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Chapter 5: Q19P (page 229)

Find the energy at the bottom of the first allowed band, for the caseβ=10 , correct to three significant digits. For the sake of argument, assume αa=1eV.

Short Answer

Expert verified

The energy at the bottom of the first band is 0.345eV.

Step by step solution

01

Define the Schrödinger equation

  • A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.
  • The time-dependent Schrodinger equation is represented as

Iħddt|ψt>=H^|ψ(t)>

02

Calculating the minimum energy of first band

The minimum energy of the first band required z,f(z)=1

And f(z) is given by

fz=cosz+βsinzzz=ka,z≤πβ=10=mαah2⇒fz=cosz+10sinzz

03

Using MATLAB to calculate the minimum energy of first band

UsingMATLABwegetz=2.6276.EnergyisgivenwithE=h2k22m=h22m(za)2 =z22ah2mβαα=z22aαβαa=1eVE=2.627672201eVE=0.345eV

Therefore the energy at the bottom of the first band is 0.345eV.

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Most popular questions from this chapter

Suppose you had three (noninteracting) particles, in thermal equilibrium in a one-dimensional harmonic oscillator potential, with a total energyE=92hω .

(a) If they are distinguishable particles (but all with the same mass), what are the possible occupation-number configurations, and how many distinct (threeparticle) states are there for each one? What is the most probable configuration? If you picked a particle at random and measured its energy, what values might you get, and what is the probability of each one? What is the most probable energy?

(b) Do the same for the case of identical fermions (ignoring spin, as we did in Section 5.4.1).

(c) Do the same for the case of identical bosons (ignoring spin).

a) Hund’s first rule says that, consistent with the Pauli principle, the state with the highest total spin (S) will have the lowest energy. What would this predict in the case of the excited states of helium?

(b) Hund’s second rule says that, for a given spin, the state with the highest total orbital angular momentum (L) , consistent with overall antisymmetrization, will have the lowest energy. Why doesn’t carbon haveL=2? Note that the “top of the ladder”(ML=L)is symmetric.

(c) Hund’s third rule says that if a subshell(n,l)is no more than half filled,
then the lowest energy level hasJ=lL-SI; if it is more than half filled, thenJ=L+Shas the lowest energy. Use this to resolve the boron ambiguity inProblem 5.12(b).

(d) Use Hund’s rules, together with the fact that a symmetric spin state must go with an antisymmetric position state (and vice versa) to resolve the carbon and nitrogen ambiguities in Problem 5.12(b). Hint: Always go to the “top of the ladder” to figure out the symmetry of a state.

We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic domain by replacing the classical kinetic energy, E=p2/2m,,with the relativistic formula, E=p2c2+m2c4-mc2. Momentum is related to the wave vector in the usual way: p=hk. In particular, in the extreme relativistic limit, E≈pc=hck.

(a) Replace h2k2n Equation 5.55 by the ultra-relativistic expression, hck, and calculateEtotin this regime.

dE=h2k22mVπ2k2dk (5.55).

(b) Repeat parts (a) and (b) of Problem 5.35 for the ultra-relativistic electron gas. Notice that in this case there is no stable minimum, regardless of R; if the total energy is positive, degeneracy forces exceed gravitational forces, and the star will expand, whereas if the total is negative, gravitational forces win out, and the star will collapse. Find the critical number of nucleons, Nc , such that gravitational collapse occurs for N>N_{C}is called the Chandrasekhar limit.

(c) At extremely high density, inverse beta decaye-+p+→n+v,converts virtually all of the protons and electrons into neutrons (liberating neutrinos, which carry off energy, in the process). Eventually neutron degeneracy pressure stabilizes the collapse, just as electron degeneracy does for the white dwarf (see Problem 5.35). Calculate the radius of a neutron star with the mass of the sun. Also calculate the (neutron) Fermi energy, and compare it to the rest energy of a neutron. Is it reasonable to treat a neutron star non relativistic ally?

The ground state of dysprosium (element 66, in the 6th row of the Periodic Table)

is listed as Is5. What are the total spin, total orbital, and grand total angular

momentum quantum numbers? Suggest a likely electron configuration for

dysprosium.

Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation

EV=(π2kB415ħ3C3)T4=7.57×10-16Jm-3K-4T4
See all solutions

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