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 9.2, Problem 9.3P
To determine
The approximate transmission probability for a particle of energy
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Problem 1.17 A particle is represented (at time=0) by the wave function
A(a²-x²). if-a ≤ x ≤+a.
0,
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4(x, 0) = {
(a) Determine the normalization constant A.
(b) What is the expectation value of x (at time t = 0)?
(c) What is the expectation value of p (at time t = 0)? (Note that you cannot
get it from p = md(x)/dt. Why not?)
(d) Find the expectation value of x².
(e) Find the expectation value of p².
Problem 4.38. Two magnetic systems in thermal contact
Consider two isolated systems of noninteracting spins with NA
energies are EA = -2µB and Eg = -2µB.
= 4 and NB
= 16. Their initial
(a) What is the total number of microstates available to the composite system?
(b) If the two systems are now allowed to exchange energy with one another, what is the probability
that system A has energy EA?
(c) What is the mean value of EA and its relative fluctuations? Calculate the analogous quantities
for system B.
Suppose 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).
Chapter 9 Solutions
Introduction To Quantum Mechanics
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- A 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_forwardProblem 2.29 Analyze the odd bound state wave functions for the finite square well. Derive the transcendental equation for the allowed energies, and solve it graphically. Examine the two limiting cases. Is there always an odd bound state?arrow_forwardFor 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.arrow_forward
- 6.2 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_forwardProblem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that barrow_forwardProblem 7.3. Consider a system consisting of a single hydrogen atom/ion, which has two possible states: unoccupied (i.e., no electron present) and occupied (i.e., one electron present, in the ground state). Calculate the ratio of the probabilities of these two states, to obtain the Saha equation, already derived in Section 5.6. Treat the electrons as a monatomic ideal gas, for the purpose of determining J-arrow_forwardConsider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure 6.10 (attached). The particle is in thermal equilibrium with a reservoir at temperature so the probabilities of its various states are determined by Boltzmann statistics. If we keep the cubic term in the Taylor series as well, the integrals in the formula for x become difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from X0 by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series for u(x).arrow_forwardStarting from equation 7.83 attached, derive a formula for the density of states of a photon gas (or any other gas of ultrarelativistic particles having two polarization states). Sketch this function.arrow_forwardDetermine the transmission coefficient for a rectangular barrier (same as Equation 2.127, only with +Vo in the region -a Vo (note that the wave function inside the barrier is different in the three cases). Partial answer: For Earrow_forwardProblem 3.10 Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not?arrow_forwardA neutron of mass m with energy E a,V(x) =+Vo I. Draw the potential sketch! Write down the Schrödinger equation for: II. region I (0 a,V(x) =+Vo ), Calculate the total probability of nucleons tunneling through the barrier?; from region I to II (with E 2 0 and E < 0). II.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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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