The Phase Shift - Quantum Mechanics - Lecture Notes | PHYSICS 137A, Study notes of Quantum Mechanics

Download The Phase Shift - Quantum Mechanics - Lecture Notes | PHYSICS 137A and more Study notes Quantum Mechanics in PDF only on Docsity! Physics 137A The Phase Shift Imagine a particle of mass m and energy E incident on a square potential V (x) = { V0 if − a < x < a 0 otherwise (1) with V0 < E. Label the region x < −a as I, −a < x < a as II and x > a as III. Since V (x) is even we can decompose the solutions into even and odd parity modes which in region II are ψII,e = cosκx and ψII,o = sinκx (2) where κ = √ 2m(E − V0) and we have set h̄ = 1 for simplicity. For now we consider only the even mode, the argument for the odd mode is completely analogous. In regions I and III, ψII,e must match onto even functions oscillating with period 2π/k where k = √ 2mE so ψI,e = A cos(kx− δe) (3) ψIII,e = A cos(kx+ δe) (4) where A and δe are constants. This satisfies the wave equation in all 3 regions and has the right parity. The only thing left is to match the function and its first derivative at x = ±a. At x = a we have cosκa = A cos(ka+ δe) (5) −κ sin κa = −kA sin(ka+ δe) (6) from which we can solve for the phase shift as tan(ka+ δe) = κ k tanκa (7) and then use Eq. (5) to solve for A. If we were to match at x = −a we would obtain the identical equations. An argument completely analogous to the above for the odd modes gives tan(ka + δo) = k κ tanκa (8)