> Element[a, Reals]]
Refine[Conjugate[z], Assumptions -> Element[a, Reals]]


both yield

E^(-I [Alpha])

as you would expect, knowing that a is real. If this assumption is true for all your calculation you might want to globally specify this assumption by executing

\$Assumptions=Element[a,Reals]


before you start over with your symbolic calculus.