What is the relationship between frequency and wave function?

[phyky]

wave function always refer to the position of particle with probability amplitude. since wave function is not a wave but field, frequency seems meaningless to it. can i imagine it is a vibration field with certain frequency f in complex plane or something?

 

[jtbell]

vibration field with certain frequency f in complex plane or something?

That's exactly what it is. A wave function generally has a complex phase factor of the form $e^{i \omega t} = \cos (\omega t) + i \sin (\omega t)$ where $\omega = 2 \pi f$.