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Chapter 5: Q32P (page 245)

Imagine two non interacting particles, each of mass , in the one dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate ⟨(x1-x2)2⟩assuming
(a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state.)

Short Answer

Expert verified

(a) ((x1-x2)2)=2hmω.

(b)(x1-x2)2=hmω

(c)(x1-x2)2=3hmω

Step by step solution

01

(a) Determining ⟨(x1-x2)2⟩ when they are distinguishable particles.

From the Previous problem :

⟨x⟩0=0⟨x⟩1=0x20=h2mωx21=3h2mω

We need to calculate:

0x1=∫-∞∞xψ0xψ1xdx=h2mω1δ0,0+0δ1,-1=h2mω

From Equation 5.21:

role="math" localid="1658402226570" x1-x22=h2mω+3h2mω-0=2hmω

02

(b) Determining ⟨(x1-x2)2⟩when they are identical bosons,

From Equation 5.21

x1-x22=2ħmω+2h2mω=hmω

03

(c) Determining ⟨(x1-x2)2⟩ when they identical fermions,

From Equation 5.21:

x1-x22=2hmω+2h2mω=3hmω

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Most popular questions from this chapter

(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

ψ(X)=C[sinkx+e-ikasina-x]0≤x≤a

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

Find the correct wave function for the case. Note what happens toψeach delta function.

(a) Construct the completely anti symmetric wave function ψ(xA,xB,xC)for three identical fermions, one in the state ψ5, one in the state ψ7,and one in the state ψ17

(b)Construct the completely symmetric wave function ψ(xA,xB,xC)for three identical bosons (i) if all are in state ψ11(ii) if two are in state ψ19and another one is role="math" localid="1658224351718" ψ1c) one in the state ψ5, one in the state ψ7,and one in the stateψ17

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?

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?
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