Download Spring 2004 PHY662 Class Notes: WKB, Maple, Perturbation Theory and more Assignments Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004, Feb. 26, 2004 26th February 2004 1 Miscellaneous 1. HWK #7 is due Tues., Mar. 2. 2. I am out of town on Mon., Mar. 1 - so I encourage you to start early on the homework. Office hours Friday? 3-4? 3. Continue to read Ch. 17 Shankar (or Griffiths Ch. 6). 4. Today: WKB final wrapup, intro to perturbation theory in physics, do start non- degenerate time-independent perturbation theory. 5. Use maple along the way. 2 Intro to maple Computer symbolic manipulation, numerical methods, and plotting routines can make your life a whole lot easier, allowing you to focus on the physics. You need to know how to do things by hand, but automation can reduce errors and improve your work. 1. Basic introduction: a sequence of input prompts and outputs. Define functions, apply fsolve, etc. See “Basic Tasks” for a reference sheet. Read the New User Tour for a better background. 2. How can we normalize (numerically) the radial wavefunction U(r) = sin 2(Πr2 ) sinh(Πr) ? 3 WKB - just a little more. 1. Discuss tunneling out of a box: to get the rate of escape, multiply tunneling probability by “attempt rate”. (a) The “attempt rate” is also a semi-classical approximation: frequency of classical oscillation within the potential. 1 (b) Note that this frequency is constant for a harmonic potential (just ω) and is 2L/v, with v = √ 2E/m for a square well potential. (c) The tunneling probability is 2 ∫ κ(x) dx, where κ(x) = √ 2m[V (x)− E] and the integral is over the forbidden region. 2. The trickiest part of the WKB approximation is where the classical momen- tum vanishes. This is where the oscillating solution needs to be connected to the exponential solution. Derive, for example, the quantization condition 2 ∫ x2 x1 dx √ 2m(E − V ) = (n + 12 )h for a bound state with classical turning points x1 and x2, by matching the exp(±i ∫ k(x) dx) with exp(− ∫ κ(x) dx) parts of the WKB solution, using an Airy function where each of these functions breaks down. Note the asymptotic forms Ai(z) ∼ (2 √ πz1/4)−1e− 2 3 z 3/2 z 0 Ai(z) ∼ [( √ π(−z)1/4]−1 sin [ 2 3 (−z)3/2 + π 4 ] z 0 . Apply this to connection formula, following Griffiths. The term ei ∫ k dx, e−i ∫ k dx can be written as a sin() with a phase shift (the wave function can be chosen to be real). By changing coordinates z = α(x− xr), α = [ 2mV ′(xr) h̄2 ]1/3 , near the classical turning point x = xr, one gets the equation d2ψ dz2 = zψ , for small z. The result from matching with aAi(z) = aAi(αx), with is (choosing a = D √ 4π αh̄ ) ψ(x) ≈ 2D√ k(x) sin[ ∫ xr x k(x′) dx′], ifx < xr D√ k(x) e −h̄−1 ∫ x xr |k(x′)| dx′ , ifx > xr. This means that the function D √ 4π αh̄ Ai[α(x− xr)] is the patching function between the two WKB solutions. For an infinite wall at x = 0, this gives a quantization condition:∫ xr 0 k(x) dx = (n− 1 4 )π, n = 1, 2, . . . . If the left turning point at xl is also “soft”, one gets the quantization condition∫ xr xl k(x) dx = (n− 1 2 )π, n = 1, 2, . . . . 2
