Simple Complex Conjugate of exponentials

Question

Often I work with complex exponentials, and I want Mathematica to understand that the conjugate of $e^{iHt}$ is $e^{-iHt}$.

Is there a simple way of doing this? Typically I use the complex-expand function to deal with mathematica not knowing if a variable is real/complex, but this time it returns my output in terms of sins & cosines, which isn't desirable.

Answer 1

Try

TrigToExp[ComplexExpand[Conjugate[E^(I H t)]]]
(*E^(-I H t)*)

You might need to add a Simplify in some cases. ComplexExpand assumes everything is real automatically.

Answer 2

Simplify[Conjugate[Exp[I h t]], h t ∈ Reals]
(* E^(-I h t) *)

Simplify can handle this, if the assumption that the exponent is purely imaginary holds. Note that this assumes that the product is real: if each variable is real, you may wish to state that instead:

Simplify[Conjugate[Exp[I h t]], {h ∈ Reals, t ∈ Reals}]

If Simplify takes too long it's also possible that you could consider an equivalent replacement rule:

Conjugate[Exp[I h t]] /. Conjugate[h t]->h t

However, you'll have to be a bit more careful with this, since the inside term can vary quite a bit based on assumptions, variable ordering, etc.

If you are not certain that $Ht$ is real, then it's not actually necessarily true that $e^{iHt}=e^{-iHt}$.

And if you are absolutely, positively, and perfectly certain that the conjugate of an exponential containing I is purely imaginary, you can do this:

Unprotect[Conjugate];
Conjugate[Exp[x_ /; Not[FreeQ[x, I]]]] := 
    Evaluate[Exp[Conjugate[x]]] /. Conjugate[y_] -> y;

That will cause issues in a variety of places, so double check the use case. You will likely need to refine the qualifier (Not[FreeQ[x,I]] is almost definitely overzealous).