The WKB Approximation-Quantum Physics and Mechanics-Lecture Slides, Slides of Quantum Mechanics

Download The WKB Approximation-Quantum Physics and Mechanics-Lecture Slides and more Slides Quantum Mechanics in PDF only on Docsity! Homework Phys 452 Wednesday Feb 29: assignment # 13 8.5, 8.6, 8.9, 8.13 docsity.com Homework Phys 452 Friday Feb 25: assignment # 12 WKB &Tunneling: Pb 8.3 Pb 8.4 Pb 8.16 WKB & turning points: Connection formulae Pb 8.5 Pb 8.6 Pb 8.9 Pb 8.13 Tuesday Mar 1: assignment # 12 docsity.com Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Turning point   0p x  docsity.com Quiz 19a Phys 452 What does physically happen to the wave function at the turning points? A. The amplitude goes to infinity B. The wave function is undefined C. The wave function collapses D. The wave function is finite and can be retrieved by continuity between the WKB approximated wave functions A. The wave function is finite and can be retrieved by patching the WKB approximated wave functions docsity.com Patching region Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V)   ( ) ( ) i p x dxC x e p x       1 ( ) ( ) p x dxD x e p x     docsity.com Phys 452 The WKB approximation Solving the Schrödinger equation in the patching region Solutions: Airy functions      p z Ai z Bi z    docsity.com Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Patching Linear approximation Patching region Overlap 1 Overlap 2 X=0 docsity.com The WKB approximation Patching Phys 452 • Continuity on overlap 1     3/2 1/4 2 sin 3 4 WKB P A x x               ( ) 0x  ( ) • Continuity on overlap 2     3/22 3 1/4 x WKB P C e x        0x  docsity.com Phys 452 The WKB approximation           1 1 ' 1 exp ' ' 2 ' 1 sin ' ' 4 x x x x D p x dx p x x D p x dx p x                      1x x 1x x Patching – downward slope Pb 8.9 • General expression for the wave function right side (scattering state) left side (bound state) docsity.com Phys 452 The WKB approximation Connection formula Different scenarios Potential with 2 walls Potential with 1 wall Potential with no walls docsity.com Quiz 19b Phys 452 Which one of these cases does NOT provide enough information to quantize the energy using the WKB approximation? A. A potential with two walls B. A potential with no walls but two turning points C. A potential with no walls but one turning point D. A potential with one wall and one turning point E. A potential with two walls and one turning point docsity.com Phys 452 The WKB approximation Connection formulas • Potential with no walls but two turning points V(x) E Turning points 1x 2x Connect the two expressions  2,x x  1,x x   2 1 1 2 x x p x dx n          Same phase modulo  docsity.com Homework Phys 452 Friday Feb 26: assignment # 13 WKB & turning points: Pb 8.9 (downward slope) Pb 8.5 Pb 8.6 Example of gravity potential Pb 8.13: logarithmic radial potential docsity.com