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.
Problem 1: Bosons, Fermions
Consider a system of five particles, inside a container where the allowed energy levels
are nondegenerate and evenly spaced. For instance, the particles could be trapped in a
one-dimensional harmonic oscillator potential. In this problem you will consider the
allowed states for this system, depending on whether particles are identical fermions,
identical bosons, or distinguishable particles.
a) Describe the ground state of this system, for each of these three cases.
b) Suppose that the system has one unit of energy (above the ground state). Describe
the allowed states of the system, for each of the three cases. How many possible
system states are there in each case?
c) Repeat part (b) for two units of energy and for three units of energy.
d) Suppose that the temperate of this system is low, so that the total energy is low
(though not necessarily zero). In what way will the behavior of the bosonic system
differ from that of the system of distinguishable…
H2) Particle in a finite well: Let us consider the following potential.
V(x) = -Vo for |x| L
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Chapter 9 Solutions
Introduction To Quantum Mechanics
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Similar questions
- 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]arrow_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_forwardQuestion 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_forward
- Problem 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_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_forwardSolve 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_forward
- Problem 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_forwardHow 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_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_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_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_forward
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