Q3P Two identical spin-zero bosons a... [FREE SOLUTION]

91影视

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 6: Q3P (page 255)

Two identical spin-zero bosons are placed in an infinite square well (Equation 2.19). They interact weakly with one another, via the potential
$V (x_{1,} x_{2}) = - a V_{0} 未 (x_{1} - x_{2}) .$ (2.19).
(where $V_{0}$ is a constant with the dimensions of energy, and a is the width of the well).
(a)First, ignoring the interaction between the particles, find the ground state and the first excited state鈥攂oth the wave functions and the associated energies.
(b) Use first-order perturbation theory to estimate the effect of the particle鈥� particle interaction on the energies of the ground state and the first excited state.

Short Answer

Expert verified

(a) Ground state:

$唯_{1}^{0} (x_{1}, x_{2}) = 唯_{1} (x_{1}) 唯_{1} (x_{2}) = \frac{2}{a} s i n (\frac{蟺 x_{1}}{a}) s i n (\frac{蟺 x_{2}}{a}); E_{1}^{0} = 2 E_{1} = \frac{蟺^{2} 魔^{2}}{m a^{2}}$

First excited state:

$唯_{2}^{0} (x_{1}, x_{2}) = \frac{1}{\sqrt{2}} [唯_{1} (x_{1}) 唯_{2} (x_{2}) + 唯_{2} (x_{1}) 唯_{1} (x_{2})] . = E_{2}^{0} = E_{1} + E_{2} = \frac{5}{2} \frac{蟺^{2} 魔^{2}}{m a^{2}}$

(b) $- \frac{32 V_{0}}{蟺} (\frac{3 蟺}{8} - \frac{5 蟺}{16}) = - 2 V_{0}$

Step by step solution

(a) Finding the ground state and the first excited state

In terms of the one-particle states (Eq. 2.28) and energies (Eq. 2.27):

$唯_{n} (x) = \sqrt{\frac{2}{a}} \sin (\frac{苍蟺}{a} x)$ (2.28).

$E_{n} = \frac{魔^{2} k_{n}^{2}}{2 m} = \frac{n^{2} 蟺^{2} 魔^{2}}{2 m a^{2}}$ (2.27).

Ground state: $唯_{1}^{0} (x_{1} x_{2}) = 唯_{1} (x_{1}) 唯_{1} (x_{2}) = \frac{2}{a} \sin (\frac{{蟺虫}_{1}}{a}) \sin (\frac{{蟺虫}_{2}}{a}); E_{1}^{0} = 2 E_{1} = \frac{蟺^{2} 魔^{2}}{m a^{2}}$

First excited state: $唯_{2}^{0} (x_{1} x_{2}) = \frac{1}{\sqrt{2}} [唯_{1} (x_{1}) 唯_{2} (x_{2}) + 唯_{2} (x_{1}) 唯_{1} (x_{2})]$

Step2: (b) estimating the effect of the particle– particle interaction on the energies of the ground state and the first excited state

$E_{1}^{1} = <唯_{1}^{0} |H^{'}| 唯_{1}^{0}> = (- a V_{0}) {(\frac{2}{a})}^{2} {鈭�}_{0}^{a} {鈭�}_{0}^{a} \sin^{2} (\frac{蟺 x_{1}}{a}) \sin^{2} (\frac{蟺 x_{2}}{a}) 未 (x_{1} - x_{2}) d x_{1} d x_{2} = - \frac{4 V_{0}}{a} {鈭�}_{0}^{a} \sin^{4} (\frac{蟺 x}{a}) d x = - \frac{4 V_{0}}{a} \frac{a}{蟺} {鈭�}_{0}^{蟺} \sin^{4} y d y = - \frac{4 V_{0}}{蟺}, \frac{3 蟺}{8} = - \frac{3}{2} V_{0} E_{2}^{1} = <唯_{2}^{0} |H^{'}| 唯_{2}^{0}>$

$= (- a V_{0}) (\frac{2}{a^{2}}) 鈭� {鈭�}_{0}^{a} {[\sin (\frac{蟺 x_{1}}{a}) \sin (\frac{2 蟺 x_{2}}{a}) + \sin (\frac{2 蟺 x_{1}}{a}) \sin (\frac{蟺 x_{2}}{a})]}^{2} 未 (x_{1} - x_{2}) d x_{1} d x_{2 .} = - \frac{2 V_{0}}{a} {鈭�}_{0}^{a} {[\sin (\frac{蟺 x}{a}) \sin (\frac{2 蟺 x}{a}) + \sin (\frac{2 蟺 x}{a}) + \sin (\frac{2 蟺 x}{a}) \sin (\frac{蟺 x}{a})]}^{2} d x = - \frac{8 V_{0}}{a} {鈭�}_{0}^{a} \sin (\frac{蟺 x}{a}) \sin (\frac{2 蟺 x}{a}) d x = - \frac{8 V_{0}}{a} . \frac{a}{蟺} {鈭�}_{0}^{蟺} \sin^{2} y \sin^{2} (2 y) d y = - \frac{8 V_{0}}{a} . 4 {鈭�}_{0}^{蟺} \sin^{2} y \sin^{2} y \cos^{2} y d y = - \frac{32 V_{0}}{蟺} {鈭�}_{0}^{蟺} (\sin^{4} y - \sin^{6} y) d y - \frac{32 V_{0}}{蟺} (\frac{3 蟺}{8} - \frac{5 蟺}{16}) = - 2 V_{0 .}$

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 the ground state and the first excited state

Step2: (b) estimating the effect of the particle– particle interaction on the energies of the ground state and the first excited state

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Quantum Physics

Physical Quantities And Units

Energy Physics

Further Mechanics and Thermal Physics

Turning Points in Physics

Study anywhere. Anytime. Across all devices.