I haven't seen much on this topic, but that's probably because - for superpositions of only a few quantum states - it can be done by inspection rather than Mathematica.

Still, I would like to know how to compute the conjugate of a wavefunction, and I am having trouble getting Mathematica to do this.

My wavefunction is:

Ψ = b1*Exp[-I*ω1*t]*ψ1 + b2*Exp[-I*ω1*t]*Exp[-I*ω*t]*ψ2

Which I then try to take the conjugate of:

Simplify[Conjugate[Ψ], t ω1 ω ∈ Reals]

But Mathematica just outputs the following:

E^(I Conjugate[t ω1])Conjugate[b1 ψ1 + b2 E^(-I t ω) ψ2]

Ideally, I'd like it so that it doesn't just say 'Conjugate', and then leave all of the expressions as they were. In texts, you would see complexes denoted by an '*' in superscript. Is there any way to see this also by a Conjugate function in Mathematica?

  • $\begingroup$ In Mathematica imaginary unit is I not i! $\endgroup$ Commented Mar 19, 2021 at 17:40
  • $\begingroup$ Oh what a rookie error. Thank you - fixed the i's but doesn't affect Mathematica's ability to perform the conjugate sadly. $\endgroup$ Commented Mar 19, 2021 at 17:44
  • $\begingroup$ See my answer, ComplexExpandmight help. $\endgroup$ Commented Mar 19, 2021 at 17:47

1 Answer 1


Change i to I

Ψ = b1*Exp[-I*ω1*t]*ψ1 + b2*Exp[-I*ω1*t]*Exp[-I*ω*t]*ψ2

Simplify[Conjugate[Ψ],Element[{t, ω1, ω}, Reals]]
(*E^(I t ω1) Conjugate[b1 ψ1 + b2 E^(-I t ω) ψ2]*)

Further Simplification might be achieved with ComplexExpand

(*b1 ψ1 Cos[t ω1] + b2 ψ2 Cos[t ω + t ω1] - 
I (-b1 ψ1 Sin[t ω1] -b2 ψ2 Sin[t ω + t ω1])*)

For complex b1,b2 try

ComplexExpand[Conjugate[Ψ], {b1, b2}]
(*ψ1 (Cos[t ω1] Re[b1] + 
Im[b1] Sin[t ω1]) + ψ2 (Cos[ t ω + t ω1] Re[b2] + Im[b2] Sin[t ω + t ω1]) - 
I (ψ1 (Cos[t ω1] Im[b1] -Re[b1] Sin[t ω1]) + ψ2 (Cos[t ω + t ω1] Im[b2] -Re[b2] Sin[t ω + t ω1]))*)
  • 1
    $\begingroup$ Thanks for the info - your first answer is helpful, as it at least gives the [Omega]1 exponential term conjugate. Is there any way that you know of where the conjugates of b1 and b2 can be shown as b1* and b2*, or is this not a Mathematica functionality? $\endgroup$ Commented Mar 19, 2021 at 17:53
  • $\begingroup$ Sorry don't know how to do that. But if you know b1,b2are complex, tell it ComplexExpand(see my modified answer) $\endgroup$ Commented Mar 19, 2021 at 18:05
  • 1
    $\begingroup$ (Ψconj = Assuming[Element[{t, ω1, ω}, Reals], ComplexExpand[Conjugate[Ψ], {b1, b2}, TargetFunctions -> Conjugate] // Simplify]) // TraditionalForm $\endgroup$
    – Bob Hanlon
    Commented Mar 19, 2021 at 19:21
  • $\begingroup$ @TubularHell Ulrich may be overthinking regarding how to achieve such formatting. Ulrich, you may simply add //FullSimplify//TraditionalForm to your last example to achieve OP’s desired form. $\endgroup$ Commented Mar 19, 2021 at 19:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.