Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 7.2, Problem 7.13P
To determine
Prove that the Equation analog to 7.27 can be interpreted as the Eigen value equation for matrix W.
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Students have asked these similar questions
Write down the energy eigenfunctions for a particle in an infinitely deep one-
dimensional square well extending from z = -L/2 to z = +L/2 and check that they are
eigenfunctions of parity operator (that maps z H -z) corresponding to the eigenvalue
(-1)7-1, where n labels the energy level.
Solve the time-independent Schrödinger equation with appropriate
boundary conditions for an infinite square well centered at the origin [V (x) = 0, for
-a/2 < x < +a/2; V (x) = 00 otherwise]. Check that your allowed energies are
consistent with mine (Equation 2.23), and confirm that your y's can be obtained from
mine (Equation 2.24) by the substitution x x - a/2.
Problem #1
(Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and
derive the differential form that reveals A as a potential:
dA < -SdT – pdV [Eqn 5.20]
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
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- The Hermitian conjugate A† of a linear operator can be defined by ⟨ψ|Aφ⟩ = ⟨A†ψ|φ⟩ . Use this definition, along with the definition of the inner product of two functions, ⟨ψ|φ⟩ = ⎰ ψ∗(x) φ(x) dx, (where the weight function w(x) is taken to be 1), to prove/show the following three statements (image).arrow_forwardBit confused you say with is the definition of a complex conjugate but all I've ever seen is |X|^2=(X*)(X). Can you provide maybe a reference or proof of this?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
- For each positive integer n, define the set A, (2 (2n – 1): ieZ>o}. (a) Prove that P = {A1.A2,.. is a partition of the set Zo of positive integers.arrow_forwardLet f(x, y) = x² + 4y² and let C be the line segment from (0, 0) to (2, 2). You are going to compute là Vf. dr two ways: first, using the method learned in section 6.2 for с evaluating line integrals, and second, using the fundamental theorem for line integrals. First way: Vf=( C can be parameterized by r(t) = (t, Then '(t) and ▼ ƒ(r(t)) = { so sv. So 2 = [² = 2 - 1² || ( Vf. dr ▼ f(r(t)). r' (t)dt dt ). > > for 0 ≤ t ≤ 2. ).arrow_forwardlog z = log r + i is holomorphic in the region r>0 and - < 0 <. 10. Show that where z = re¹ with - < 0 < where is the Laplacian Ә əz əz A = 4 əz əz dx² 8² dy² 11. Use Exercise 10 to prove that if f is bolomorphic in the open set , then the real and imaginary parts of f are harmonic; that is, their Laplacian is zero.arrow_forward
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- Problem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation (for the infinite square well) with E = 0 or E < 0. (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation and showing that you cannot meet the boundary conditions.)arrow_forward6.2 Let the "uniform" ensemble of energy E be defined as the ensemble of all systems of the given type with energy less than E. The equivalence between (6.29) and (6.27) means that we should obtain the same thermodynamic functions from the "uniform" ensemble of energy E as from the microcanonical ensemble of energy E. In particular, the internal energy is E in both ensembles. Explain why this seemingly paradoxical result is true.arrow_forwardDemonstrate that e#ikz are solutions to both H and p, (momentum) for a free particle. Do you expect a difference for a bound particle where V (z) + 0?arrow_forward
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