# What happens if the wave function is multiplied by i?

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 $$e^{i\pi/2}=i$$.
• 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 – FGSUZ Jul 11 at 21:58