Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Question
Chapter 5, Problem 91CE
To determine
The uncertainty in position of the particle.
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Problem 3. Consider the two example systems from quantum mechanics. First, for a
particle in a box of length 1 we have the equation
h² d²v
2m dx²
EV,
with boundary conditions (0) = 0 and (1) = 0.
Second, the Quantum Harmonic Oscillator (QHO)
V = EV
h² d²
2m da² +ka²)
1
+kx²
2
(a) Write down the states for both systems. What are their similarities and differences?
(b) Write down the energy eigenvalues for both systems. What are their similarities
and differences?
(c) Plot the first three states of the QHO along with the potential for the system.
(d) Explain why you can observe a particle outside of the "classically allowed region".
Hint: you can use any state and compute an integral to determine a probability of
a particle being in a given region.
What do we need to do to average over Θ and ф to get the probability that the electron is inside a shell of radius r and thickness dr?
PROBLEM 2. Calculate the probabilities of measurement of different mo-
menta p for a particle with the wave function
(x) = Ceka sin (gx),
%3D
where C is a normalization constant.
Chapter 5 Solutions
Modern Physics
Ch. 5 - Prob. 1CQCh. 5 - Prob. 2CQCh. 5 - Prob. 3CQCh. 5 - Prob. 4CQCh. 5 - Prob. 5CQCh. 5 - Prob. 6CQCh. 5 - Prob. 7CQCh. 5 - Prob. 8CQCh. 5 - Prob. 9CQCh. 5 - Prob. 10CQ
Ch. 5 - Prob. 11CQCh. 5 - Prob. 12CQCh. 5 - Prob. 13CQCh. 5 - Prob. 14CQCh. 5 - Prob. 15CQCh. 5 - Prob. 16CQCh. 5 - Prob. 17CQCh. 5 - Prob. 18CQCh. 5 - Prob. 19ECh. 5 - Prob. 20ECh. 5 - Prob. 21ECh. 5 - Prob. 22ECh. 5 - Prob. 23ECh. 5 - Prob. 24ECh. 5 - Prob. 25ECh. 5 - Prob. 26ECh. 5 - Prob. 27ECh. 5 - Prob. 28ECh. 5 - Prob. 29ECh. 5 - Prob. 30ECh. 5 - Prob. 31ECh. 5 - Prob. 32ECh. 5 - Prob. 33ECh. 5 - Prob. 34ECh. 5 - Prob. 35ECh. 5 - Prob. 36ECh. 5 - Prob. 37ECh. 5 - Prob. 38ECh. 5 - Prob. 39ECh. 5 - Prob. 40ECh. 5 - Prob. 41ECh. 5 - Prob. 42ECh. 5 - Obtain expression (5-23) from equation (5-22)....Ch. 5 - Prob. 44ECh. 5 - Prob. 45ECh. 5 - Prob. 46ECh. 5 - Prob. 47ECh. 5 - Prob. 48ECh. 5 - Prob. 49ECh. 5 - Prob. 50ECh. 5 - Prob. 51ECh. 5 - Prob. 52ECh. 5 - Prob. 53ECh. 5 - Prob. 54ECh. 5 - Prob. 55ECh. 5 - Prob. 56ECh. 5 - Prob. 57ECh. 5 - Prob. 58ECh. 5 - Prob. 59ECh. 5 - Prob. 60ECh. 5 - Prob. 61ECh. 5 - Prob. 62ECh. 5 - Prob. 63ECh. 5 - Prob. 64ECh. 5 - Prob. 65ECh. 5 - Prob. 66ECh. 5 - Prob. 67ECh. 5 - Prob. 68ECh. 5 - Prob. 69ECh. 5 - Prob. 70ECh. 5 - Prob. 71ECh. 5 - In a study of heat transfer, we find that for a...Ch. 5 - Prob. 73CECh. 5 - Prob. 74CECh. 5 - Prob. 75CECh. 5 - Prob. 76CECh. 5 - Prob. 77CECh. 5 - Prob. 78CECh. 5 - Prob. 79CECh. 5 - Prob. 80CECh. 5 - Prob. 81CECh. 5 - Prob. 82CECh. 5 - Prob. 83CECh. 5 - Prob. 84CECh. 5 - Prob. 85CECh. 5 - Prob. 86CECh. 5 - Prob. 87CECh. 5 - Prob. 88CECh. 5 - Consider the differential equation...Ch. 5 - Prob. 90CECh. 5 - Prob. 91CECh. 5 - Prob. 92CECh. 5 - Prob. 93CECh. 5 - Prob. 94CECh. 5 - Prob. 95CECh. 5 - Prob. 96CECh. 5 - Prob. 97CECh. 5 - Prob. 98CE
