1. (6 pts) Free energy of a two state system.
(a) Find an expression for the free energy as
a function of
τ
=k
B
T of a system with two states, one at energy 0 and one at energy
ε
. (b)
From the free energy, find expressions for the energy and entropy of the system.
2. (5 Pts) Quantum concentration.
Consider one particle confined to a cube of side L; the
concentration in effect is
3
L
/
1
n
=
. Find the kinetic energy of the particle when in the
ground orbital. There will be a value of the concentration for which this zeropoint
quantum kinetic energy is equal to the temperature
τ
=k
B
T. (At this concentration the
occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a
higher occupancy than any other orbital.) Show that the concentration n
o
equal to the
quantum concentration
2
/
3
2
Q
2
M
n
π
τ
=
h
, within a factor of the order of unity.
3. (6 pts) Onedymensional gas.
Consider an ideal gas of N particles each of mass M,
confined to a onedimensional line of length L. Find the partition function, Helmholtz
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 Spring '07
 Lynch
 Physics, Atom, Energy, Work, Statistical Mechanics, Entropy, Fundamental physics concepts, electron concentration

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