Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Question
Chapter 35, Problem 24P
To determine
The ground state energy of seven identical non interacting fermions in one dimensional box.
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Students have asked these similar questions
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.
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
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 V
Chapter 35 Solutions
Physics for Scientists and Engineers
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
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Similar questions
- 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.arrow_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_forwardFor 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_forward
- Two 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_forwardFor 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_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
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