Is the normalisation constant of a wavefunction real?

[spaghetti3451]

Homework Statement

Consider the wavefunction $\Psi (x, t) = c\ \psi (x) e^{-iEt/ \hbar}$ such that $\int | \Psi (x, t) |^{2} dx = 1$.

I would like to prove to myself that the normalisation factor $c$ is a real number.

Homework Equations

The Attempt at a Solution

$\int | \Psi (x, t) |^{2} dx = 1$

$\int c\ \psi (x)\ e^{-iEt/ \hbar}\ c^{*}\ \psi^{*}(x)\ e^{iEt/ \hbar}\ dx = 1$

$\int c\ \psi (x)\ c^{*}\ \psi^{*}(x)\ dx = 1$

$\int |c|^{2}\ |\psi (x)|^{2}\ dx = 1$

This isn't getting me anywhere, though!

 

[Orodruin]

You cannot prove this, quantum mechanics is fine with redefining all phases by the same phase factor and the physics does not depend on the phase of a state. It only depends on the relative phase of interfering states. However, you may choose the normalisation constant to be real for this very reason.

 

[spaghetti3451]

Thanks!

I knew that the phase factors are tunable, but I was not able to see that this could account for the plausibility of a real normalisation constant.

Now, it's all clear!