Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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132 CHAPTER 6. DENSITY MATRIX FORMALISM.<br />
intensive st<strong>at</strong>e variables their correct value, then to take the thermodynamic<br />
limit N → ∞, and finally consider the results as a function of ɛ and take the<br />
limit ɛ → 0. In practice, one should make N large and choose ɛ as small as<br />
possible, but large compared to the average spacing of the energy levels. We<br />
will discuss this question by looking <strong>at</strong> a simple example.<br />
A quantum system in one dimension has single-particle eigenst<strong>at</strong>es with<br />
energy nω, with n = 1, 2, 3, · · ·. The energy of N particles is given by ( N i ni)ω.<br />
Hence the entropy is equal to<br />
S(U, N) = −kB<br />
<br />
n1,n2,.,nN<br />
C 1<br />
ɛ √ π e− (U−( N i ni )ω)2<br />
ɛ2 <br />
log C 1<br />
ɛ √ π e− (U−( N i ni )ω)2<br />
ɛ2 <br />
(6.76)<br />
We assume th<strong>at</strong> the particles are not identical. Next we set xi = ni ω<br />
Nɛ and<br />
assume th<strong>at</strong> ɛ ≫ ω<br />
N , in agreement with the st<strong>at</strong>ement th<strong>at</strong> we can only take<br />
the limit ɛ → 0 after taking the limit N → ∞. The quantity ω<br />
N is the spacing<br />
between the possible energy values per particle and we have to choose ɛ much<br />
larger than this value. The summ<strong>at</strong>ions are replaced by integr<strong>at</strong>ions and we<br />
have<br />
S(U, N) = −kB<br />
N <br />
Nɛ<br />
ω<br />
<br />
· · ·<br />
C 1<br />
ɛ √ U<br />
e−( ɛ<br />
π −Nx1−···−NxN ) 2<br />
<br />
log C 1<br />
ɛ √ <br />
− (<br />
π<br />
U<br />
dx1 · · · dxN<br />
ɛ − Nx1 − · · · − NxN) 2<br />
<br />
(6.77)<br />
Now we define x = <br />
i xi and replace the integr<strong>at</strong>ion variables by x, x2, · · · , xN.<br />
Since all the variables xi have to be positive, we know th<strong>at</strong> for a given value<br />
of x all other variables are limited by N i=2 xi x. Since the integrand only<br />
depends on the value of x, the integr<strong>at</strong>ion over x2, · · · , xN is just a geometrical<br />
factor g(x). Therefore, the entropy is equal to<br />
N <br />
ɛN<br />
S(U, N) = −kB<br />
dxg(x)<br />
ω<br />
<br />
− ( U<br />
<br />
− Nx)2<br />
ɛ<br />
C 1<br />
ɛ √ U<br />
e−( ɛ<br />
π −Nx)2<br />
<br />
log C 1<br />
ɛ √ π<br />
The coefficient C follows from Tr ρ = 1:<br />
or<br />
C −1 = Tr<br />
C −1 = <br />
n1,n2,.,nN<br />
(6.78)<br />
1<br />
ɛ √ (U−H)2<br />
e− ɛ<br />
π 2 (6.79)<br />
1<br />
ɛ √ π e− (U−( N i n i )ω)2<br />
ɛ 2 (6.80)