In this section we computed the single-particle translational partition function, Ztr, by summing over all definite-energy wavefunctions. An alternative approach, however, is to sum over all possible position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip in a factor of 1/h3 to get a unitless number that actually counts the independent wavefunctions. Thus, we might guess the formula
Ztr=h31∫d3rd3pe−Etr/kT,
where the single integral sign actually represents six integrals, three over the position components (denoted d3r) and three over the momentum components (denoted d3p). The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume V. By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text. (The only time this formula would not be valid would be when the box is so small that we could not justify converting the sum in equation 6.78 to an integral.)