WeekProb_Nov23 - Physics 7c Discussion Problem 1 The...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online