I want to look at the complex wave function $\psi$ in quantum mechanics.

If a complex number $a + bi$ is multiplied by $i$ it is rotated by 90 degree in the complex plane.

What does this mean for a complex scalar field, i.e. the wave function $\psi$?

I know, that $i\psi$ is as well a solution of the Schrödinger equation, because it is a linear differential equation.

My questions are:

  • What are the physical consequences, if one multiplies the wave function by $i$?
  • Are there any examples of actual experiments/phenomena, where something like that happens?
  • How can I depict the multiplication of the wave function by $i$?

There's no physical consequence in multiplying a wave function $\psi$ by a phase factor $e^{i\phi}$. Since in quantum mechanics we're interested in probabilities where the wave function appears always as $|\psi|^2$, the two wave functions $$\psi\qquad e^{i\phi}\psi$$ yield exactly the same physical results.

Multiplying by $i$ falls in this category since $e^{i\pi/2}=i$.

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    $\begingroup$ That's true. I only want to add that phase is only important when dealing with many wavefunctions. If two wavefunctions interact, their relative phase is important. But for a lonely $\psi$ it is meaningless $\endgroup$ – FGSUZ Jul 11 at 21:58
  • $\begingroup$ @FGSUZ Yeah, that's a very good point. $\endgroup$ – Local Mathmatician Jul 11 at 22:07

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