# Question about the uncertainty principle and unit cell in phase space

1. Oct 19, 2012

### Icefrog

In statistical mechanics, nearly all the textbooks say that the volume of the smallest cell in the phase space of a N-particle system is h$^{rN}$ where h is the Planck Constant, r is the degree of freedom.

Also these books say that this comes from the uncertainty principle.

However, the uncertainty principle is ΔxΔpx$\geqℏ/2$ .
So why they take h instead of ℏ/2 in calculating the volume?

Many thanks!

Last edited: Oct 19, 2012
2. Oct 19, 2012

### Bill_K

Actually, the books I read all say it does not. Historically, a finite cell size h was put in by Boltzmann to prevent the number of states from becoming infinite. Its value was determined experimentally by Sackur and Tetrode when they investigated the low temperature entropy of a monatomic gas. This work was pre-quantum mechanics, and so Planck's constant was not involved.

Now we know that particles are described quantum mechanically by wave functions. For a cube of volume V = L3, we impose periodic boundary conditions and find the allowed values of k to be spaced at intervals Δk = 2π/L, or Δp = 2πħ/L, implying V(Δp)3 = h3.

Last edited: Oct 19, 2012