I am stuck on #2.

Note: the attached document has the question in greater detail. I am stuck on #2.

In lecture we estimated the partition function for a quantum mechanical particle in a box using the approximation

Q =∞

n=1

e−βh2n2/(8mL2) ≈ ∞

0

dx e−βh2x2/(8mL2).

We argued that this approximation is a good one if the spacing between energy levels is small compared to kBT.Here you will explore the basis for this reasoning in more detail.

(a) *D1* Consider first a helium atom in a box of size L = 1 m at room temperature.Calculate βh2/(8mL2).

(b) *D1* For the helium atom of part (a), calculate the Boltzmann factor b(n) = exp[−βh2n2/(8mL2)] for n = 1, n = 2, n = 3, n = 10, n = 1000, n = 10000, n = 109, n = 1010, and n = 1011. (Your answers need to be accurate only up to a few decimal places.)

(c) *D1* In light of your answers to part (b), give a simple estimate for the numerical value of the sum

106

n=1,2,3...

e−βh2n2/(8mL2)

1

for the helium atom.Again, your answer should be accurate up to a few decimal places.