Physics for Scientists and Engineers With Modern Physics
9th Edition
ISBN: 9781133953982
Author: SERWAY, Raymond A./
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 41, Problem 16P
(a)
To determine
The uncertainty in momentum of the quantum particle.
(b)
To determine
The energy of the particle.
(c)
To determine
Compare the energy with the actual ground state energy.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
An electron is confined to a region of space of the size
of an atom (0.1 nm). (a) What is the uncertainty in the
momentum of the electron? (b) What is the kinetic energy
of an electron with a momentum equal to Ap? (c) Does
this give a reasonable value for the kinetic energy of an
electron in an atom?
For a quantum particle of mass m in the ground state of a square well with length L and infinitely high walls, the uncertainty in position is Δx ≈ L. (a) Use the uncertainty principle to estimate the uncertainty in its momentum.(b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then ⟨p2⟩ ≈ (Δp)2. Estimate the energy of the particle. (c) State how the result of part (b) compares with the actual ground-state energy.
The speed of a proton is measured to within an uncertainty of 1 × 103m/s. Calculate the length of the smallest region of space in which the electron can be confined.
Chapter 41 Solutions
Physics for Scientists and Engineers With Modern Physics
Ch. 41.1 - Prob. 41.1QQCh. 41.2 - Prob. 41.2QQCh. 41.2 - Prob. 41.3QQCh. 41.5 - Prob. 41.4QQCh. 41 - Prob. 1OQCh. 41 - Prob. 2OQCh. 41 - Prob. 3OQCh. 41 - Prob. 4OQCh. 41 - Prob. 5OQCh. 41 - Prob. 6OQ
Ch. 41 - Prob. 7OQCh. 41 - Prob. 8OQCh. 41 - Prob. 9OQCh. 41 - Prob. 10OQCh. 41 - Prob. 1CQCh. 41 - Prob. 2CQCh. 41 - Prob. 3CQCh. 41 - Prob. 4CQCh. 41 - Prob. 5CQCh. 41 - Prob. 6CQCh. 41 - Prob. 7CQCh. 41 - Prob. 8CQCh. 41 - Prob. 1PCh. 41 - Prob. 2PCh. 41 - Prob. 3PCh. 41 - Prob. 4PCh. 41 - Prob. 5PCh. 41 - Prob. 6PCh. 41 - Prob. 7PCh. 41 - Prob. 8PCh. 41 - Prob. 9PCh. 41 - Prob. 10PCh. 41 - Prob. 11PCh. 41 - Prob. 12PCh. 41 - Prob. 13PCh. 41 - Prob. 15PCh. 41 - Prob. 16PCh. 41 - Prob. 17PCh. 41 - Prob. 18PCh. 41 - Prob. 19PCh. 41 - Prob. 20PCh. 41 - Prob. 21PCh. 41 - Prob. 22PCh. 41 - Prob. 23PCh. 41 - Prob. 24PCh. 41 - Prob. 25PCh. 41 - Prob. 26PCh. 41 - Prob. 27PCh. 41 - Prob. 28PCh. 41 - Prob. 29PCh. 41 - Prob. 30PCh. 41 - Prob. 31PCh. 41 - Prob. 32PCh. 41 - Prob. 33PCh. 41 - Prob. 34PCh. 41 - Prob. 36PCh. 41 - Prob. 37PCh. 41 - Prob. 38PCh. 41 - Prob. 39PCh. 41 - Two particles with masses m1 and m2 are joined by...Ch. 41 - Prob. 41PCh. 41 - Prob. 42PCh. 41 - Prob. 43APCh. 41 - Prob. 44APCh. 41 - Prob. 45APCh. 41 - Prob. 46APCh. 41 - Prob. 47APCh. 41 - Prob. 48APCh. 41 - Prob. 49APCh. 41 - Prob. 50APCh. 41 - Prob. 51APCh. 41 - Prob. 52APCh. 41 - Prob. 53APCh. 41 - Prob. 54APCh. 41 - Prob. 56APCh. 41 - Prob. 57APCh. 41 - Prob. 58APCh. 41 - Prob. 59CPCh. 41 - Prob. 60CPCh. 41 - Prob. 61CPCh. 41 - Prob. 62CPCh. 41 - Prob. 63CP
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- (2nx sin \1.50. 2nz Consider the case of a 3-dimensional particle-in-a-box. Given: 4 = sin(ny) sin 2.00. What is the energy of the system? O 6h?/8m O 4h²/8m O 3h2/8m O none are correctarrow_forwardThe normalized wavefunction of an electron in a linear accelerator is ψ = (cos χ)eikx + (sin χ)e–ikx, where χ (chi) is a parameter. (a) What is the probability that the electron will be found with a linear momentum (a) +kħ, (b) −kħ? (c) What form would the wavefunction have if it were 90 per cent certain that the electron had linear momentum +kħ? (d) Evaluate the kinetic energy of the electron.arrow_forwardConsider a physical quantity q=2a+ 3b-4c, and a = 50 +2.5, b = 25 +3.1 and c = 80 +3.5 and a, b and c are random and independent, what is the value and absolute uncertainty of q?arrow_forward
