Notes on Miscellaneous, WKB Semiclassical Approximation | PHY 662, Study notes of Quantum Mechanics

Download Notes on Miscellaneous, WKB Semiclassical Approximation | PHY 662 and more Study notes Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004, Feb. 10, 2004 19th February 2004 1 Miscellaneous 1. HWK #6 not quite complete. 2. Read Ch. 8, Griffiths. 2 WKB - semiclassical approximation 1. A method (mostly) for 1D problems. It is very useful for understanding tunnel- ing. 2. WhenE > V (x), the Schrodinger equation was wave type solutions, e±ikx, with k(x) = √ 2m h̄2 (E − V ). When E < V (x), the solutions are of the form e±κx, with κ = √ 2m h̄2 (V − E). This is strictly true for uniform potentials V (x) = const. 3. Now apply a type of perturbation expansion. Formulated either as an expansion in h̄ or that the classical momentum changes slowly compared to the particle’s wavelength (almost everywhere). 4. The trickiest part of this approximation is where the classical momentum van- ishes. This is where the oscillating solution needs to be connected to the expo- nential solution. Leads to connection functions: (a) These are simple if the potential changes very rapidly (over a distance much less than the particle “wavelength”). Apply continuity or vanishing of the wave function. (b) Example: quantization condition in a well with sharp sides gives 2 ∫ x2 x1 dx √ 2m(E − V ) = nh. [come back to after Merzbacher]. (c) Otherwise, need to be careful, use connection formula. These give, 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. 1