Ground State Paradox of Free Electrons in Solids

3 GROUND STATE PARADOX OF FREE ELECTRONS IN SOLIDS 

For the solid states the infinitely high potential barrier stands as a valid approximation for the electronic stationary behavior however, this is 

Note that if one would like to consider the Hamiltonian with some potential that mimics the infinite barrier in asymptotic limit, i.e., as is the case of IV / a , a - 00, with a - the width of the free barrier, will soon conclude that the corresponding term in energy is not divergent in a very narrow domain, namely for o e [A / 2, A /12] that is in obvious contradiction with forming of stationary waves in the well therefore the only acceptable Hamiltonian in the case of infinite well is that restricted to the kinetic term only. 

Unfolding the ordinary variational procedure, one starts with imposing the normalization condition on the trial wave-function  

On the other side, the energy computation with the A constant expression, i.e., in the normalization condition of the wave-function, yields the paradoxical result  

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Ground State Paradox of Free Electrons in Solids.232... 

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