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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.)

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=23EtotV=23h2kF510蟺2m=(3蟺2)2/3h25mp5/3(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.

Etot=h2V2蟺2m鈭�0kFK4dk=h2kF5V10蟺2m=h2(3蟺2Nd)5/310蟺2mV-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=(9蟺4)2/3h2d5/3GmM2N1/3

(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 (seeProblem 5.36).

Show that most of the energies determined by Equation 5.64are doubly degenerate. What are the exceptional cases? Hint: Try it for N=1,2,3,4.... , to see how it goes. What are the possible values of cos(ka)in each case?

(a) Using Equations 5.59 and 5.63, show that the wave function for a particle in the periodic delta-function potential can be written in the form

蠄(X)=C[sinkx+e-ikasina-x]0鈮�x鈮�a

(b) There is an exception; At the top of a band where z is an integer multiple of蟺yiels蠄(x)=0 yields .

Find the correct wave function for the case. Note what happens to蠄each delta function.
See all solutions

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