Please answer the following question A potential well is defined as follows U(x)

Question

Please answer the following question

A potential well is defined as follows U(x) + infinity for x L. Consider an object with mass m and energy E, such that E L. Do not apply any boundary conditions. If you introduce any new variables you must explicitly define them. Apply the boundary conditions at x = 0 and as x rightarrow infinity. Does this determine any of the arbitrary constants? If so which ones? Apply the proper boundary and continuity conditions and find an equation that must be satisified by the energy. Explain how one would go about determining the allowed values of energy.

Explanation / Answer

-h^2/2m f'' + Vf =Ef ...E>0 ...here h = h cross = h/2pi

a) v= 0 => E f1 = -h^2/2m f1'' where f is the wavefunction

so f1 = Asin k1x + Bco k1x were k1 =sqrt(2mE)/h

b) V =Uo => (E-Uo)f2 = -h^2/2m f2'' => f2 = Ce^k2x + De^-k2x where k2 = sqrt(2m(Uo-E))/h

A B C D are variables

c) at x= 0, f1= 0 => B =0....

and at x=L ..f1 = f2 also f1' = f2' => Asin (k1L) = Ce^k2L + D e^-k2L also Ak1 cos k1L = Ck2 e^k2L - Dk2 e^-k2L

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