Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 3.2, Problem 3.4P
(a)
To determine
Show that the sum of two hermition operator is hermition.
(b)
To determine
The condition for
(c)
To determine
The condition for the two product of two hermition operator is hermition.
(d)
To determine
Show that the position and Hamiltonian operator are hermition.
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
Find the following commutators by applying the operators to an arbitrary function f(x)
[e, x + d²/dx²]
[x³ — d/dx, x+d²/dx²]
(a) Show that the sum of two hermitian operators is hermitian.
(b) Suppose Ô is hermitian, and a is a complex number. Under what condition
(on a) is a Q hermitian?
(c) When is the product of two hermitian operators hermitian?
(d) Show that the position operator (f = x) and the hamiltonian operator (H =
-(h2/2m)d²/dx2? + V (x)) are hermitian.
Q#03: A and B are Hermitian operators and
AB – BA = iC. Prove that C is a Hermitian operator.
Chapter 3 Solutions
Introduction To Quantum Mechanics
Ch. 3.1 - Prob. 3.1PCh. 3.1 - Prob. 3.2PCh. 3.2 - Prob. 3.3PCh. 3.2 - Prob. 3.4PCh. 3.2 - Prob. 3.5PCh. 3.2 - Prob. 3.6PCh. 3.3 - Prob. 3.7PCh. 3.3 - Prob. 3.8PCh. 3.3 - Prob. 3.9PCh. 3.3 - Prob. 3.10P
Ch. 3.4 - Prob. 3.11PCh. 3.4 - Prob. 3.12PCh. 3.4 - Prob. 3.13PCh. 3.5 - Prob. 3.14PCh. 3.5 - Prob. 3.15PCh. 3.5 - Prob. 3.16PCh. 3.5 - Prob. 3.17PCh. 3.5 - Prob. 3.18PCh. 3.5 - Prob. 3.19PCh. 3.5 - Prob. 3.20PCh. 3.5 - Prob. 3.21PCh. 3.5 - Prob. 3.22PCh. 3.6 - Prob. 3.23PCh. 3.6 - Prob. 3.24PCh. 3.6 - Prob. 3.25PCh. 3.6 - Prob. 3.26PCh. 3.6 - Prob. 3.27PCh. 3.6 - Prob. 3.28PCh. 3.6 - Prob. 3.29PCh. 3.6 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Prob. 3.32PCh. 3 - Prob. 3.33PCh. 3 - Prob. 3.34PCh. 3 - Prob. 3.35PCh. 3 - Prob. 3.36PCh. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - Prob. 3.39PCh. 3 - Prob. 3.40PCh. 3 - Prob. 3.41PCh. 3 - Prob. 3.42PCh. 3 - Prob. 3.43PCh. 3 - Prob. 3.44PCh. 3 - Prob. 3.45PCh. 3 - Prob. 3.47PCh. 3 - Prob. 3.48P
Knowledge Booster
Similar questions
- (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ĝ, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q. with eigenvalue q. (b) Check that f (x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the operator d2/dx2, with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).arrow_forward(a) Show that for a Hermitian bounded linear operator Ĥ : H → H, all of its eigen- values are real and the eigenvectors corresponding to different eigenvalues are orthogonal. Hint: start by calculating (ø|H|ø) for an eigenstate lø).arrow_forwardTaking into account that the product of two Hermitian operators Aˆ and Bˆ can be written as a) Prove that kˆ and hˆ are Hermitians.arrow_forward
- Consider the following operators on a Hilbert space V³ (C): 0-i 0 ABAR-G , Ly i 0-i , Liz 00 √2 0 i 0 LE √2 010 101 010 What are the corresponding eigenstates of L₂? 10 00 0 0 -1 What are the normalized eigenstates and eigenvalues of L₂ in the L₂ basis?arrow_forward: The Hamiltonian for the one-dimensional simple harmonic oscillator is: mw? 1 ÎĤ =- + 2m Use the definition of the simple harmonic oscillator lowering operator 1 -î + iv mwh mw V2 and its Hermitian conjugate to: (a) Evaluate (â', â] (b) Show that = Vi e n = Vmwh () and Ĥ = ħw(âtâ + }) à ât %3D (c) Evaluate [âtâ, â] (d) Find (î),(f), (x²) and (p2) for the nth stationary state of the harmonic oscillator. Check that the uncertainty principle is satisfied.arrow_forwardSuppose I have an operator A, and I discover that Â(2²) = 5 sinx and Â(sin x) = 5x². (a) Find Â(2² - sin x) (b) Name one eigenfunction and one eigenvalue of A.arrow_forward
- The Hamiltonian of a certain system is given by [1 0 H = ħw]0 LO 0 1 Two other observables A and B are represented by 1 0 0 , B = b]0 2 0 lo o 0- i 0 A = a|-i 0 0 0 1 w, a, b are positive constant. a. Find the eigenvalues and normalized eigenvectors of H b. Suppose the system is initially in the state 2c lµ(0) >= -c 2c where c is a real constant. Determine the normalized state |(t) >. c. What are the eigenvectors of B? d. Find the expectation values of A and B in the state |p(t) >, and hence determine if A and B are conservative observablesarrow_forwardI).Show that if Aˆ is a Hermitian operator in a function space, then so is the operator Aˆn, where n is any positive integer. ii).Now consider a function space of the set of functions f(x) of a real variable x ∈ (−∞, +∞),such that f(x) → 0 as x → ±∞. For a unit weight function w(x) = 1, determine if thefollowing operators are Hermitian:i) id/dx, andii) id5/dx5.arrow_forwardShow that the eigenvalues of an anti-Hermitian linear bounded operator  : H → H are either purely imaginary or equal to zero. Note that for anti- Hermitian operators Â, we have A = -A.arrow_forward
- If A, B and C are Hermitian operators then 1 2i verfy whether the relation Hermitian or not. [AB] isarrow_forwardThe Hamiltonian of a certain system is given by 0 0 H = ħw|0 _0 0 0 1 1 Two other observables A and B are represented by A = a|-i 0 0 0 0 1 [1 0 0 B = b|0 _2 0 Lo o 2 w, a, b are positive constant. a. Find the eigenvalues and normalized eigenvectors of H b. Suppose the system is initially in the state 2c ly(0) >= 2c where c is a real constant. Determine the normalized state |4(t) >. c. What are the eigenvectors of B? d. Find the expectation values of A and B in the state [Þ(t) >, and hence determine if A and B are conservative observablesarrow_forwardAn Operator O is said to be linear if O{c1 f1(x)+ c2 f2(x)} =c1 O f1(x) +c2 O f2(x). Check the linearity of O (psi of x)= x(d/dx)(psi of x) , O(psi of x)= exp(psi of x) and O(psi of x) = x3 (psi of x)arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON