# Conjugate of a Wavefunction

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?

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

## 1 Answer

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

ComplexExpand[Conjugate[Ψ]]
(*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]))*)

• 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? Commented Mar 19, 2021 at 17:53
• Sorry don't know how to do that. But if you know b1,b2are complex, tell it ComplexExpand(see my modified answer) Commented Mar 19, 2021 at 18:05
• (Ψconj = Assuming[Element[{t, ω1, ω}, Reals], ComplexExpand[Conjugate[Ψ], {b1, b2}, TargetFunctions -> Conjugate] // Simplify]) // TraditionalForm Commented Mar 19, 2021 at 19:21
• @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. Commented Mar 19, 2021 at 19:27