Essential University Physics (3rd Edition)
3rd Edition
ISBN: 9780134202709
Author: Richard Wolfson
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Question
Chapter 35, Problem 59P
(a)
To determine
To Show: The continuity of wave function and their derivatives lead to the equations
(b)
To determine
To Show: The equations
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
A particle is in a three-dimensional box. The y length of the box is twice the x length, and the z length is one-third of the y length.
(a) What is the energy difference between the first excited level and the ground level?
(b) Is the first excited level degenerate?
(c) In terms of the x length, where is the probability distribution the greatest in the lowest-energy level?
∆E ∆t ≥ ħTime is a parameter, not an observable. ∆t is some timescale over which the expectation value of an operator changes. For example, an electron's angular momentum in a hydrogen atom decays from 2p to 1s. These decays are relativistic, however the uncertainty principle is still valid, and we can use it to estimate uncertainties.
∆E doesn't change in time, so when an excited state decays to the ground state (infinite lifetime, so no energy uncertainty), the energy uncertainty has to go somewhere. Usually, it’s in the frequency of a photon giving a width (through E = hν) to the transition line in an spectroscopy experiment. The linewidth of the 2p state in 9Be+ is 19.4 MHz. What is its lifetime? (Note: in the relativistic atom–photon system, the Hamiltonian is independent of time and both energy and its uncertainty are conserved.)
∆E ∆t ≥ ħTime is a parameter, not an observable. ∆t is some timescale over which the expectation value of an operator changes. For example, an electron's angular momentum in a hydrogen atom decays from 2p to 1s. These decays are relativistic, however the uncertainty principle is still valid, and we can use it to estimate uncertainties.
The lifetime of hydrogen in the 2p state to decay to the Is ground state is 1.6 x 10-9 s. Estimate the uncertainty ∆E in energy of this excited state. What is the corresponding linewidth in angstroms?
Chapter 35 Solutions
Essential University Physics (3rd Edition)
Ch. 35.1 - Prob. 35.1GICh. 35.2 - Prob. 35.2GICh. 35.3 - Prob. 35.3GICh. 35.3 - Prob. 35.4GICh. 35.3 - Prob. 35.5GICh. 35.4 - Prob. 35.6GICh. 35 - Prob. 1FTDCh. 35 - Prob. 2FTDCh. 35 - Prob. 3FTDCh. 35 - Prob. 4FTD
Ch. 35 - Prob. 5FTDCh. 35 - Prob. 6FTDCh. 35 - Prob. 7FTDCh. 35 - What did Einstein mean by his re maxi, loosely...Ch. 35 - Prob. 9FTDCh. 35 - Prob. 10FTDCh. 35 - Prob. 12ECh. 35 - Prob. 13ECh. 35 - Prob. 14ECh. 35 - Prob. 15ECh. 35 - Prob. 16ECh. 35 - Prob. 17ECh. 35 - Prob. 18ECh. 35 - Prob. 19ECh. 35 - Prob. 20ECh. 35 - Prob. 21ECh. 35 - Prob. 22ECh. 35 - Prob. 23ECh. 35 - Prob. 24ECh. 35 - Prob. 25ECh. 35 - Prob. 26ECh. 35 - Prob. 27ECh. 35 - Prob. 28ECh. 35 - Prob. 29ECh. 35 - Prob. 30ECh. 35 - Prob. 31ECh. 35 - Prob. 32PCh. 35 - Prob. 33PCh. 35 - Prob. 34PCh. 35 - Prob. 35PCh. 35 - Prob. 36PCh. 35 - Prob. 37PCh. 35 - Prob. 38PCh. 35 - Prob. 39PCh. 35 - Prob. 40PCh. 35 - Prob. 41PCh. 35 - Prob. 42PCh. 35 - Prob. 43PCh. 35 - Prob. 44PCh. 35 - Prob. 45PCh. 35 - Prob. 46PCh. 35 - Prob. 47PCh. 35 - Prob. 48PCh. 35 - Prob. 49PCh. 35 - Prob. 50PCh. 35 - Prob. 51PCh. 35 - Prob. 52PCh. 35 - Prob. 53PCh. 35 - Prob. 54PCh. 35 - Prob. 55PCh. 35 - Prob. 56PCh. 35 - Prob. 57PCh. 35 - Prob. 58PCh. 35 - Prob. 59PCh. 35 - Prob. 60PCh. 35 - Prob. 61PPCh. 35 - Prob. 62PPCh. 35 - Prob. 63PPCh. 35 - Prob. 64PP
Knowledge Booster
Similar questions
- the wave functions px and dxz are linear combinations of the spherical harmonic functions, which are eigenfunctions of the operators H, 12, Iz for rotation in three dimensions. The combinations have been chosen to yield real functions. Are these functions still eigenfunctions of lz? answer this question by applying the operator to the functionsarrow_forwardConsider a very simplistic model of atomic nucleus in 1D: a proton is completely localized in a 1D box of width L = 1.00 × 10¬14m. In other words, the proton wavefunction outside of the "nucleus" is zero. Note that L represents a typical nuclear radius. (A) What are the energies of the ground and the first excited states? If the proton makes a transition from the first excited state to the ground state, what is the angular frequency of the emitted photon? (B) What is the probability that the proton in its ground state (i.e., the lowest energy state) is not found in the distance L/12 around each boundary of the box? (C) Using the uncertainty principle, derive a minimum possible value on the momentum uncertainty in the second state above the ground state. (D) Compare your answer to the previous question (B) to probability distribution one would obtain for a classical particle. First argue about how the probability distribution would look for a classical object in its ground state. How…arrow_forwardA particle of mass m is moving around the z-axis in an orbit of radius r in two dimensions. It is shown on the figure as a wave function depending on the quantum number m of the particle. a) Find the normalization coefficient Φo by normalizing the wave function b) Using the operator expression, find the Lz eigen (eigen) value for the wave function.arrow_forward
- An electron is confined to move in the xy plane in a rectangle whose dimensions are Lx and Ly. That is, the electron is trapped in a two dimensional potential well having lengths of Lx and Ly. In this situation, the allowed energies of the electron depend on two quantum numbers nx and ny and are given by E = h2/8me (nx2/Lx2 + ny2/Ly2)Using this information, we wish to find the wavelength of a photon needed to excite the electron from the ground state to the second excited state, assuming Lx = Ly = L. (a) Using the assumption on the lengths, write an expression for the allowed energies of the electron in terms of the quantumnumbers nx and ny. (b) What values of nx and ny correspond to the ground state? (c) Find the energy of the ground state. (d) What are the possible values of nx and ny for the first excited state, that is, the next-highest state in terms of energy? (e) What are the possible values of nx and ny for thesecond excited state?…arrow_forwardSeven electrons are trapped in a one-dimensional infinite potential well of width L.What multiple of h2/8mL2 gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.arrow_forward∆E ∆t ≥ ħTime is a parameter, not an observable. ∆t is some timescale over which the expectation value of an operator changes. For example, an electron's angular momentum in a hydrogen atom decays from 2p to 1s. These decays are relativistic, however the uncertainty principle is still valid, and we can use it to estimate uncertainties. When heavy particles are made in colliders, they typically decay, which gives a linewidth to their measured mass via the uncertainty in energy. The ∆ particle lives very briefly (often called resonances instead of particles), and its lifetime is 5.58 x 10-24 s. Its mass is 1.232 GeV/c2. Predict the width of its energy resonance in MeV.arrow_forward
- A particle of mass m is confined to a 3-dimensional box that has sides Lx,=L Ly=2L, and Lz=3L. a) Determine the sets of quantum numbers n_x, n_y, and n_z that correspond to the lowest 10 energy levels of this box.arrow_forwardWhat is the solution of the time-dependent Schrödinger equation (x, t) for the total energy eigenfunction 4(x) = √2/a sin(3mx/a) for an electron in a one-dimensional box of length 1.00 x 10-10 m? Write explicitly in terms of the parameters of the problem. Give numerical values for the angular frequency and the wave- length of the particle.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_forward
- Chapter 39, Problem 009 Suppose that an electron trapped in a one-dimensional infinite well of width 144 pm is excited from its first excited state to the state with n 9. (a) What energy must be transferred to the electron for this quantum jump? The electron then de- excites back to its ground state by emitting light. In the various possible ways it can do this, what are the (b) shortest, (c) second shortest, (d) longest, and (e) second longest wavelengths that can be emitted? (a) Number Units (b) Number Units (c) Number Units (d) Number Units (e) Number Unitsarrow_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 2m dx² EV, with boundary conditions (0) = 0 and (1) = 0. Second, the Quantum Harmonic Oscillator (QHO) V = EV h² d² 2m da² +ka²) 1 +kx² 2 (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_forwardA particle is trapped in an infinite one-dimensional well of width L. If the particle is in its ground state, evaluate the probability to find the particle (a) between x = x = L/3; (b) between x = L/3 and x = x = 2L/3 and x = L. O and 2L/3; (c) between %3Darrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Physics for Scientists and Engineers with Modern ...
Physics
ISBN:9781337553292
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning