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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition · 469 pages

ISBN: 9780131118928

Chapter 5: Q15P (page 223)

Find the average energy per free electron $(E_{t o t} / N d)$ , as a fraction of the
Fermi energy. Answer: $(3 / 5) E F$

Short Answer

Expert verified

The average energy per free electron is $\frac{E_{t o t} / N q}{E_{F}} = \frac{3}{5} E F$

Step by step solution

01

Formula used

Find ratio of average energy per free electron and Fermi energy.

We know:

$E_{t o t} = \frac{h^{2} {(3 π^{2} Nq)}^{5 / 3}}{10 π^{2} m} V^{- 2 / 3}$ . …(i)

$E_{F} = \frac{h^{2}}{2 m} {(3 p π^{2})}^{2 / 3}$ …(ii)

02

Finding the average energy per free electron

Rearranging the terms in the equation 1:

$E_{t o t} = \frac{h^{2} V}{2 π^{2} m} \int_{0}^{k F} K^{4} d k = \frac{h^{2} k_{F}^{5} V}{10 π^{2} m} V^{- 2 / 3} = \frac{h {(3 π^{2} Nd)}^{5 / 3}}{10 π^{2} m} V^{- 2 / 3}$

Density is given by:

$ρ = \frac{N d}{V}$

Now, average energy per free electron is:

$\frac{E_{t o t} / N q}{E_{F}} = \frac{h^{2} {(3 π^{2} Nq)}^{5 / 3}}{10 π^{2} m} \frac{1}{N q} \frac{2 m}{{(3 π^{2} Nq / V)}^{2 / 3}} = \frac{3}{5} E F$

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Most popular questions from this chapter

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

$\frac{E}{V} = (\frac{π^{2} k_{B}^{4}}{15 ħ^{3} C^{3}}) T^{4} = (7.57 \times 10^{- 16} {Jm}^{- 3} K^{- 4}) T^{4}$

Check the equations 5.74, 5.75, and 5.77 for the example in section 5.4.1

The density of copper is $8.96 g / {cm}^{3},$ and its atomic weight is $63.5 g / mole$

(a) Calculate the Fermi energy for copper (Equation 5.43). Assume d = 1, and give your answer in electron volts.

$E_{F} = \frac{ħ^{2}}{2 m} {(3 {ρπ}^{2})}^{2 / 3}$ (5.43).

(b) What is the corresponding electron velocity? Hint: Set $E_{F} = (1 / 2) m v^{2}$ Is it safe to assume the electrons in copper are nonrelativistic?

(c) At what temperature would the characteristic thermal energyrole="math" localid="1656065555994" $(k_{B} T, w h e r e k_{B} k B$ is the Boltzmann constant and T is the Kelvin temperature) equal the Fermi energy, for copper? Comment: This is called the Fermi temperature, $T_{F}$

. As long as the actual temperature is substantially below the Fermi temperature, the material can be regarded as “cold,” with most of the electrons in the lowest accessible state. Since the melting point of copper is 1356 K, solid copper is always cold.

(d) Calculate the degeneracy pressure (Equation 5.46) of copper, in the electron gas model.

$P = \frac{2}{3} \frac{E_{t o t}}{V} = \frac{2}{3} \frac{ħ^{2} k_{F}^{5}}{10 π^{2} m} = \frac{{(3 π^{2})}^{2 / 3} ħ^{2}}{5 m} ρ^{5 / 3}$

(a) Figure out the electron configurations (in the notation of Equation

5.33) for the first two rows of the Periodic Table (up to neon), and check your

results against Table 5.1.

${(1 s)}^{2} {(2 s)}^{2} {(2 p)}^{2}$ (5.33).

(b) Figure out the corresponding total angular momenta, in the notation of

Equation 5.34, for the first four elements. List all the possibilities for boron,

carbon, and nitrogen.

${}^{2 S + 1}{L_{J}}$ (5.34).

Certain cold stars (called white dwarfs) are stabilized against gravitational collapse by the degeneracy pressure of their electrons (Equation 5.57). Assuming constant density, the radius R of such an object can be calculated as follows:

$P = \frac{2}{3} \frac{E_{tot}}{V} = \frac{2}{3} \frac{h^{2} k_{F}^{5}}{10 π^{2} m} = \frac{{(3 π^{2})}^{2 / 3} h^{2}}{5 m} p^{5 / 3} (5.57)$

(a) Write the total electron energy (Equation 5.56) in terms of the radius, the number of nucleons (protons and neutrons) N, the number of electrons per nucleon d, and the mass of the electron m. Beware: In this problem we are recycling the letters N and d for a slightly different purpose than in the text.

$E_{tot} = \frac{h^{2} V}{2 π^{2} m} \int_{0}^{k_{F}} K^{4} dk = \frac{h^{2} k_{F}^{5} V}{10 π^{2} m} = \frac{h^{2} {(3 π^{2} Nd)}^{5 / 3}}{10 π^{2} m} V^{- 2 / 3} (5.56)$

(b) Look up, or calculate, the gravitational energy of a uniformly dense sphere. Express your answer in terms of G (the constant of universal gravitation), R, N, and M (the mass of a nucleon). Note that the gravitational energy is negative.

(c) Find the radius for which the total energy, (a) plus (b), is a minimum.

$R = {(\frac{9 π}{4})}^{2 / 3} \frac{h^{2} d^{5 / 3}}{{GmM}^{2} N^{1 / 3}}$

(Note that the radius decreases as the total mass increases!) Put in the actual numbers, for everything except , using d=1/2 (actually, decreases a bit as the atomic number increases, but this is close enough for our purposes). Answer:

(d) Determine the radius, in kilometers, of a white dwarf with the mass of the sun.

(e) Determine the Fermi energy, in electron volts, for the white dwarf in (d), and compare it with the rest energy of an electron. Note that this system is getting dangerously relativistic (see Problem 5.36).

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