/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q32P Imagine two non interacting part... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 5: Q32P (page 245)

Imagine two non interacting particles, each of mass , in the one dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate $鉄� (x_{1} - x_{2})^{2} 鉄�$ assuming
(a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state.)

Short Answer

Expert verified

(a) $((x_{1} - x_{2})^{2}) = \frac{2 h}{m 蝇} .$

(b) $(x_{1} - x_{2})^{2} = \frac{h}{m 蝇}$

(c) $(x_{1} - x_{2})^{2} = \frac{3 h}{m 蝇}$

Step by step solution

01

(a) Determining ⟨(x1-x2)2⟩ when they are distinguishable particles.

From the Previous problem :

$鉄� x {鉄�}_{0} = 0 鉄� x {鉄�}_{1} = 0 {(x^{2})}_{0} = \frac{h}{2 m 蝇} x^{2} 1 = \frac{3 h}{2 m 蝇}$

We need to calculate:

$(0 |x| 1) = {鈭�}_{- 鈭�}^{鈭�} x 蠄_{0} (x) 蠄_{1} (x) d x = \sqrt{\frac{h}{2 m 蝇}} (\sqrt{1} 未_{0, 0} + \sqrt{0} 未_{1, - 1}) = \sqrt{\frac{h}{2 m 蝇}}$

From Equation 5.21:

role="math" localid="1658402226570" $({(x_{1} - x_{2})}^{2}) = \frac{h}{2 m 蝇} + \frac{3 h}{2 m 蝇} - 0 = \frac{2 h}{m 蝇}$

02

(b) Determining ⟨(x1-x2)2⟩when they are identical bosons,

From Equation 5.21

$({(x_{1} - x_{2})}^{2}) = \frac{2 魔}{m 蝇} + 2 \frac{h}{2 m 蝇} = \frac{h}{m 蝇}$

03

(c) Determining ⟨(x1-x2)2⟩ when they identical fermions,

From Equation 5.21:

$({(x_{1} - x_{2})}^{2}) = \frac{2 h}{m 蝇} + \frac{2 h}{2 m 蝇} = \frac{3 h}{m 蝇}$

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

Most popular questions from this chapter

The ground state of dysprosium (element 66, in the 6th row of the Periodic Table)

is listed as ${}^{5}{I_{s}}$ . What are the total spin, total orbital, and grand total angular

momentum quantum numbers? Suggest a likely electron configuration for

dysprosium.

(a) Calculate $< (1 / |r_{1} - r_{2}|) >$ for the state $蠄_{0}$ (Equation 5.30). Hint: Do $d^{3} r_{2}$ integral

first, using spherical coordinates, and setting the polar axis along $r_{1}$ , so

that

$蠄_{0} (r_{1}, r_{2}) = 蠄 100 (r_{1}) 蠄 100 (r_{2}) = \frac{8}{{蟺补}^{3}} e^{- 2 (r_{1} + r_{2}) / a}$ (5.30).

$|r_{1} - r_{2}| = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} {肠辞蝉胃}_{2}} .$

The $胃_{2}$ integral is easy, but be careful to take the positive root. You鈥檒l have to

break the $r_{2}$ integral into two pieces, one ranging from 0 to $r_{1}$ ,the other from $r_{1} t o 鈭�$

Answer: 5/4a.

(b) Use your result in (a) to estimate the electron interaction energy in the ground state of helium. Express your answer in electron volts, and add it to $E_{0}$ (Equation 5.31) to get a corrected estimate of the ground state energy. Compare the experimental value. (Of course, we鈥檙e still working with an approximate wave function, so don鈥檛 expect perfect agreement.)

$E_{0} = 8 (- 13.6 e V) = - 109 e V$ (5.31).

We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic domain by replacing the classical kinetic energy, $E = p^{2} / 2 m,$ ,with the relativistic formula, $E = \sqrt{p^{2} c^{2} + m^{2} c^{4}} - {mc}^{2}$ . Momentum is related to the wave vector in the usual way: $p = h k$ . In particular, in the extreme relativistic limit, $E 鈮� pc = h ck .$

(a) Replace $h^{2} k^{2}$ n Equation 5.55 by the ultra-relativistic expression, $h ck$ , and calculate $E_{tot}$ in this regime.

$dE = \frac{h^{2} k^{2}}{2 m} \frac{V}{蟺^{2}} k^{2} dk$ (5.55).

(b) Repeat parts (a) and (b) of Problem 5.35 for the ultra-relativistic electron gas. Notice that in this case there is no stable minimum, regardless of R; if the total energy is positive, degeneracy forces exceed gravitational forces, and the star will expand, whereas if the total is negative, gravitational forces win out, and the star will collapse. Find the critical number of nucleons, Nc , such that gravitational collapse occurs for $N > N_{C}$ is called the Chandrasekhar limit.

(c) At extremely high density, inverse beta decay $e^{-} + p^{+} 鈫� n + v$ ,converts virtually all of the protons and electrons into neutrons (liberating neutrinos, which carry off energy, in the process). Eventually neutron degeneracy pressure stabilizes the collapse, just as electron degeneracy does for the white dwarf (see Problem 5.35). Calculate the radius of a neutron star with the mass of the sun. Also calculate the (neutron) Fermi energy, and compare it to the rest energy of a neutron. Is it reasonable to treat a neutron star non relativistic ally?

Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let 蟽 be the number of free electrons per unit area.

(a) If $蠄_{a}$ and $蠄_{b}$ are orthogonal, and both normalized, what is the constant A in Equation 5.10?

(b) Ifrole="math" localid="1658225858808" $蠄_{a} = 蠄_{b}$ (and it is normalized), what is A ? (This case, of course, occurs only for bosons.)

See all solutions

Recommended explanations on Physics Textbooks

Solid State Physics

Read Explanation

Radiation

Read Explanation

Engineering Physics

Read Explanation

Magnetism

Read Explanation

Thermodynamics

Read Explanation

Fluids

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.