Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Chapter 5, Problem 15CQ
To determine
To Sketch:Plausible wave function.
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Students have asked these similar questions
For a particle in a finite potential well, is it correct to say that each bound state of definite energy is also a state of definite wavelength? Is it a state of definite momentum? Explain.
A quantum mechanical particle is confined to a one-dimensional infinite potential well
described by the function V(x) = 0 in the region 0 < x < L, V(x) = ∞ elsewhere.
The normalised eigenfunctions for a particle moving in this potential are:
Yn(x)
=
√
2
Nπ
sin -X
L L
where n = 1, 2, 3, ..
a) Write down the expression for the corresponding probability density function. Sketch
the shape of this function for a particle in the ground state (n = 1).
b) Annotate your sketch to show the probability density function for a classical particle
moving at constant speed in the well. Give a short justification for the shape of your
sketch.
c) Briefly describe, with the aid of a sketch or otherwise, the way in which the quantum
and the classical probability density functions are consistent with the correspondence
principle for large values of n.
The wavefunction for the particle in a one-dimensional infinite potential well is given by
V(x, t) = VI
2
e-iEnt/h En
n?n?h?
sin
2mL2
with 0
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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- A particle with zero (total) energy is described by the wavefunction, Ψ(x) =A cos((n?x/L)): −L/4≤ x ≤ L/4 = 0 : elsewhere. Determine the normalization constant A. Calculate the potential energy of the particle. What is the probability that the particle will be found between x= 0 and x=L/8?arrow_forwardA one-dimensional infinite potential well has a length of 2L. What are the energy eigenvalues? Calculate the ground state energy if ten protons are confined in the box. Assume that the protons don’t interact with each other. If the ten protons are replaced by ten neutral hydrogen atoms, what is the total ground state energy resulting from the confinement? Again, assume that the hydrogen atoms do not interact with each other. You can treat the mass of proton and hydrogen atom to be identical.arrow_forward3n (2x –), find Ynormalized, the normalized wave function for a 1-dimensional particle- Given 4 = cos in-a-box where the box boundaries are at x=0 and x=2. The potential energy is zero when 0arrow_forwardConsider an infinite well, width L from x=-L/2 to x=+L/2. Now consider a trial wave-function for this potential, V(x) = 0 inside the well and infinite outside, that is of the form (z) = Az. Normalize this wave-function. Find , .arrow_forwardAssume that an electron is confined in a one-dimensional quantum well with infinite walls, draw the wave functions for the first 3 levels, ψ1, ψ2, ψ3. Also, show the probability density functions corresponding to these three levels?arrow_forwardPlease don't provide handwritten solution ..... Determine the normalization constant for the wavefunction for a 3-dimensional box (3 separate infinite 1-dimensional wells) of lengths a (x direction), b (y direction), and c (z direction).arrow_forwardPROBLEM 2. Consider a spherical potential well of radius R and depth Uo, so that the potential is U(r) = -Uo at r R. Calculate the minimum value of Uc for which the well can trap a particle with l = 0. This means that SE at Uo > Uc has at least one bound ground state at l = 0 and E < 0. At Ug = Uc the bound state disappears.arrow_forwarda) For a particle in a one-dimensional box of length L, do the energy levels move up ordown if the box gets longer? Explain your answer clearly.b) Consider a particle in a two-dimensional box of side lengths a and b, where b = 2a.In which direction is it more expensive (in terms of energy) to add nodes? Explain youranswer clearly.c) Consider a particle in a two-dimensional box of side lengths a and b, where b = 2a.Write down the quantum numbers corresponding to the rst (lowest) 5 energy levels of thissystem. Note any degeneracies.arrow_forwardFind the expectation value of the kinetic energy for the particle in the state, (x,t)=Aei(kxt). What conclusion can you draw from your solution?arrow_forwardConsider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)arrow_forwardFor (i) the infinite square well, (ii) the finite square well and (iii) the quantum harmonic oscillator, carefully sketch each of the following: a) The potential and the energy levels of the two lowest energy eigenstates. b) The wavefunctions for the two lowest energy eigenstates. c) The probability densities for the two lowest energy eigenstates. In the case of the finite square well, assume that the well admits exactly two energy eigenstates.arrow_forwardConsider a particle of mass m, located in a potential energy well.one-dimensional (box) with infinite height walls. The wave function that describes this system is:Ψn(x) = K sin (nπx /L), for 0 ≤ x ≤ LΨn(x) = 0 for any other value.K is a constant and n = 1,2,3,... Determine K*K = │K│2arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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