Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Question
Chapter 5, Problem 24P
(a)
To determine
The uncertainty in the electron’s momentum in terms of r.
(b)
To determine
The electron’s kinetic, potential, and total energies in terms of
(c)
To determine
The value of total energy.
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An electron is revolving around a proton in a circular orbit of radius r. The proton is assumed to be
stationary. The total energy of this system is
p? 1 e?
E
2m 4TE, r
where p and m denote the momentum and mass of the electron, respectively. Take the radius r to be an estimate
of the uncertainty in position Ar, and the uncertainty in momentum Ap to be an estimate of p. Suppose that
ArAp = ħ when the system is in the ground state.
Show that the ground state energy is given by
1 me4
e 8h?
E1
Give the numerical value for E, in electronvolts. Discuss if your results are consistent with Bohr's model for
the hydrogen atom.
An electron of momentum p is at a distance r from a stationary proton. The electron has kinetic energy K = p2/2me . The atom has potential energy UE = -kee2/r and total energy E = K + UE. 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 vector momentum is zero, but its average squared momentum is approximately equal to the squared uncertainty in its momentum as given by the uncertainty principle. Treating the atom as a one-dimensional system, (a) estimate the uncertainty in the electron’s momentum in terms of r. Estimate the electron’s (b) kinetic energy and (c) total energy in terms of r. The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find (d) that value of r and (e) the resulting total energy. (f) State how your answers compare with the predictions of the Bohr theory.
Using the Bohr model, calculate the speed of the electron when it is in the first excited state, n = 2. The Bohr radius ₁ 5.29 x 10-11 m.
Assume the electron is non-relativistic.
Chapter 5 Solutions
Modern Physics
Ch. 5.1 - A 0.20-kg ball is thrown upward. How much work is...Ch. 5.5 - Prob. 5ECh. 5 - Prob. 1QCh. 5 - Prob. 2QCh. 5 - Prob. 3QCh. 5 - Prob. 4QCh. 5 - Prob. 5QCh. 5 - Prob. 7QCh. 5 - Prob. 8QCh. 5 - Prob. 9Q
Ch. 5 - Prob. 10QCh. 5 - Prob. 11QCh. 5 - Prob. 1PCh. 5 - Prob. 2PCh. 5 - Prob. 3PCh. 5 - Prob. 4PCh. 5 - Prob. 5PCh. 5 - Prob. 6PCh. 5 - Prob. 7PCh. 5 - Prob. 8PCh. 5 - Prob. 9PCh. 5 - Prob. 10PCh. 5 - Prob. 11PCh. 5 - Prob. 12PCh. 5 - Prob. 13PCh. 5 - Prob. 14PCh. 5 - Show that the group velocity for a nonrelativistic...Ch. 5 - Prob. 16PCh. 5 - Prob. 17PCh. 5 - Prob. 18PCh. 5 - Prob. 19PCh. 5 - Prob. 20PCh. 5 - Prob. 21PCh. 5 - Prob. 22PCh. 5 - Prob. 23PCh. 5 - Prob. 24PCh. 5 - Prob. 25PCh. 5 - Prob. 26PCh. 5 - Prob. 27PCh. 5 - Prob. 28PCh. 5 - Prob. 29PCh. 5 - Prob. 30PCh. 5 - Prob. 31PCh. 5 - Prob. 32PCh. 5 - Prob. 33PCh. 5 - Prob. 34PCh. 5 - Prob. 35PCh. 5 - Prob. 36PCh. 5 - Prob. 37P
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