m,A particle of mass 2.00 × 10-10 kg is confined in a hollow cubicalthree-dimensional box, each edge of which has a length, 2.00 × 10-10and for which the potential energy function is zero inside, and infiniteoutside, the box. The total energy of the particle is 2.47 × 10-37 J. Find (b) (i)If the same particle is instead confined in one dimension betweeninfinitely high potential barriers, with the same energy and in thesame size region as above, find the single quantum number thatcharacterises the wavefunction of the particle when it occupies thisenergy level.

Solution for the Particle in a 1D Box(a) Recap of 3D Box (not required for part (b))We can calculate…

Answered: m, A particle of mass 2.00 × 10-10 kg…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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For an infinite potential well of length L, determine the difference in probability that a particle might be found between x = 0.25L and x = 0.75L between the n = 3 state and the n = 5 states.
Question 1An electron is trapped in a region between two infinitely high energy barriers. In the regionbetween the walls the potential energy of the electron is zero. The normalised wavefunction of the electron in the region between the walls is ?(?) = ? sin ??, where ? =0.50 nm−1/2and ? = 1.18 nm−1.(a) Calculate the energy of the electron in this state of motion.Question 2In some physical situations, the behaviour of an electron can be approximated as if theelectron were bound to an equilibrium position by a spring force (?(?) = −??, ?(?) =??22,where ? is the spring constant). Suppose such an electron were in the first excited state,with a wave function ?(?) = ???−??2, where ? is a constant, ? = √??/2ℏ and ?represents the distance of the electron from its equilibrium position.(a) Calculate the energy of the electron in terms of ? and its mass ?.(b) If the electron behaved like a classical oscillating particle, the largest value of ?would be ??. Show that ?? = √3ℏ. (??)−1/4.
An electron is trapped in an infinitely deep one-dimensional well of width 10 nm. Initially, the electron occupies the n = 4 state. Calculate the photon energy required to excite the electron in the ground state to the first excited state.
An electron that is trapped in a one-dimensional infinite potential well of width L is excited from the ground state to the first excited state. Does the excitation increase, decrease, or have no effect on the probability of detecting the electron in a small length of the x axis (a) at the center of the well and (b) near one of the well walls?
(a) A quantum dot can be modelled as an electron trapped in a cubic three-dimensional infinite square well. Calculate the wavelength of the electromagnetic radiation emitted when an electron makes a transition from the third lowest energy level, E3, to the lowest energy level, E₁, in such a well. Take the sides of the cubic box to be of length L = 3.2 x 10-8 m and the electron mass to be me = 9.11 x 10-³¹ kg. for each of the E₁ and E3 energy (b) Specify the degree of degeneracy levels, explaining your reasoning.
An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy difference E43 between the levels n = 4 and n = 3? (c) Show that no pair of adjacent levels has an energy difference equal to 2E43.
If you double the width of a one-dimensional infinite potential well, (a) is the energy of the ground state of the trapped electron multiplied by 4, 2, , , or some other number? (b) Are the energies of the higher energy states multiplied by this factor or by some other factor, depending on their quantum number?
A beam of electrons with kinetic energy 350 eV is incident normal to the surface of a KC1 crystal that has been cut so that the spacing d between adjacent atoms in the planes parallel to the surface is 0.315 nm. (a) There are multiple diffraction peaks from this scattering corresponding to different inte- gers n in the Bragg condition. Show that there is a maximum n above which diffraction peaks are not possible, and find this integer nmax. (b) Calculate the angles at which diffraction peaks will occur for all orders that are possible.

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