- Q:-3(a)-An electron moves in the x direction with a speed of 2.8 x 106 m/s. We can measure its speed to a precision of 2 %. With what precision can we simultaneously measure its x coordinate? (b)- Repeat the calculations of the previous example in the case of a pitched baseball (m = 0.135 kg) moving at a speed of (44.5 ms). Again assume that its speed can be measured to a precision of 2%arrow_forwardQuantum tunneling was applied in 1928 by physicist George Gamow (and others) to explain the alpha emission of nuclear radiation, as we will see in the topic of nuclear physics. The alpha particle, whose mass is m = 6.64 x 10^(-27) kg, remains attached to the nucleus due to a strong interaction with the other nuclear constituents (nucleons) which overcomes the electrostatic repulsion between them, resulting in a graph of potential energy shown in figure A, where R is the range of the strong force and is of the order of the nuclear radius. However, note that there may be states for the alpha particle with energy E > 0 that can “tunnel” the Coulomb potential barrier. Consider a one-dimensional approximation for the potential energy well of an alpha particle in a 15 fm wide Uranium core (equivalent to the core diameter), in which the Coulomb barrier was modeled as a 20 fm wide rectangular barrier and 30 MeV high (see figure B). Knowing that the longest de Broglie wavelength for an…arrow_forwardProblem 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 EV, 2m dx² with boundary conditions (0) = 0 and V(1) = 0. Second, the Quantum Harmonic Oscillator (QHO) = h² d² +kr²V = EV 2m dg²+ka² 1/ k2²) v (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.arrow_forward
- A quantum particle (mass m) is confined in a 1-dimensional box represented by the interval 0 ≤ x < L. Its wave function is ó(x) = { Na(L- x) when 0 < aarrow_forwardA certain atom remains in an excited state for about 51.7 ns before emitting a 2.15-eV photon and transitioning to the ground state. What is the uncertainty in the frequency of the photon in Hz?arrow_forward(a) Calculate the minimum uncertainty in momentum (expressed in MeV/?) for a proton confined to a nucleus of diameter 5.0 fm. (1 fm = 1 * 10-15 m). (b) A proton (rest mass 938.3 MeV/?2) in a nucleus of radius 6.0 fm has a kinetic energy of 5.6 MeV. If the proton were represented by a de Broglie wave, how many wavelengths could fit across the diameter of that nucleus? (1 fm = 1 * 10-15 m). (c) Electrons (rest mass 0.51 MeV/?2) moving with a speed of 1.60 × 105 m/s are described by a wave packet of width 2.65 nm. What range of values will most likely result from a measurement of the speed of the electrons? Please use:ℎ? 1240 eVnmℏ? 197 eVnmarrow_forwardThe speed of an electron is measured to within an uncertainty of 2.0 × 104 m/s. What is the size of the smallest region of space in which the electron can be confined?arrow_forwardAn 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.arrow_forwardAn 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 p2 E e2 2m 4T€, 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 involving only m, e, ħ, and €o. Give the numerical value for E1 in electronvolts. Discuss if your results are consistent with Bohr's model for the hydrogen atom. h when the system is in the ground state. Obtain the expression for the ground state energy E1,arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning
Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning