Q7P Use the WKB approximation to fin... [FREE SOLUTION]

91影视

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 8: Q7P (page 334)

Use the WKB approximation to find the allowed energies of the harmonic oscillator.

Short Answer

Expert verified

The allowed energies of the harmonic oscillator are $(\frac{1}{2}, \frac{3}{2} \frac{, 5}{2}, 鈰�) 鈩� 蝇 .$

Step by step solution

To find the energies of harmonic oscillator.

Consider a harmonic oscillator with mass and frequency $蝇$ . The associated potential is

$V (x) = \frac{1}{2} m 蝇^{2} x^{2} w h i c h c o r r e s p o n d s t o a p o t e n t i a l w e l l w i t h n o v e r t i c a l w a l l s . T h u s, t h e E q u a t i o n s a y s {鈭�}_{x_{1}}^{x_{2}} p (x) d x = (n - \frac{1}{2}) 蟺魔, (n = 1, 2, 3, . . .) 鈥� 鈥� 鈥� . (1) where x_{1} and x_{2} are the turning points satisfying E = \frac{1}{2} {m蝇}^{2} {x_{1}}^{2} = \frac{1}{2} {m蝇}^{2} {x_{2}}^{2}, (or) x_{1} = x_{2} = \frac{1}{蝇} \sqrt{\frac{2 E}{m}} . In this case, p (x) = \sqrt{2 m (E - \frac{1}{2} {m蝇}^{2} x^{2})} p (x) = m蝇 \sqrt{{x_{2}}^{2} - x^{2}} Thus, {鈭�}_{x_{1}}^{x_{2}} p (x) dx = 2 m蝇 {鈭�}_{0}^{x_{2}} \sqrt{{x_{2}}^{2} - x^{2}} dx This integral is done using the substitution x 鈮� x_{2} \sin 胃, so {鈭�}_{x_{1}}^{x_{2}} p (x) dx = \frac{蟺}{2} m蝇 {x_{2}}^{2} = \frac{蟺贰}{蝇}, and the quantization condition (Equation (1)) yields E_{n} = (n - \frac{1}{2}) 魔蝇 = (\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, . . .) 魔蝇 . In this particular case the WKB approximation actually delivers the exact allowed energies,$