Derivation of Bloch's theorem

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Why are the solutions satisfying ##\psi(x+l)=\lambda\,\psi(x)## (4.191) the only physically admissible solutions? (##l## is the period of the periodic potential.)

We may argue that the probability of finding an electron at ##x##, ##|\psi(x)|^2##, must be the same at any indistinguishable position:

##|\psi(x+l)|^2=|\psi(x)|^2##

This implies

##\psi(x+l)=\lambda\,\psi(x)##
##|\lambda|^2=1##
##\lambda=e^{iKl}##, which is the same as (4.196)

It seems that we can get (4.196) in a shorter way this way as compared to how it is done below by using the characteristic equation of matrix ##a## and the Wronskian determinant.

So it seems that the book justifies (4.191) in a different way from the argument using probability.

EDIT: I realise the probability argument does not justify the use of (4.191) either. Rather, it is justified by symmetry: Since the potential is periodic, the expectation values of all dynamical variables must be identical in every period. The only way this can happen is when (4.191) and (4.196) are true.

Still, it's unclear why the book uses such a long way to derive (4.196), without justifying the use of (4.191).

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Still, it's unclear why the book uses such a long way to derive (4.196), without justifying the use of (4.191).

View attachment 93523
It has been always my question that what is the reason for using such a long way to drive Bloch Theorem. However I thought this long way is to obtain the phase factor ##e^{iKl}##. Could you please specify exactly how it can be obtained from symmetry considerations?
 
665
24
It has been always my question that what is the reason for using such a long way to drive Bloch Theorem. However I thought this long way is to obtain the phase factor ##e^{iKl}##. Could you please specify exactly how it can be obtained from symmetry considerations?
Since the potential is periodic, every cell (of length ##l##) is indistinguishable from each other and the expectation values of all dynamical variables must be identical in every cell. The sufficient conditions are that (4.191) and (4.196) are true (because wave functions that differ by a constant phase factor have the same expectation values).

But I wonder if they are the necessary conditions too (ie., if two wave functions have the same expectation values of all dynamical variables, then they must differ by a constant phase factor).
 
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