An electron of momentum p is at a distance r from a stationary proton. The two particles are interacting via the electric Coulomb potential. If the electron is bound to the proton to form a hydrogen atom, its average position is at the proton, but the uncertainty in its position is approximately equal to the radius, r, of its orbit. The electron's average momentum will be zero, but the uncertainty in its momentum will be given by the uncertainty principle. Treat the atom as a one-dimensional system to solve the following questions. (a) Estimate the uncertainty in the electron's momentum in terms ofr. (b) Estimate the electron's kinetic, potential, and total energies in terms of r. (c) The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find that value of r and the resulting total energy. Compare your answer with the predictions of the Bohr theory.

An electron of momentum p is at a distance r from a stationary proton. The two particles are interacting via the electric Coulomb potential. If the electron is bound to the proton to form a hydrogen atom, its average position is at the proton, but the uncertainty in its position is approximately equal to the radius, r, of its orbit. The electron's average momentum will be zero, but the uncertainty in its momentum will be given by the uncertainty principle. Treat the atom as a one-dimensional system to solve the following questions. (a) Estimate the uncertainty in the electron's momentum in terms ofr. (b) Estimate the electron's kinetic, potential, and total energies in terms of r. (c) The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find that value of r and the resulting total energy. Compare your answer with the predictions of the Bohr theory.

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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Question

Problem 2:
An electron of momentum p is at a distancer from a stationary proton. The two particles are interacting
via the electric Coulomb potential. If the electron is bound to the proton to form a hydrogen atom, its
average position is at the proton, but the uncertainty in its position is approximately equal to the radius,
r, of its orbit. The electron's average momentum will be zero, but the uncertainty in its momentum will
be given by the uncertainty principle. Treat the atom as a one-dimensional system to solve the following
questions.
(a) Estimate the uncertainty in the electron's momentum in terms of r.
(b) Estimate the electron's kinetic, potential, and total energies in terms of r.
(c) The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find that
value of r and the resulting total energy. Compare your answer with the predictions of the Bohr theory.

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