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Simplify[Conjugate[Exp[I h t]], h t \[Element] 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 \[Element] Reals, t \[Element] 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).

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).

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

Simplify can handle this, if the assumption that the exponent is purely imaginary holds.

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]]]] := Exp[-x]

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).

Simplify[Conjugate[Exp[I h t]], h t \[Element] 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 \[Element] Reals, t \[Element] 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).

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

Simplify can handle this, if the assumption that the exponent is purely imaginary holds.

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 $e^{iHt}=e^{-iHt}$.

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

Simplify can handle this, if the assumption that the exponent is purely imaginary holds.

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]]]] := Exp[-x]

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).