what is phase angle of wave function $\phi \,$?

Question

this is wave function: $$\Psi{(\vec r, t)}=\Psi_0 e^{i(\vec k \cdot \vec r-\omega t)}$$ $$\Psi{(\vec r, t)}=A e^{i(\phi + \vec k \cdot \vec r-\omega t)}$$.

1. what is phase angle $\phi$ of wave function?

2. is there any graph that shows what is phase angle of waves?

3. how measure phase angle of light (double slit experiment)?

Answer 1

To address your questions 1 and 2: this graph shows the real part of $\Psi{(\vec r, t)}=A e^{i(\vec k \cdot \vec r-\omega t)} $ in blue and the real part of $\Psi{(\vec r, t)}=A e^{i(\phi + \vec k \cdot \vec r-\omega t)} $ in purple. Obviously $\Psi$ is a function of two variables, so you can regard the graph either as keeping $\vec r$ constant and varying $t$ or keeping $t$ constant and varying $\vec r$.

The quantity $\phi$ is just the phase difference between the two waves e.g. the distance between the peaks shown by the arrow on the diagram.

The absolute value of $\phi$ has no physical significance because you can measure $\phi$ from any reference point you want. However the difference in $\phi$ between two wavefunctions has a very important physical meaning because it determines how the waves will interfere.

To address your question 3: actually the mention of the double slit experiment is spot on. The slits split the incoming light (or electrons or whatever) into two sources, call these $\Psi_a$ and $\Psi_b$, and if you take some point on the screen, this point will receive light from $\Psi_a$ and from $\Psi_b$, but the phase of the two waves, $\phi_a$ and $\phi_b$ won't be the same.

There isn't any physical meaning to the absolute phase of $\Psi_a$ and $\Psi_b$, $\phi_a$ and $\phi_b$, but if $\phi_a - \phi_b$ is an even multiple of $\pi$ ($2\pi$, $4\pi$, etc) the waves will be in sync and you'll get constructive interference and a bright area. If the phase difference is an odd multiple of $\pi$ ($\pi$, $3\pi$ etc) the waves will interfere destructively and you get a dark spot. This is exactly why you get the pattern of alternating bright and dark bands in the two slit experiment - it's because the phase difference, $\phi_a - \phi_b$, varies as you move along the screen.

So no experiment can measure the absolute value of $\phi_a$ or $\phi_b$, because the absolute value has no physical significance. However the double slit experiment can measure the phase difference $\phi_a - \phi_b$.