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