Physics for Scientists and Engineers, Vol. 1
6th Edition
ISBN: 9781429201322
Author: Paul A. Tipler, Gene Mosca
Publisher: Macmillan Higher Education
expand_more
expand_more
format_list_bulleted
Question
Chapter 35, Problem 24P
To determine
The ground state energy of seven identical non interacting fermions in one dimensional box.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
a) Consider two single-particle energy states of the fermion system, A and B, for which EA = μ- x
and EB = μ + x. Show that the probability that state A is occupied is equal to the probability that
state B is unoccupied. In other words, show that the Fermi and Dirac distribution,
ƒ(E)
1
eß(E-μ) +1
is "symmetric" about the point E = μ.
b) Write simplified, approximate expressions for f(E) when
value of E is very close to μ.
(i) E < µ, (ii) E» µ
2
and (iii) the
Consider a collection of fermions at T = 293 K. Find the probability that a single-particle state will be occupied if that state’s energy is (a) 0.1 eV less than EF; (b) equal to EF; (c) 0.1 eV greater than EF.
Problem 4. Consider two indistinguishable, noninteracting spin-1/2 fermions in a one-
dimensional infinite square well potential of length L.
(a) What is the ground-state energy of the two-particle system?
(b) What is the ground-state quantum state vector?
(c) What is the first excited state energy of the two-particle system?
(d) What are the quantum state vectors of the first excited state?
(e) What is the degeneracy of the first excited state?
(f) Discuss qualitatively how the excited-state energies change if we consider the particles
to be interacting through the Coulomb potential.
Chapter 35 Solutions
Physics for Scientists and Engineers, Vol. 1
Ch. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7PCh. 35 - Prob. 8PCh. 35 - Prob. 9PCh. 35 - Prob. 10P
Ch. 35 - Prob. 11PCh. 35 - Prob. 12PCh. 35 - Prob. 13PCh. 35 - Prob. 14PCh. 35 - Prob. 15PCh. 35 - Prob. 16PCh. 35 - Prob. 17PCh. 35 - Prob. 18PCh. 35 - Prob. 19PCh. 35 - Prob. 20PCh. 35 - Prob. 21PCh. 35 - Prob. 22PCh. 35 - Prob. 23PCh. 35 - Prob. 24PCh. 35 - Prob. 25PCh. 35 - Prob. 26PCh. 35 - Prob. 27PCh. 35 - Prob. 28PCh. 35 - Prob. 29PCh. 35 - Prob. 30PCh. 35 - Prob. 31PCh. 35 - Prob. 32PCh. 35 - Prob. 33PCh. 35 - Prob. 34PCh. 35 - Prob. 35PCh. 35 - Prob. 36PCh. 35 - Prob. 37PCh. 35 - Prob. 38P
Knowledge Booster
Similar questions
- For 3D free electron gas, the density of states counts the number of degenerate electron states dn per energy interval dE around a given energy E as g(E): = dn dE 3 (2m₂)2V 1 E2 2π²ħ³ At absolute zero temperature, N electrons can fill up all low lying energy levels (following Pauli exclusion principle) up to a given energy level E called Fermi energy. From the density of states, what is the relation between the total electron states N below a given energy E? Use this result to show that the Fermi energy EF is given by - - 2010 (307² M)³ ħ² 3π²N\3 EF 2me Varrow_forwardProblem 1: Fermi temperature of the sun At the center of the sun, the temperature is approximately 107K and the concentration of electrons is approximately 1032 per cubic meter. Would it be (approximately) valid to treat these electrons as a classical gas (using Boltzmann statistics), or as a degenerate Fermi gas (with T × 0), or neither?arrow_forwardConsider a three-dimensional infinite-well modeled as a cube of dimensions L x L x L. The length L is such that the ground state energy of one electron confined to this box is 0.50eV. (a) Write down the four lowest energy states and evaluate their corresponding degeneracy. (b) If 15 (total) electrons are placed in the box, find the Fermi energy of the system. (c) What is the total energy of the 15-electron system? (d) How much energy would be required to lift an electron from Fermi energy of part (b) to the first excited state? Need full detailed answers and explanations to understand the concept.arrow_forward
- For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is. 0.01 eV less than μarrow_forwardTwo nonintercating gaseous particles form an open system. There are three nondegenerate single particle states of energies Є, 2Є, 3Є in the system. Find the canonical partition functions Z0, Zl, Z2 and the grand canonical partition function Z if the particles are (a) fermions (b) bosons What is the average number of particles per state in each case? Calculate the internal energy U? (related to statistical physics)arrow_forward(d) Fermions are represented by Dirac spinors and obey the Dirac equation. The Dirac equation is (i", - m)=0 where, in the so-called chiral basis, the gamma matrices are: x=(-15) - where i runs over 1,2,3 and o are the Pauli spin matrices. i. In this basis, calculate the 'fifth' gamma matrix 75 = iyºy¹z²z³. ii. Determine the result of the projection operator (1+75) acting on the spinor - (3). X =arrow_forward
- For 3D free electron gas, the density of states counts the number of degenerate electron states dn per energy interval de around a given energy E as g(E) = dn dE 3 (2m₂)²V 1 -E2 2π²ħ³ At absolute zero temperature, N electrons can fill up all low-lying energy levels (following Pauli exclusion principle) up to a given energy level E called Fermi energy. In terms of the Fermi energy, the density of states can now be written as, 3N 263/2 2E g(E) -E¹/2 Using this simplified expression, show that the average energy of an electron in a free-electron gas is given by, 3 (E) = { Ef 1 EF Clue: interpret g(E)/N as probability density for E, and thus (E) = √³ Eg(E)dE NJOarrow_forwardProblem 35-28: What is the ground-state energy in electron volts of 10 noninteracting fermions each with a mass of 5.02E-27 kg in a one- dimensional box of length L 3.90E-10 m ? (Because the quantum number associated with spin can have two values, each spatial = state can hold two fermions.) 0.044 eVarrow_forward). Atoms, which can be assumed to be hard spheres of radius R, are arranged in an fcc lattice with lattice constant a, such that each atom touches its nearest neighbours. Take the center of one of the atoms as the origin. Another atom of radius r (assumed to be hard sphere) is to be accommodated at a position 0,,0 without distorting the lattice. The maximum value of Rarrow_forward
- Plot the Fermi-Dirac probability of occupation function fFD(E) for T = 0, 10, 100, 200, 300 and 400K.arrow_forwardAn atomic nucleus can be crudely modeled as a gas of nucleons with a number density of 0.18 fm -3(where 1 fm = 10-15m). Because nucleons come in two different types (protons and neutrons), each with spin 1/2, each spatial wavefunction can hold four nucleons. Calculate the Fermi energy of this system, in MeV. Also calculate the Fermi temperature, and comment on the result.arrow_forwardQ. No 02: a) Consider a system of two identical particles occupying any of three energy levels A, B and C having energies E, 2E and 3E, respectively. The level A is doubly degenerate (A1 and A2) and the system is in thermal equilibrium. Find the possible configurations and the corresponding energy in the following cases: (i) the particles obey Fermi statistics; (ii) the particles obey Bose statistics; and (iii) the particles are distinguishable and obey Boltzmann statistics.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:PEARSON
Introduction 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