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- Particle is described by the wave function Y = 0,x 0 a) Calculate A. b) Take L as 10 nm and calculate the probability of finding the particle in the region 1 nmarrow_forwardTry to normalize the wave function ei(kx-ωt) . Why can’t it be done over all space? Explain why this is not possiblearrow_forwardProblem 1 An electron is described by the wave function with the following x dependence: (x) = 0 for a 0, where C is a constant. (a) Determine the value of C is the wave-function is normalised. (b) Where is the electron most likely to be found? (c) Calculate the expectation value of position for the electron. (d) What is the uncertainty in finding the position of the electron?arrow_forwardFind the distance between points P and P,. P = (-4,4, 8), P2 = (5,8, 0) : %3D P = (7, 8, 9), P2 = (8,9, 10) : Р Р %3 (4, —8,5), Р, 3 (0,0, 0) :arrow_forwardThe normalized solution to the Schrodinger equation for a particular potential is ψ = 0 for x <0, and Ψ(x)=(2/a^{3/2}) x exp(-ax) for x>=0. What is the probability of finding a particle at this potential between x = a - 0.027a and x = a + 0.027a?arrow_forward8-6. If you were to use a trial function of the form p(x) = (1+cax²)e-ax²/2, where a = (ku/h²)1/2 and c is a variational parameter, to calculate the ground-state energy of a harmonic oscillator, what do you think the value of c will come out to be? Why?arrow_forwardDetermine the probability of an electron in the region of x = 0.490L and 0.510L in a box of length L in the energy level n = 1.arrow_forwardAn electron outside a dielectric is attracted to the surface by a force, F = -A/x2, where x is the perpendicular distance from the electron to the surface, and A is a positive constant. Electrons are prevented from crossing the surface, as there aren't any quantum states in the dielectric for them to occupy. Suppose that the surface is infinite, so that the problem is 1-dimensional. Write the Schrodinger equation for an electron outside of the surface (x > 0) and determine the appropriate boundary condition at x = 0. Obtain a formula for the allowed energy levels of the system. (Hint: Compare the equation for the wave function Ψ(x) with that satisfied by the wave functinon u(r) = rR(r) for a hydrogenic atom.)arrow_forwardAn electron has a wavefunction ψ(x)=Ce-|x|/x0 where x0 is a constant and C=1/√x0 for normalization. For this case, obtain expressions for a. ⟨x⟩ and Δx in terms of x0. b. Also calculate the probability that the electron will be found within a standard deviation of its average position, that is, in the range ⟨x⟩-∆x to ⟨x⟩+∆x, and show that this is independent of x0.arrow_forwardAn electron (m = 9.109*10^-31 kg) is confined in a one-dimensional infinite square well of width L = 10 nm. determine energy for first excited energy statearrow_forwardThe wave function for the simple harmonic oscillator is vo(x) = Coe normalization constant, Co. Find the [Hint: [° e-kx dx = 4 2hn А. 4 ma С. D. Vmw. OA. О в Oc. OD. B.arrow_forwardConsider a particle in a box of length L= 1 for the n= 2 state. The wave function is defined as: (x) = sin (27x) %3| Normalize the wave function.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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