/*! 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} Q15P Show that two noncommuting opera... [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 3: Q15P (page 113)

Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if $\hat{P}$ and $\hat{Q}$ have a complete set of common eigenfunctions, then $[\hat{P} . \hat{Q}] f = 0$ for any function in Hilbert space.

Short Answer

Expert verified

A whole set of common eigenfunctions cannot be shared by two noncommuting operators.

Step by step solution

01

Concept used

The same complete set of common eigenfunctions:

$\hat{P} f_{n} = 位_{n} f_{n} and \hat{Q} f_{n} = 渭_{n} f_{n}$

02

Calculation

Assuming the operators $\hat{P}$ and $\hat{Q}$ have the same complete set of common eigenfunctions, that is:

$\hat{P} f_{n} = 位_{n} f_{n} and \hat{Q} f_{n} = 渭_{n} f_{n}$

And suppose the set $f_{n}$ is complete, so that any function in Hilbert space $f {x)$ can be expressed as a linear combination, that is:

$f = 鈭� c_{n} f_{n}$

Solving for the above function,

$[\hat{P}, \hat{Q}] f = (\hat{P} \hat{Q} - \hat{Q} \hat{P}) 鈭� c_{n} f_{n} = \hat{P} (鈭� c_{n} 渭_{n} f_{n}) - \hat{Q} (鈭� c_{n} 位_{n} f_{n}) = 鈭� c_{n} 渭_{n} 位_{n} f_{n} - 鈭� c_{n} 位_{n} 渭_{n} f_{n} = 0$

If two operators have the same set of eigenfunctions, the commutator will be zero.

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 Hermitian conjugate (or adjoint) of an operator $\hat{Q}$ is the operator ${\hat{Q}}^{鈥�}$ such that

$鉄� f 鈭� \hat{Q} g 鉄� = 鉄� {\hat{Q}}^{鈥�} f 鈭� g 鉄� 鈥�(蹿辞谤补濒濒 f and g).$

(A Hermitian operator, then, is equal to its Hermitian conjugate: $\hat{Q} = {\hat{Q}}^{鈥�}$ )

(a)Find the Hermitian conjugates of x, i, and $d / dx$ .

(b) Construct the Hermitian conjugate of the harmonic oscillator raising operator, $a_{+}$ (Equation 2.47).

(c) Show that ${(\hat{Q} \hat{R})}^{鈥�} = {\hat{R}}^{鈥�} {\hat{Q}}^{鈥�}$ .

An anti-Hermitian (or skew-Hermitian) operator is equal to minus its Hermitian conjugate:

${\overset{铃�}{Q}}^{t} = - \overset{铃�}{Q}$

(a) Show that the expectation value of an anti-Hermitian operator is imaginary. (b) Show that the commutator of two Hermitian operators is anti-Hermitian. How about the commutator of two anti-Hermitian operators?

The Hamiltonian for a certain three-level system is represented by the matrix

$H = (\begin{array}{l} a & 0 & b \\ 0 & c & 0 \\ b & 0 & a \end{array})$ , where a, b, and c are real numbers.

(a) If the system starts out in the state $| & (0) 鉄� = (\begin{array}{l} 0 \\ 1 \\ 0 \end{array})$ what is $| & (t)$ ?

(b) If the system starts out in the state $| & (0) 鉄� = (\begin{array}{l} 0 \\ 0 \\ 1 \end{array})$ what is $| & (t)$ ?

(a) Write down the time-dependent "Schr枚dinger equation" in momentum space, for a free particle, and solve it. Answer: $\exp (- i p^{2} t / 2 m h) 桅 (p, 0) .$

(b) Find role="math" localid="1656051039815" $桅 (p, 0)$ for the traveling gaussian wave packet (Problem 2.43), and construct $桅 (p, t)$ for this case. Also construct ${| 桅 (p, t) |}^{2}$ , and note that it is independent of time.

(c) Calculaterole="math" localid="1656051188971" $<p>$ androle="math" localid="1656051181044" $<p^{2}>$ by evaluating the appropriate integrals involving $桅$ , and compare your answers to Problem 2.43.

(d) Show thatrole="math" localid="1656051421703" $< H > = {< p >}^{2} / 2 m + {< H >}_{0}$ (where the subscript denotes the stationary gaussian), and comment on this result.

The Hamiltonian for a certain two-level system is

$\hat{H} = \overset{藱}{o} (1 > < 1 - 2 > < 2 + 1 > < 2 + 2 > < 1)$

where $1 >, 2 >$ is an orthonormal basis and localid="1658120083298" $\overset{藱}{o}$ is a number with the dimensions of energy. Find its eigenvalues and eigenvectors (as linear combinations oflocalid="1658120145851" $1 >$ and $2 >$ . What is the matrix H representing $\hat{H}$ with respect to this basis?

See all solutions

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Force

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Fluids

Read Explanation

Energy Physics

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.