Question

(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in this potential energy region. If the particle has total energy E < 0, in the classical view it could only exist in region 2 (0 < x < L). Recall that E-K + U, and kinetic energy K is always a positive quantity. In the classical view, a particle with E < 0 cannot be found at x > L, since its kinetic energy there must be negative In Quantum Mechanics, we learned that there could be some non-zero probability for the wave- function of the particle with E < 0 to extend beyond x > L. However, it would have to drop off fairly quickly with increasing values or in order to be normalizable when x → +00 If the particle has energy E > 0, in the classical view it could exist anywhere in the region r > 0 For this reason, we will call any state with E > 0 an "unbound state" (a) (5 points) Setting up the wave equation for unbound states E >0 i. Write down the Time-Independent Schrödinger Equation (T.I.S.E.) valid for the wave function ψ2(x) in region 2 (0 < x < L). Write the corresponding time-independent Schrödinger equation for b3(x) in region 3. For region 1 (x < 0), due to U(x < 0) - +00, we must have ψ1(x)-0 for all x < 0 ii. Show that the general solution of the form V2(r)-Aeik2 + Be-ik2 satisfies T.LSE for region 2 ifk2- 2m(Et for region 2 if k iii. Show that the general solution of the form V3(x)-Ceikar + De-ik3r satisfies T.LS.E for region 3 if k

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