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r2 - 23 Jul 2007 - 22:23:09 - ThanhTruongYou are here: TWiki >  Main Web > LectureNotes > UndergraduatePChem > Chapter5 > ParticleFiniteWell

Particle in a Finite Potential Well

In the first problem, we solve the problem in an infinite potential box.  That is not very realistic.   In reality, the potential well has a finite depth.  What difference does it make to the solution of the problem?

(Insert figure)

In this case, the potential energy V(x) of the system has the form:

\[V(x) = \left \lbrace \begin{array}{l} V_0 \; \; \; for \: x <-a \; \; \; (Region \: I) \\ 0 \; \; \; \; \; \;-a \leq x \leq a \; \; (Region \: II) \\V_0 \; \; \;for \: a < x \; \; \; (Region \: III)\end{array} \right. \]

The general solution to the Schrödinger equation for this system in each region is

\[\begin{array} {l} \psi_1(x) = Ae^{k_1x} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x < -a) \\ \psi_2(x)= B \sin (\alpha x) + C \cos (\alpha x) \; \; \; \; \; \; (-a \leq x \leq a) \\ \psi_3(x)=De^{-k_1x} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (a < x) \end{array} \]
 

where

\[\alpha = \frac {\sqrt {2mE}} {\hbar} \; \; \; \; \; and \; \; \; \; \; k_1 = \frac {\sqrt{2m(V_0-E)}} {\hbar}\]

-- ThanhTruong - 23 Jul 2007

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