# Box Normalization

Question

Free particle in 1D where V(x) = 0. There is a general boundary condition $$\psi(x+L)=e^{i\theta}\psi(x)$$ used for box normalization which has arbitrary phase theta. $$E=k^2\hbar/(2m)$$ is true for free particle energies.

Attempt

Comparing with the condition $$\psi(x+L)=\psi(x)$$ I don't see how I will get different energies E since L is still the maximum wavelength, therefore $$\lambda = L/n = 2\pi/k$$ or $$k = 2n\pi/L$$ for n = 1, 2, ...; and then energies $$E_n$$ can be computed.

How do I get theta dependence into the energies for the case $$\psi(x+L)=e^{i\theta}\psi(x)$$? Or maybe the better question is do I need theta dependence in the energies for a correct solution? Shouldn't the phase of a wave function have no physical significance?

Given the k above is true then my normalized eigenfunctions would be $$\psi_n(x) = L^{-1/2} \exp(i(2\pi n/L)x+i\theta)$$? ...But I'm not sure that k is correct.

Can anyone clear this up for me? Much thanks.

Last edited: