Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 11.3, Problem 11.11P
(a)
To determine
Derive the classical modes.
(b)
To determine
Derive energy per unit volume.
(c)
To determine
Derive emission rate.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Problem 1:
Estimate the probability that a hydrogen atom at room temperature is in one of its first
excited states (relative to the probability of being in the ground state). Don't forget to
take degeneracy into account. Then repeat the calculation for a hydrogen atom in the
atmosphere of the star y UMa, whose surface temperature is approximately 9500K.
Consider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on.
A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators?
B) Obtain the average energy of N oscillators at temperature T from the partition function.
C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this?
D) Find the Helmholtz free energy from this system.
E) Derive the expression that gives the entropy of this system for the temperature.
Single Photon in a Gaussian Wavepacket. Consider a plane-wave wavepacket (Sec. 2.6A)
containing a single photon traveling in the z direction, with complex wavefunction
U(r, t) = a(t -)
(13.1-18)
where
a(t) = exp(-) exp(j27unt).
(13.1-19)
(a) Show that the uncertainties in its time and z position are o, = 7 and o, = cơ, respectively.
(b) Show that the uncertainties in its energy and momentum satisfy the minimum uncertainty rela-
tions:
(13.1-20)
0,0p = h/2.
(13.1-21)
Equation (13.1-21) is the minimum-uncertainty limit of the Heisenberg position-momentum
uncertainty relation provided in (A.2-7) of Appendix A.
(A.2-7)
Heisenberg
Uncertainty Relation
2.6 A. Temporal and Spectral Description
Although a polychromatic wave is described by a wavefunction e(r, t) with nonhar-
monie time dependence, it may be expanded as a superposition of harmonie func-
tions, each of which represents a monochromatic wave. Since we already know how
monochromatic waves propagate in free space and through various…
Chapter 11 Solutions
Introduction To Quantum Mechanics
Ch. 11.1 - Prob. 11.1PCh. 11.1 - Prob. 11.2PCh. 11.1 - Prob. 11.3PCh. 11.1 - Prob. 11.4PCh. 11.1 - Prob. 11.5PCh. 11.1 - Prob. 11.6PCh. 11.1 - Prob. 11.7PCh. 11.1 - Prob. 11.8PCh. 11.1 - Prob. 11.9PCh. 11.3 - Prob. 11.10P
Ch. 11.3 - Prob. 11.11PCh. 11.3 - Prob. 11.12PCh. 11.3 - Prob. 11.13PCh. 11.3 - Prob. 11.14PCh. 11.3 - Prob. 11.15PCh. 11.3 - Prob. 11.16PCh. 11.4 - Prob. 11.17PCh. 11.5 - Prob. 11.18PCh. 11.5 - Prob. 11.19PCh. 11.5 - Prob. 11.20PCh. 11.5 - Prob. 11.21PCh. 11.5 - Prob. 11.22PCh. 11 - Prob. 11.23PCh. 11 - Prob. 11.24PCh. 11 - Prob. 11.25PCh. 11 - Prob. 11.26PCh. 11 - Prob. 11.27PCh. 11 - Prob. 11.28PCh. 11 - Prob. 11.29PCh. 11 - Prob. 11.30PCh. 11 - Prob. 11.31PCh. 11 - Prob. 11.33PCh. 11 - Prob. 11.35PCh. 11 - Prob. 11.36PCh. 11 - Prob. 11.37P
Knowledge Booster
Similar questions
- 8.5 Calculate the grand partition function for a system of N noninteracting quantum mechanical harmonic oscillators, all of which have the same natural frequency wn. Do this for the following two cases: (a) Boltzmann statistics (b) Bose statistics.arrow_forwardThe three lowest energy levels of a hydrogen atom are -13.6 eV, -3.4 eV, and -1.5 eV. Assume that there is only one way to occupy any one of these levels. Calculate the relative probability that a hydrogen atom in thermal equilibrium in a star, at temperature T = 9674 K, is in its first excited state (at -3.4 eV) relative to its ground state (at -13.6 eV). Write your answer in exponential form. Recall that Boltzmann's constant can be written as 8.617 x 10-5 eV K-1.arrow_forwardFor an ideal gas of classical non- interacting atoms in thermal equilibrium, the Cartesian component of the velocity are statistically independent. In three dimensions, the probability density distribution of the velocity is: where σ² = kBT m P(Vx, Vy, Vz) = (2nо²)-³/² exp 20² 1. Show that the probability density of the velocity is normalized. 2. Find an expression of the arithmetic average of the speed. 3. Find and expression of the root-mean-square value of the speed. 4. Estimate the standard deviation of the speed.arrow_forward
- Relation Between Spectral Width and Coherence Time. Show that the coherence time T. defined in (12.1-10) is related to the spectral width Ave defined in (12.1-18) by the simple inverse relation Te = 1/Ave. Hint: Use the definitions of Av, and Te, the Fourier-transform relation between S(v) and G(r), and Parseval's theorem provided in (A.1-7) [Appendix A]. 19(7)l* dr (12.1-10) Coherence Time Te ( s(u) du Ave (12.1-18) s*(u) dv Parseval's Theorem. The signal energy, which is the integral of the signal power If(t)P, equals the integral of the energy spectral density F(v)l², so that sO)P dt = IF(»)P du. (A.1-7) Parseval's Theorem -00 S(v) = G(7) exp(-j2mvT) dr. (12.1-17) Power Spectral Density |arrow_forward1.7.12 Classically, orbital angular momentum is given by L = r xp, where p is the linear momentum. To go from classical mechanics to quantum mechanics, replace p by the operator -V (Section 14.6). Show that the quantum mechanical angular momentum operator has Cartesian components (in units of h). a ay a Ly=-i(22 -x- az Lx -i a az a əx L₂=-i (x-²) ayarrow_forwardFor Problem 11.23, how do I prove the following? This problem is in a chapter titled, "Atomic transitions and Radiation." It is also in quantum mechanics.arrow_forward
- It is stated without proof with respect to Bragg’s law that when the atoms are not sym- metrically disposed to the incident and reflected beams (Fig. 8.3(b)), the path difference (AB + BC) = 2dhkl sin θ . Prove, using very simple geometry, that this is indeed the case.arrow_forwardStarting from equation 7.83 attached, derive a formula for the density of states of a photon gas (or any other gas of ultrarelativistic particles having two polarization states). Sketch this function.arrow_forwardSuppose an Einstein Solid is in equilibrium with a reservoir at some temperature T. Assume the ground state energy is 0, the solid is composed of N oscillators, and the size of an energy "unit" is e. (a) Find the partition function for a single oscillator in the solid, Z1. Hint: use the general series summation formula 1+ x + x? + x³ + ... = 1/ (1- x) (b) Find an expression for , the average energy per oscillator in the solid, in terms of kT and e. (c) Find the total energy of the solid as a function of T, using the expression from part (b). (d) Suppose e = 2 eV and T = 25°C. What fraction of the oscillators is in the first excited state, compared to the ground state (assuming no degeneracies of energy levels)?arrow_forward
- Consider an object containing 6 one-dimensional oscillators (this object could represent a model of 2 atoms in an Einstein solid). There are 4 quanta of vibrational energy in the object. (a) How many microstates are there, all with the same energy? (b) If you examined a collection of 38000 objects of this kind, each containing 4 quanta of energy, about how many of these objects would you expect to find in the microstate 000004?arrow_forwardProblem 4.45. A simple partition function The partition function of a hypothetical system is given by In Z = «T4V, (4.172) where a is a constant. Evaluate the mean energy E, the pressure P, and the entropy S.arrow_forwarda formula was generated for the velocity of an electron that generates a stable waveform. To use that equation you need a constant k. This constant is 2.307×10-28 and the charges are entered as unitless as positive. (That is, if one charge be4ing entered into the formula is a proton, it is entered into the formula as +1.) The mass must be in kg and the distance must be in meters. If all these are entered correctly, the velocity comes out in meters per second. The radius of the orbit of an electron in the Bohr atom of hydrogen is 52.9 pm. What is the velocity of that electron?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
University Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Lecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON