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Unformatted text preview: Physics 7c Discussion Problem
1. The Inﬁnite Square Well in 2-D (Separation of Variables Review) A particle of mass m is in an inﬁnite 2-D square well potential given by: V (x, y ) = 0 for 0 < x < a and 0 < y < b and ∞ everywhere else. a. Write down the two dimension TIME DEPENDENT Schroedinger equation. b. Split the time and spatial parts by writing Ψ(x, y, t) = ψ (x, y ) exp(−iωt). Find a simple equation that ψ (x, y ) must satisfy if the original Ψ(x, y, t) is a solution to the Schroedinger equation. c. The technique used in part b, is a standard separation of variables technique. It is useful because ALL solutions to the Schroedinger equation can be written as a sum F (x, y, t) = i ci ψ (x, y ) exp(−iωi t). The ωi ≡ Ei deﬁne energy values for which the the solution would just stationarily oscillate up and down. Let’s do the separation of variables once more to characterize the energy states. Write ψ (x, y ) = f (x)g (y ), and derive an equation in the form E − V = x-only-part + y-only-part. Find the the x and y parts. d. Because E (and V in the allowed region) is just a constant for every solution, we can write E = Ex + Ey (V is zero in the allowed region) and split the equations by letting Ex = x-piece from part c, and Ey = y-piece. e. Solve each of the equations separately, with the hint that that they involve sines or cosines whose argument is quantized by noting V’s eﬀect in imposing boundry conditions on f (x) and g (y ). f. Normalize the complete solution ψ (x, y ) to ﬁnd the amplitude (in front of the π /2 π /2 sines/cosines of part e. You may use 0 sin2 (x)dx = 0 cos2 (x)dx = π/4) 1 ...
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