Answer: The partition function is given by , Z = 1 / (1 - e^(-hv/kT))
Explanation:
To determine the partition function for a harmonic oscillator at temperature T, we need to sum over all possible energy states of the system.
The partition function, denoted as Z, can be calculated using the formula:
Z = Σe^(-Ei/kT)
where Σ represents the summation over all energy states (i), Ei is the energy of the i-th state, k is the Boltzmann constant, and T is the temperature in Kelvin.
In the case of a harmonic oscillator, the energy levels are given by Ei = i * hv, where h is Planck's constant and v is the vibrational frequency.
So, the partition function for the harmonic oscillator is:
Z = Σe^(-i*hv/kT)
To evaluate this sum, we can use the geometric series formula, which states:
Σ(r^i) = 1 / (1 - r)
where r is the common ratio.
In our case, r = e^(-hv/kT), so we can substitute this into the formula:
Z = 1 / (1 - e^(-hv/kT))
This equation represents the partition function for a harmonic oscillator at temperature T.