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Chapter 5: Q33P (page 246)

Suppose you have three particles, and three distinct one-particle state唯aX,唯bX,and唯cxare available. How many different three-particle states can be constructed (a) if they are distinguishable particles, (b) if they are identical bosons, (c) if they are identical fermions? (The particles need not be in different states -唯aX1,唯aX2唯ax3would be one possibility, if the particles are distinguishable.)

Short Answer

Expert verified

(a)27

(b)10

(c)1

Step by step solution

01

(a)if they are distinguishable

Each particle has 3 possible states: 3 脳 3 脳 3 = 27.

02

(b) if they are identical bosons

All in same state: aaa, bbb, ccc鈬�3.

2 in one state: aab, aac, bba, bbc, cca, ccb鈬�6 (each symmetrized).

3 different states: abc (symmetrized)鈬�1.

Total: 10.

If they are identical fermions

Only abc (antisymmetrized) 鉄� 1.

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Most popular questions from this chapter

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?

Suppose you had three particles, one in state蠄a(x), one in state蠄b(x), and one in state蠄c(x). Assuming 蠄a,蠄b, and蠄c are orthonormal, construct the three-particle states (analogous to Equations 5.15,5.16, and 5.17) representing

(a) distinguishable particles,

(b) identical bosons, and

(c) identical fermions.

Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be completely antisymmetric, in the same sense. Comment: There's a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is蠄a(x1),蠄b(x1),蠄c(x1) , etc., whese second row is蠄a(x2),蠄b(x2),蠄c(x2) , etc., and so on (this device works for any number of particles).

Imagine two noninteracting particles, each of mass m, in the infinite square well. If one is in the state蠄n(Equation 2.28 ), and the other in state 蠄1(l鈮�n), calculate localid="1658214464999" (x1-x2)2, assuming (a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions.

(a) Construct the completely anti symmetric wave function 蠄(xA,xB,xC)for three identical fermions, one in the state 蠄5, one in the state 蠄7,and one in the state 蠄17

(b)Construct the completely symmetric wave function 蠄(xA,xB,xC)for three identical bosons (i) if all are in state 蠄11(ii) if two are in state 蠄19and another one is role="math" localid="1658224351718" 蠄1c) one in the state 蠄5, one in the state 蠄7,and one in the state蠄17

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?
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