Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 9, Problem 9.14P
To determine
The bound state energy for the potential in Problem 2.52 using the WKB approximation.
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For Problem 9.18, how do I determine part A & B? This is from a chapter titled, "Electron Spin." This chapter is part of quantum mechanics.
H2) Particle in a finite well: Let us consider the following potential.
V(x) = -Vo for |x| L
%3D
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 9 Solutions
Introduction To Quantum Mechanics
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- Question 2 2.1 Consider an infinite well for which the bottom is not flat, as sketched here. If the slope is small, the potential V = 6 |x|/ a may be considered as a perturbation on the square- well potential over -a/2 ≤x≤a/2. -8 W V(x) a/2 -a/2 X Calculate the ground-state energy, correct to first order in perturbation theory. Given (0) = √²/co COS Ground state of box of size a: = Ground state energy: E(0) = 4²k² 2ma². 0 Yarrow_forwardProblem 3.36. Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle." (a) Show that the chemical potential is N+ - kT ln N (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?arrow_forwardProblem 6.25 Express the expectation value of the dipole moment pe for an electron in the hydrogen state 1 4 = (211 +210) √2 in terms of a single reduced matrix element, and evaluate the expectation value. Note: this is the expectation value of a vector so you need to compute all three components. Don't forget Laporte's rule!arrow_forward
- 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.arrow_forwardProblem 2.14 In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? Hint: Classically, the energy of an oscillator is E = (1/2) ka² = (1/2) mo²a², where a is the amplitude. So the “classically allowed region" for an oscillator of energy E extends from –/2E/mw² to +/2E/mo². Look in a math table under “Normal Distribution" or "Error Function" for the numerical value of the integral, or evaluate it by computer.arrow_forwardA particle of mass in moving in one dimension is confined to the region 0 < 1 < L by an infinite square well potential. In addition, the particle experiences a delta function potential of strengtlh A located at the center of the well (Fig. 1.11). The Schrödinger equation which describes this system is, within the well, + A8 (x – L/2) v (x) == Ep(x), 0 < x < L. !! 2m VIx) L/2 Fig. 1.11 Find a transcendental equation for the energy eigenvalues E in terms of the mass m, the potential strength A, and the size L of the system.arrow_forward
- How do I prove that the transition is successful in problem 7.22?arrow_forwardSuppose a particle of mass m and charge q is in a one-dimensional harmonic oscillator potential with natural frequency wo. For times t > 0 a time-dependent potential of the form V₁(x,t) = εx cos(wt) is turned on. Assume the system starts in an initial state In). 1. Find the transitionn probability from initial state (n) to a state \n') with n' ‡ n. 2. Find the transition rate (probability per unit time) for the transition (n) → \n'). Note: (n'|x|n)= = ħ 2mwo (√√n +18n',n+1 + √ñdn',n−1).arrow_forward4.7 Let (x.t) be the wave function of a spinless particle corresponding to a plane wave in three dimensions. Show that (x.-) is the wave function for the plane wave with the momentum direction reversed. b. Let x(n) be the two-component eigenspinor of a-n with eigenvalue +1. Using the explicit form of x(n) (in terms of the polar and azimuthal angles $ and y that characterizen) verify that -io₂x() is the two-component eigenspinor with thearrow_forward
- conditions.) Problem 2.4 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) = ∞ 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. Droblo m 25 Celaulnte lu) .2arrow_forward2.29 Consider a particle in one dimension bound to a fixed center by a 6-function potential of the form V(x) = -vod(x) where v is real and positive. Find the wave function and the binding energy of the ground state. Are there excited bound states?arrow_forwardThe potential energy within the embedded atom method (EAM) formalism is ex- pressed as (5) A special case of EAM potentials is the Finnis-Sinclair (FS) type of potential where -√√₁. Derive the pairwise forces within a dimer (between two isolated atoms), separated by the distance, ro, in for FS potentials. F(pi) = U = [Fi(P₁) + = ΣV(rij), Pi = = [ f(rij) jfiarrow_forward
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