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 7.2, Problem 7.8P
(a)
To determine
Show that
(b)
To determine
Show that
(c)
To determine
Show that
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
The following problem arises in quantum mechanics (see Chapter 13, Problem 7.21). Find the number of ordered triples of nonnegative integers a, b, c whose sum a+b+c is a given positive integer n. (For example, if n = 2, we could have (a, b, c) = (2, 0, 0) or (0, 2, 0) or (0, 0, 2) or (0, 1, 1) or (1, 0, 1) or (1, 1, 0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in 3 boxes, and follow the method of the diagram in Example 5.
Let V = 2xyz3 +3 In(x² + 2y² + 3z²) V in free space. Evaluate each of the following quantities at P(3, 2, –1): (a) V; (b) |V|; (c) E; (d) |E|; (e) a (f) D.
Problem 3: Chemical potential of an Einstein solid.
Consider an Einstein solid for which both N and q are much greater than 1. Think of each
ocillator as a separate “particle".
a) Show that the chemical potential is
H = -kT In (**e)
b) Discuss this result in the limits N » q and N « q, concentrating on the question of how
much S increases when another particle carrying no energy is added to the system. Does
the formula make intuitive sense?
Chapter 7 Solutions
Introduction To Quantum Mechanics
Ch. 7.1 - Prob. 7.1PCh. 7.1 - Prob. 7.2PCh. 7.1 - Prob. 7.3PCh. 7.1 - Prob. 7.4PCh. 7.1 - Prob. 7.5PCh. 7.1 - Prob. 7.6PCh. 7.2 - Prob. 7.8PCh. 7.2 - Prob. 7.9PCh. 7.2 - Prob. 7.10PCh. 7.2 - Prob. 7.11P
Ch. 7.2 - Prob. 7.12PCh. 7.2 - Prob. 7.13PCh. 7.3 - Prob. 7.15PCh. 7.3 - Prob. 7.16PCh. 7.3 - Prob. 7.17PCh. 7.3 - Prob. 7.18PCh. 7.3 - Prob. 7.19PCh. 7.3 - Prob. 7.20PCh. 7.3 - Prob. 7.21PCh. 7.3 - Prob. 7.22PCh. 7.4 - Prob. 7.23PCh. 7.4 - Prob. 7.24PCh. 7.4 - Prob. 7.25PCh. 7.4 - Prob. 7.26PCh. 7.4 - Prob. 7.27PCh. 7.4 - Prob. 7.28PCh. 7.4 - Prob. 7.29PCh. 7.5 - Prob. 7.31PCh. 7.5 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - Prob. 7.38PCh. 7 - Prob. 7.39PCh. 7 - Prob. 7.40PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - Prob. 7.46PCh. 7 - Prob. 7.47PCh. 7 - Prob. 7.49PCh. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - Prob. 7.52PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.56PCh. 7 - Prob. 7.57P
Knowledge Booster
Similar questions
- The harmonic oscillator eigenfunction ₁(x) = = 4a³\1/4 e-ax²/2, where a = μω ħ (a) Find (x²) for an oscillator in this state, and express your result in terms of u, w, and ħ. 1 (b) In an eigenstate it is always true that (T) = (V) for a harmonic oscillator. If ↑ =-=-=-=A² 2μ 1 and ✩ = µw²x², find (p²) when the system is in the state 1₁ (x). (Note: You can use the result found in part (a), or compute it directly from the form of the operator p.) 2arrow_forwardThe first four Hermite polynomials of the quantum oscillator areH0 = 1, H1 = 2x, H2 = 4x2 − 2, H3 = 8x3 − 12x. Let p(x) = 12x3 − 8x2 − 12x + 7. Using the basis H = {H0, H1, H2, H3}, find the coordinate vector ofp relative to H. That is, find [p]H. This is a textbook question, not a graded questionarrow_forwardStarting with the equation of motion of a three-dimensional isotropic harmonic ocillator dp. = -kr, dt (i = 1,2,3), deduce the conservation equation dA = 0, dt where 1 P.P, +kr,r,. 2m (Note that we will use the notations r,, r2, r, and a, y, z interchangeably, and similarly for the components of p.)arrow_forward
- Find A and B so that f(x, y) = x2 + Ax + y? + B has a local minimum at the point (4, 0), with z-coordinate 5. A = B =arrow_forwardThe state u,> \u₂ > and lu,> from a complete set of orthogonal basis for a givne system. The state ₁ and ₂ > are defined as 14₁) = (1/√₁2² Y√2² ½ √₂) √2/√2 1+2) - (Y√5.a 1/√5) ,0, Are these state are normalized?arrow_forwardwhere and kyk₂ I 2 k 2m E 2 ħ² 2m ħ² (V-E) 3 Show that the solutions for region II can also be written as 2/₁₂ (²) = Ccas (₁₂²) + D sin (k₂²) for Z≤ 1W/ 4 Since the potential well Vez) is symmetrical, the possible eigen functions In You will be symmetrical, so Yn will be either even or odd. a) write down the even solution for region. II b) write down the odd solution region for on II In problem 2, explain why A=G=0₁arrow_forward
- Consider a 1-dimensional quantum system of one particle Question 01: in which the particle is under a potential V(x) = mw?a?, with m being the mass of the particle and w being a parameter (you may take it as angular fre- quency) with inverse dimension of time. The particle may be found in the region -0 < x < o. Varify that the lowest two states of the system are mutually orthonormal.arrow_forwardA triangle in the xy plane is defined with corners at (x, y) = (0,0), (0, 2) and (4, 2). We want to integrate some function f(x, y) over the interior of this triangle. Choosing dx as the inner integral, the required expression to integrate is given by: Select one: o Sro S-o f(x, y) dx dy x=0 2y y=0 O S-o So F(x, y) dæ dy O o S f(x, y) dy dæ O So So F(x, y) dx dy x/2 =0arrow_forwardQ4. Given three agents with states x¡, i = (1,2,3). The agents are connected together and they update their states in continuous domain based on the following rule: 3 š¼(t) = 2 (*;(t) – x;(t). j=1 a) Represent the state progress as a linear system i = Ax, where x = [x1, X2, X3]" and with the initial condition x(0) = [x1(0), x2(0), x3(0)]'. b) Analyze the properties of A, i.e., find its eigenvalues, eigenvectors, determinant, etc. c) Find the Jordan normal form of the matrix A. Find e4t analytically. Write down the analytical solution of x(t) and calculate lim x(t). Analyze behavior of the system as time goes to infinity. t→∞ How would the state progress with time?arrow_forward
- For the function f(z) = 2+2 of complex variable z, which of the following statements is incorrect? z2-2z Select one: Oa. z=0 is a simple pole with residue -1 Ob. z=2 is a simple pole with residue 2 ○ c. Both the first two options are correct O d. None of the first two options are correctarrow_forwardDo periodic functions, where f (x = 0) = f (x = L), form a vector space? If they do, explicitly show that they satisfy all eight properties required of a vector space. If not, which property fails? Show how it fails.arrow_forwardA system with j = 35 is in the state |ψ⟩= 1/√2 |35,35⟩ + 1/2 |35,34⟩ − 1/2 |35,−20⟩. The state is written in |j,m⟩ notation (m is the Jz projection). Find ⟨Jz⟩ and ∆Jz for this state. Find ⟨Jx⟩ and ∆Jx for this state. (Note: This must be done by hand with all work shown; also do this in bracket notation instead of working out the matrices)arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
University 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 Learning
Lecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-Wesley
College 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