An Electron in a Square Well One-Dimensional Case

A free electron can have any value of energy. However, the state of an electron changes if its motion is limited by external interactions. 

We consider first the one-dimensional case. The square well potential is used in quantum physics to represent a situation when a particle is confined to some region of space by forces holding it in the region. This model represents an infinite potential barrier placed at the ends of an interval 0 x L. The infinite square well potential is written as 

Two condihons have to be satisfied. The wave function must vanish at both ends of the interval I, that is tp (x) = 0. Further, the localization of the electron somewhere inside the interval is the sure event, that is the probability of this event has to be 1, 

A probability to find the electron within the segment dx equals to xp dx, 2 

The probability distribution is not a uniform one, that is probabilihes to find the electron in different places within the segment are different. For n = 1 the probability is maximal at x = 1/2, it decreases towards ends of the segment and 

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