Transcription of Bound States in One Dimension - University of Illinois ...
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Bound States in One Dimension - University of Illinois ...
Bound States in One DimensionIn this chapter we will concern ourselves with obtaining stationary state so-lution of the time independent Schr odinger Equation: h22m 2 x2 (x)+V(x) (x)=E (x)(1)for particles (such as electrons) Bound in one dimensional potential wells. Ingeneral these solutions can be chosen to be real rather than complex functionswhich can often be considerable simplification. To see this lets take the complexconjugate of the time independent SE given in Eq. (1): h22m 2 x2 (x)+V(x) (x)=E (x)(2)Because all the factors which operate on (x) are real, they don t change undercomplex conjugation and hence (x)and (x) are equally valid solutions of thesame potentialV(x) with the same energyE. Since the SE is a homogenouslinear differential equation, we can always form new valid solutions by takinglinear combinations of valid solutions. In particular, the solution r(x)whichisconstructed out of a general solutions (x)and (x) according to r(x) (x)+ (x)(3)is an intrinsically real.
the wave function vanishes outside the box, it must vanish on the walls of the box atx =0andx = a. Since we know that the wave function is comprised of sinusoidal functions we are searching for a wave function with a nodes at x =0 and x = a. Only a pure sine function has a node at the origin. The distance between nodes of a sinusoidal function ...
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