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Chapter 5: Q10P (page 214)

Discuss (qualitatively) the energy level scheme for helium if (a) electrons were identical bosons, and (b) if electrons were distinguishable particles (but with the same mass and charge). Pretend these 鈥渆lectrons鈥� still have spin 1/2, so the spin configurations are the singlet and the triplet.

Short Answer

Expert verified

(a)The ground state (Eq. 5.30) is spatially symmetric, so it goes with the symmetric (triplet) spin configuration.

(b) The ground state (Eq. 5.30) and all excited states (Eq. 5.32) come in both ortho and para form.

Step by step solution

01

(a) If electrons were identical bosons

The ground state (Equation 5.30) fits into a symmetric (triplet) spin configuration because it is spatially symmetric.

Thus, orthohelium, a degenerate triple, is the ground state. Ortho (triplet) and para excited states (Equation 5.32) are two types of excited states (singlet). The energy level of the orthohelium state is higher than that of the comparable (non-degenerate) para-state since the former is linked to the symmetric space wavefunction.

蠄0r1,r2=蠄100r1蠄100r2=8蟺a3e-2r1+r2/a 鈥�(5.30)

蠄n鈩�m蠄100 ...(5.32).

02

(b) Spinning the configurations are the singlet and the triplet

Both orthogonal and paragon variants of the ground state (Eq. 5.30) and all stimulated states (Eq. Everything is degenerate four times, or alternatively, we don't know what happens in a symmetric spatial composition, so we can't determine which is more energetic鈥攁t least ortho or para.

蠄0r1,r2=蠄100r1蠄100r2=8蟺a3e-2r1+r2/a 鈥�(5.30).

蠄n鈩�m蠄100 ...(5.32).

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Most popular questions from this chapter

(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.

1s22s22p2(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.

LJ2S+1 (5.34).

(a) If 蠄aand蠄b are orthogonal, and both normalized, what is the constant A in Equation 5.10?

(b) Ifrole="math" localid="1658225858808" 蠄a=蠄b (and it is normalized), what is A ? (This case, of course, occurs only for bosons.)

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鈮�pc=hck.

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

dE=h2k22mV蟺2k2dk (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+鈫�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?

Find the average energy per free electron (Etot/Nd), as a fraction of the

Fermi energy. Answer:(3/5)EF

(a) Calculate<1/r1-r2>for the state蠄0(Equation 5.30). Hint: Dod3r2integral

first, using spherical coordinates, and setting the polar axis alongr1, so

that

蠄0r1,r2=蠄100r1蠄100r2=8蟺补3e-2r1+r2/a(5.30).

r1-r2=r12+r22-2r1r2肠辞蝉胃2.

The胃2integral is easy, but be careful to take the positive root. You鈥檒l have to

break ther2integral into two pieces, one ranging from 0 tor1,the other fromr1to鈭�

Answer: 5/4a.

(b) Use your result in (a) to estimate the electron interaction energy in the ground state of helium. Express your answer in electron volts, and add it toE0(Equation 5.31) to get a corrected estimate of the ground state energy. Compare the experimental value. (Of course, we鈥檙e still working with an approximate wave function, so don鈥檛 expect perfect agreement.)

E0=8-13.6eV=-109eV(5.31).
See all solutions

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