Statistical Mechanics Problem Set 3 -...
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Statistical Mechanics Problem Set 3 Due date: Oct 7, 2015 Noon 1-3 Pathria 3.35 3.44 4-6 Pathria 4.3 4.5 4.10 4.13 7-9 Pathria 5.1 5.4 5.5 10 Grand Partition Function Show that ln G = PV k B T = μN 11 Harmonic Oscillator Evaluate the matrix elements of the density matrix, ρ = exp(-βH), H = p 2 /2m + 1 2 mω 2 q 2 of a one-domensional harmonic oscillator in the q-representation. Discuss, in particular the limiting case of ~ω/k B T = β~ω 1. Hint: The eigenfunction ψ n (q) for the eigenvalue E n =(n + 1 2 )~ω is given by ψ n (q)= mω π~ 1/4 H n (ξ ) √ 2 n n! e - 1 2 ξ 2 , ξ = mω ~ q. The Hermite polynomials H n (ξ ) are defined as H n (ξ )=(-1) n e ξ 2 d dξ n e -ξ 2 = e xi 2 √ π Z ∞ -∞ (-2iu) n e -u 2 +2iξu du. 12 Statistical Potential for Anyon We consider an anyon system in two spatial dimensions with two electrons which interact with external magnetic-flux tubes. The nonrelativistic Hamiltonian of the two-electron system is H = 2 X i=1 1 2m p i - e c A(r i ) 2 , where e is the charge of the electron and A is the vector potential written in the form A(r)= Φ 2π ∇θ. Here Φ ≡ αhc/e is the flux tube with α controlling its magnitude and θ denotes the relative angle between the two electrons. (a) Solve the Schr¨ odinger equation for the Hamiltonian by decomposing into the center-of-mass and relative-coordinate parts, and by using the gauge transformation A(r) → A(r) -∇θ. Show that the wavefunction is given Ψ α ≡ X n e iP ·R/~ ψ α = X n e iP ·R/~ e i(n+α)θ J |n| (r) with corresponding eigenvalue E = P 2 /4m + p 2 /m. J m (x) is the Bessel function. Here Bose/Fermi statistics requires that n + α ≡ l be even/odd. 1
Transcript of Statistical Mechanics Problem Set 3 -...
Statistical Mechanics
Problem Set 3
Due date: Oct 7, 2015 Noon
1 - 3 Pathria 3.35 3.44
4 - 6 Pathria 4.3 4.5 4.10 4.13
7 - 9 Pathria 5.1 5.4 5.5
10 Grand Partition Function
Show that
lnG =PV
kBT= µN
11 Harmonic Oscillator
Evaluate the matrix elements of the density matrix, ρ = exp(−βH), H = p2/2m + 12mω
2q2 of a one-domensionalharmonic oscillator in the q-representation. Discuss, in particular the limiting case of ~ω/kBT = β~ω � 1. Hint: Theeigenfunction ψn(q) for the eigenvalue En = (n+ 1
2 )~ω is given by
ψn(q) =(mωπ~
)1/4 Hn(ξ)√2nn!
e−12 ξ
2
, ξ =mω
~q.
The Hermite polynomials Hn(ξ) are defined as
Hn(ξ) = (−1)neξ2
(d
dξ
)ne−ξ
2
=exi
2
√π
∫ ∞−∞
(−2iu)ne−u2+2iξudu.
12 Statistical Potential for Anyon
We consider an anyon system in two spatial dimensions with two electrons which interact with external magnetic-fluxtubes. The nonrelativistic Hamiltonian of the two-electron system is
H =
2∑i=1
1
2m
(pi −
e
cA(ri)
)2,
where e is the charge of the electron and A is the vector potential written in the form
A(r) =Φ
2π∇θ.
Here Φ ≡ αhc/e is the flux tube with α controlling its magnitude and θ denotes the relative angle between the twoelectrons.
(a) Solve the Schrodinger equation for the Hamiltonian by decomposing into the center-of-mass and relative-coordinateparts, and by using the gauge transformation A(r)→ A(r)−∇θ. Show that the wavefunction is given
Ψα ≡∑n
eiP ·R/~ψα =∑n
eiP ·R/~ei(n+α)θJ|n|(r)
with corresponding eigenvalue E = P 2/4m + p2/m. Jm(x) is the Bessel function. Here Bose/Fermi statisticsrequires that n+ α ≡ l be even/odd.
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(b) The system’s partition function Z = Tr e−βH takes the form Z = 2Aλ−2T Z, where Z is the partition function of asingle particle in the relative-coordinate problem, λT is the thermal wavelength, and A is the area of the system.Calculate Z and show
Z =1
2
∞∑l=−∞
∫ ∞0
dxe−xI|l−α|(x),
where x ≡ mr2/2β~2 = πr2/λ2T , and I|l−α|(x) is modified Bessel function of the first kind.
(c) Compare Z with the partition function of a classical system with H = p2
2µ + V where µ is the reduced mass andobtain the statistical potential V .
(d) Show that when δ = 0 the potential reduces to those of ordinary fermions and bosons, i.e. e−βV = 1∓ e−2x.
(e) Compute numerically and plot the statistical potential for (i) α = 2j + 1 + δ with |δ| < 0.5 (quasifermions) (ii)and for α = 2j + δ with |δ| < 0.5 (quasibosons). Observe the difference
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