763628SCONDENSED MATTER PHYSICS Problem Set 5
Spring 2012
1. Fluctuation–dissipation theorem
Consider a particle moving in one dimensions in a viscous material
with velocity
v̇ = −bv + ξ(t)
(1)
where b is a damping constant. In the following, we demonstrate that
in addition to damping, there necessarily exists a time–dependent
1
force F (t) = m
ξ(t) acting on the particle. Let us assume that for
the average value holds hξ(t)i = 0, and that hξ(t)ξ(t0 )i = cδ(t − t0 )
meaning that the forces at different times are uncorrelated.
a)
Show that
−bt
Z
t
ξ(τ )ebτ dτ
v(t) = e
−∞
is a solution to Eq. (1).
b)
Show that
hv 2 (t)i =
c)
c
.
2b
Show that the equipartition theorem
2
1
2 mhv i
= 21 kT
follows from the thermal distribution p(v) ∝ e−E/kT with E =
1
2
2 mv .
d)
Using the results from b) and c), show that
hξ(t)ξ(t0 )i =
e)
2bkT
δ(t − t0 ) .
m
The above equations apply also to many other systems. As an
example, let us consider a circuit consisting of a resistor and a
solenoid:
δV
L
R
˙ VR = IR, and EL = 1 LI 2 . Use the
Recall that VL = LI,
2
analogy to show that in series with the resistor there must exist
a noise voltage δV for which holds hδV (t)δV (t0 )i = 2kT Rδ(t −
t0 ). (Nyqvist 1928)
2. Molecular dynamics discretized
In discussing molecular dynamics, we brought up the equation
s
~ n+1 − 2R
~n + R
~ n−1
~ n+1 − R
~ n−1
~n
R
R
F
6bkT ~
l
l
l
l
= l −b l
+
θ
2
dt
ml
2dt
ml dt
with θ~ = 2 p1 − 12 , p2 − 21 , p3 − 21 and the pi ’s are random numbers
between 0 and 1. Check that the discretization is correct.
3. Product wave functions
a) Show that a product of wave functions obeying
h̄2 2
−
∇ + U (~r) ψl (~r) = El ψl (~r)
2m
satisfies
N X
h̄2 2
−
∇ + U (~rl ) Ψ(~r1 , . . . , ~rN ) = EΨ(~r1 , . . . , ~rN )
2m l
l=1
P
with E = l El .
b) The product wave function is actually not acceptable for many
electrons, because the Pauli principle demands that wave functions be antisymmetric under interchange of any two coordinates. The correct form of a wave function is
N
Ψ=
Y
1 X
(−1)s
ψsl (~rl ) ,
N! s
l=1
where the sum over s denotes a sum over all the permutations
of N integers, (−1)s gives the sign of the permutation, and sl
denotes entry l in the permutation. Verify that the energy of
Ψ is the same as in a).
4. Ground states
Consider a free Fermi gas with N electrons. Find the energy of the
ground state as N varies from 1 through 15.
5. Pressure
Find the pressure of the ideal Fermi gas in three dimensions at zero
temperature. You may use dE = SdT − pdV .