The general solution of particle in a 1D infinite potential well(width L) is given by: $$\psi(x,t)=\sum_n a_{n}.\sqrt{\frac{2}{L}}.\sin\bigg(\frac{n\pi x}{L}\bigg).\exp\bigg(-\frac{ iE_n.t}{\hbar}\bigg)$$ Here each of the term is an eigenfunction of the Schrodinger equation. If $\phi_n(x)=\sqrt{\frac{2}{L}}\sin\bigg(\frac{n\pi x}{L}\bigg)$ is a solution of schrodinger equation, then so is $\phi_n(x)=\sqrt{\frac{2}{L}}\sin\bigg(\frac{n\pi x}{L}\bigg).\exp(i\delta_n)$ . So the general solution should be - $$\psi(x,t)={\sum_n} a_{n}.\sqrt{\frac{2}{L}}\sin\bigg(\frac{n\pi x}{L}\bigg).\exp\bigg(-\frac{ iE_n.t}{\hbar}+i\delta_n\bigg),$$ where $\delta_n$ is real. Shouldn't this be the correct form? Why everyone neglects the phase?

  • $\begingroup$ Do you intend $\delta_n$ to be a specific expression? Because right now, you have placed no restrictions on it, which means that the wavefunction is in general not normalized if $\delta_n$ has any nonzero real part. $\endgroup$ – probably_someone Sep 5 '19 at 9:40

The coefficient $a_n$ is usually not presumed to be real. If we go with this convention, the phase $\delta_n$ can be absorbed into $a_n$ without a loss of generality.

The point is that an overall phase is physically irrelevant. So, when one talks about a specific energy eigenfunction such as a particular $\phi_n$, one doesn't care about an overall phase. You are right in noticing that when we put a bunch of individual eigenfunctions together in a linear combination, the relative phases among them matter. But, as I said, this relative phase is absorbed into the complex coefficients $a_n$ in a natural manner.

By the way, I am assuming that you meant $\exp{(i\delta_n)}$ with $\delta_n\in\mathbb{R}$ and not $\exp{(\delta_n)}$ because $\delta_n$ is not a phase if you meant $\exp{(\delta_n)}$. As pointed out in a comment, you cannot really say that if $\phi_n$ is a solution then $\phi_n\exp{(\delta_n)}$ is also a solution unless $\delta_n$ is purely imaginary (in which case, $-i\delta_n$ would be the phase and not $\delta_n$).


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