Significance of Phase Factor in Quantum Mechanics Measurements

[facenian]

If a system is in state |\psi> before a meassurement, then after we find a value a_n the system is in state
\frac{P_n|\psi>}{\sqrt{<\psi|P_n|\psi>}}

so for instance if |\psi>=c_1|a_1>+c_2|a_2>+c_3|a_3> after we find a_1 the system is in the state |\psi>'=e^{i\theta}|a_1> where \theta= is the argument of complex a_1.
I know that |\psi>' and |a_1> represent two different vectors of the Hilbert space of states but the same physical state.
My question is, is there any way that the phase factor e^{\theta} manifest itself in a physical experiment? or put in another way, does this factor have any physical significance?
I ask this because some systems when rotated 360 degrees transforms like |a> \rightarrow -|a> and this sign can be detected exparimentally

 

[olgranpappy]

If a system is in state |\psi> before a meassurement, then after we find a value a_n the system is in state
\frac{P_n|\psi>}{\sqrt{<\psi|P_n|\psi>}}

so for instance if |\psi>=c_1|a_1>+c_2|a_2>+c_3|a_3> after we find a_1 the system is in the state |\psi>'=e^{i\theta}|a_1> where \theta= is the argument of complex a_1.

you mean c_1.

I know that |\psi>' and |a_1> represent two different vectors of the Hilbert space of states but the same physical state.
My question is, is there any way that the phase factor e^{\theta} manifest itself in a physical experiment? or put in another way, does this factor have any physical significance?

No, I don't think so.

I ask this because some systems when rotated 360 degrees transforms like |a> \rightarrow -|a> and this sign can be detected exparimentally

 

[Ben Niehoff]

Phase factors can be detected in interference experiments. However, the absolute phase cannot be detected; only phase differences cause interference.

 

[facenian]

Thank you for your answers I think that know I've Got it