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**Free Electron Model: Why periodic boundary conditions and what is "L"?**

Right, hello!

The quantum free electron model for electrons in solids (in One dimension) says we need to use periodic boundary conditions such that if Y(x) is the wavefunction, then Y(x) = Y(x+L).

Where L seems to be the width of the infinite square potential well used to derive the wavefunction and associated energy eigenvalues.

Now I realise that the periodic boundary conditions relates to the periodic lattice but, if L is the length of the 1D metal, then surely x+L is outside of the metal?

If L actually relates to some very small length inside the lattice, perhaps the size of a unit cell, then why when deriving the Fermi Energy do we use n=N/L where N is the total number of electrons in the metal and n is the number per unit length.

So my question really is, what is "L"?

Thanks

PS how do I use greek and other symbols here?