Conjugate of a Wavefunction

Question

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?

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

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]))*)