Quantum Mechanics Homework 4: Solutions for Problems P1-P4, Assignments of Quantum Mechanics

Download Quantum Mechanics Homework 4: Solutions for Problems P1-P4 and more Assignments Quantum Mechanics in PDF only on Docsity! Quantum Mechanics A (PHY 5645): Homework 4 DUE: Friday Oct 31 P1(30 points): Consider the potential shown below. Apply the WKB method and the matching conditions to calculate the transmission coefficient for a wave at energy E (shown) incident from the left. The transmission coefficient is T = |ψtrans√ptrans|2 |ψinc√pinc|2 . Express your answer in terms of the parameter θ = e 1 h̄ ∫ x2 x1 |p(x′)|dx′ . (Hint: Unlike in the problem which we discussed in class, to successfully match the WKB solutions, you will also need to use the second, exponentially growing, solution of the differ- ential equation: ∂2ψ ∂x2 − xψ = 0. The two linearly independent solutions are Ai(x) = 1 π ∫ ∞ 0 dt cos ( t3 3 + xt ) (1) Bi(x) = 1 π ∫ ∞ 0 dt ( sin ( t3 3 + xt ) + ext− t3 3 ) (2) The asymptotic expansion of Bi(x) is lim x→∞Bi(x) → 1√ πx 1 4 e 2 3 x 3 2 (3) lim x→−∞Bi(x) → 1√ π(−x) 14 cos ( 2 3 (−x) 32 + π 4 ) (4)