Probability function being imaginary?

[Ananthan9470]

Suppose ψ₁ and ψ₂ are two eigenfunctions of a particle and ε₁ and ε₂ are the corresponding eigenvalues. If the state is in the superposition Ψ = αψ₁ + βψ₂ at time t=0, it evolves in time by the equation Ψ = αψ₁e^{i ħ/ε₁ t} + βψ₂e^{i ħ/ε₂ t}. I am trying to understand the probability amplitude Ψ*Ψ.If you take Ψ* and multiply with Ψ in the cross terms of ψ₁ and ψ₂ there will be the exponential part containing the imaginary component i. How is this possible? Shouldnt the probability function, being the square of the wave function always be real so that we have measurable quantity associated with the wave function? If the probability function stops being real, what does that mean in the real world? Thanks!

 

[blue_leaf77]

There will be two cross terms one of which is the complex conjugate of the other, if you add them the imaginary part will vanish.

 

[Ananthan9470]

There will be two cross terms one of which is the complex conjugate of the other, if you add them the imaginary part will vanish.

Thanks a lot! Makes sense!