English version
UNIVERSITETET I OSLO
Det matematisk-naturvitenskapelige fakultet
Eksamen i: FYS 203 Statistisk mekanikk
Eksamensdag: Fredag 25.mai 2001
Tid for eksamen: 0900 - 1500
Oppgavesettet er på ?? sider
Tillatte hjelpemidler:
Clark: Physical and Mathematical Tables
Øgrim: Størrelser og enheter i fysikken
Olivber and Boyd: Science Data Book
Tabeller i fysikk for den videregående skole
Rottmann: Matematisk formelsamling
Godkjent numerisk electronisk kalkulator
Kontrollér at oppgavesettet er komplett før du begynner å besvare spørsmålene
Problem 1: Classically Confined Fermi Gas
A Fermi gas containing N noninteractiong particles with spin S is confined in z-direction
by the harmonic-oscillator potential
1
U (z) = mω02 z 2
2
(1)
where m is the mass of the particle, while ω0 is the oscillation frequency.
In this problem, we assume that motion along z-axis obeys classical mechanics.
a: Show that the density of states as a function of the particle energy ε, D(ε), has the form
D(ε) = Ag0 ε/~ω0 ,
(2)
g0 = (2S + 1)m/2π~2
(3)
where
and A is the area of the system.
Hints:
Rβ
(i) By definition, for any γ belonging to the interval β > γ > α, α dx δ(x − γ) = 1. If γ
is outside this interval, then the integral is 0. This definition is often expressed in terms
of the Heaviside step function Θ(ξ) as
½
Z β
1, ξ > 0
dx δ(x − γ) = Θ(β − γ)Θ(γ − α) where Θ(ξ) =
.
(4)
0, ξ < 0
α
1
(ii) An auxiliary intergral:
Z ∞
Z
∞
du
0
0
µ
¶
u2
v2
E
dv Θ E − a − b
=√ .
2
2
ab
(5)
b: Show that the chemical potential µ at zero temperature (let us denote it is as µ0 ) as a
function of particle sheet concentration, ns ≡ N/A, is given by the expression
µ(0) ≡ µ0 = (2ns ~ω0 /g0 )1/2 .
Hint:
Z
∞
0
f (x) dx
≈
ex−z + 1
Z
z
f (x) dx +
0
π2 0
f (z) + . . .
6
(6)
at z À 1 .
(7)
c: Analyze temperature dependence of the chemical potential at high and at low temperatures.
Find proper criterion for crossover between these cases.
Hint: At large negative z, z < 0, |z| À 1,
Z ∞
Z z
f (x) dx
z
e−x f (x) dx .
≈e
x−z
e
+1
0
0
(8)
d: Find internal energy U , Landau free energy Ω, entropy S and specific heat CV of the gas.
Show that Ω = −U/2. Derive equation of state.
Hint: In quasi two-dimensional systems the pressure is defined as a derivative with respect
to the area A rather than with respect to the volume.
Problem 2: Quantum-Mechanical Confinement
Now let us assume that the the motion along z-direction in the potential (1) obeys quantum
mechanics.
The quantum-mechanical state is specified by the quantum numbers n, p⊥ (p⊥ =
{px , py }) where n is the discrete quantum number describing quantized motion in zdirection. The state with a given n is called the nth transverse mode.
The energy eigenvalues are
µ
¶
1
p2
εn (p) = ~ω0 n +
+ ⊥.
2
2m
a: Find density of states as a function of the particle energy ε. Show that its profile is a set of
equidistant steps of the same height,
µ
¶
X
1
D(ε) = Ag0
Θ(ε − εn ) , εn ≡ ~ω0 n +
.
(9)
2
n
2
Hint:
The motion in the x − y plane remains classical.
b: Derive the equation for the chemical potential µ as a function of particle sheet concentration,
ns . Show that at T = 0 this equation reads as
µ
¶
1
nmax (nmax + 1)
µ
1
ns
= +
−
,
(10)
~ω0
2 nmax + 1 g0 ~ω0
2
where nmax + 1 is the number of occupied transverse modes having εn ≤ µ. Find nmax .
