Q4P Suppose you don鈥檛 assume聽 Haa... [FREE SOLUTION]

91影视

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 9: Q4P (page 345)

Suppose you don鈥檛 assume $H_{a a} = H_{b b} = 0$
(a) Find $c_{a} (t)$ and $c_{b} (t)$ in first-order perturbation theory, for the case
.show that , to first order in .
(b) There is a nicer way to handle this problem. Let
.
Show that
where
So the equations for are identical in structure to Equation 11.17 (with an extra
(c) Use the method in part (b) to obtain in first-order
perturbation theory, and compare your answer to (a). Comment on any discrepancies.

Short Answer

Expert verified

${|c_{a}|}^{2} = [1 - \frac{i}{h} {鈭�}_{0}^{t} H_{a a} (t') d t'] [1 - \frac{i}{h} {鈭�}_{0}^{t} H_{a a} (t') d t'] = 1 {[1 - \frac{i}{h} {鈭�}_{0}^{t} H_{a a} (t') d t']}^{2} = 1 {|c_{a}|}^{2} = [1 - \frac{i}{h} {鈭�}_{0}^{t} H_{a a} (t') d t'] [1 - \frac{i}{h} {鈭�}_{0}^{t} H_{a a} (t') d t'] = 0 S o {|c_{a}|}^{2} + {|c_{b}|}^{2} = 1 (t o f i r s t o r d e r)$

(b) role="math" localid="1658488366910" $d_{a} = - \frac{i}{h} e^{\frac{i}{n} {鈭�}_{0}^{t} H (t) d t} c_{b} H_{a b} e^{i 蝇_{0} t} d_{b} = e^{\frac{i}{n} {鈭�}_{0}^{t} H (t) d t} (\frac{i}{h} H_{b b}) c_{b} + e^{\frac{i}{h} {鈭�}_{0}^{t} H (t) d t} c_{b}$

(c) $d_{b} = e^{\frac{i}{n} {鈭�}_{0}^{t} H (t) d t} (\frac{i}{h} H_{b b}) c_{b} + e^{\frac{i}{h} {鈭�}_{0}^{t} H (t) d t} c_{b}$

Step by step solution

(a) Finding ca(t) and cb(t)

$c_{a} = - \frac{i}{h} [c_{a} H_{a a} + c_{b} H_{a b} e^{i 蝇_{0} t}] 鈬� c_{b} = - \frac{i}{h} [c_{a} H_{a a} + c_{b} H_{a b} e^{i 蝇_{0} t}]\}$

$c_{b} = - \frac{i}{h} [c_{a} H_{a a} + c_{b} H_{a b} e^{- i {(E_{b} - E_{b})}^{t / h}}]$

Initial conditions: $c_{a} (t) = 1, c_{b} (0) = 0$ ,

Zero order:.

First order:

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

(a) Finding ca(t) and cb(t)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Torque and Rotational Motion

Measurements

Modern Physics

Conservation of Energy and Momentum

Particle Model of Matter

Study anywhere. Anytime. Across all devices.