Hint:
Z
∞
0
dx
= ln (ez + 1) .
+1
ex−z
(11)
c: Consider the case of low concentration for which nmax = 0 at T = 0.
- Show that at T = 0 the chemical potential is given by the equation
µ − ~ω0 /2 = ns /g0 .
(12)
- Find the range of applicability for this expression (i.e. the inequilities limiting temperature T
and concentration ns ).
- Find finite-temperature corrections to the chemical potential at low concentrations.
d: Find internal energy U , Landau free energy Ω, entropy S and specific heat CV at low
concentration and temperature.
e: Using grand canonical ensemble derive the expression for mean square fluctuation in the
particle number,
µ
¶
∂N
2
h(∆N ) i = T
.
(13)
∂µ T,A
Apply this expression for calculation of the fluctuation in the particle number for the discussed
above case of low concentration and temperature.
Hint: Calculate ∂hN i/∂µ and make use of the equality hN i = −∂Ω/∂µ.
Problem 3: Stochastic processes
Consider an electron (spin S = 1/2) bounded to an impiruty center and assume that the
temperature T is much less than the energy difference between the levels of orbital motion.
Consequently, only spin degrees of freedom are excited.
3
Let the system be placed in external magnetic field B, the additional energy in magnetic
field is ±βB where ± corresponds to the values ±1/2 of the spin component along the field.
Here β = |e|~/2mc is the Bohr magneton, c is the velocity of light since the Gaussian system
is used. The magnetic moment of the states with Sz = ±1/2 is ∓β.
a: Find the stationary probabilities for the states with Sz = ±1/2, P± .
- Find average magnetic moment of the system as function of temperature T and magnetic
field B.
b: Find mean square fluctuation of magnetic moment, h(∆M )2 i and the ratio h(∆M )2 i/hM i2 .
- Show that
β2
.
(14)
h(∆M 2 )i =
cosh2 (βB/T )
- Discuss limiting cases of large and low temperatures. Explain these results qualitatively.
c: Write down the Master Equation for the probability Pi (t) to find the system in the state
i = ± at time t.
- Using the result for the stationary case find the relation between the transition probabilities W+− and W−+ for the transition from the state “+” to the state “-” and for the reverse transition, respectively. Express the probabilities W+− and W−+ through the quantity
W = W+− + W+− .
- Derive the Master Equation for the population difference, P(t) ≡ w+ (t) − w+ (t).
d: Write down the Master Equation for the conditional probability, P (f, t|i, t0 ), to find the
system in the state f at time t under the conditions that at time t0 it was in the state i.
- Specify initial conditions to this equation.
- Show that solution of this equation with proper initial condition is
P (f, t|i, t0 ) = wf + (δif − wf )e−W |t−t0 |
(15)
where wf is the stationary probability to find the system in the state f .
Hint: It is convenient to express the set of Master Equations for the quantities P (f, t|i, t0 )
at different i and f as an equation for the 2 × 2 matrix P̂ with matrix elements Pf i (t|t0 ) ≡
P (f, t|i, t0 ). The solution of the matrix Master
Equation ∂ P̂ /∂T = Ŵ P̂ at t ≥ t0 can
P
then be searched in in the matrix as P̂ ∝ k Ĉ (k) e−λk (t−t0 ) . Here λk are eigenvalues of the
matrix W , while Ĉ (k) are time-independent matrices. They are determined from the intial
conditions and from the fact that at t − t0 → ∞ the conditional probabilities tend to the
stationary ones.
4
e: Using Eq. (15) show that the correlation function h∆M (t)∆M (0)i has the form
h∆M (t)∆M (0)i = h(∆M )2 i e−W |t|
where h(∆M )2 i is given by Eq. (14) while W = W+− + W−+ .
- Find the fluctuation spectrum.
5
(16)