A system of N non interacting classical particles , each of mass m is in a two dimensional harmonic potential of the form V(r)= k (x²+y²) where k is a positive constant. the canonical partition function of the system at temperature T is?

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Answer

Understanding the System
In a two-dimensional harmonic potential, the potential energy is given by
V(r) = k(x² + y²),
where k is a positive constant. The system consists of N non-interacting classical particles, each with mass m.
Canonical Partition Function
The canonical partition function $ Z $ for a single particle in two dimensions is defined as:
$$ Z_1 = \frac{1}{h^2} \int e^{-\beta H(x,y)} \, dx \, dy $$
where $ H(x,y) = T + V(x,y) $ is the Hamiltonian, $ \beta = \frac{1}{k_B T} $, and $ h $ is the Planck constant.
Calculating the Hamiltonian
For our system, the Hamiltonian is:
$$ H(x,y) = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + k(x^2 + y^2) $$
where $ p_x $ and $ p_y $ are the momenta in the x and y directions.
Evaluating the Partition Function
To compute $ Z_1 $:
- The integral can be separated into the x and y components.
$$ Z_1 = \left(\frac{1}{h} \int e^{-\beta \left(\frac{p_x^2}{2m} + kx^2\right)} \, dp_x \, dx\right) \times \left(\frac{1}{h} \int e^{-\beta \left(\frac{p_y^2}{2m} + ky^2\right)} \, dp_y \, dy\right) $$
- Each integral resembles the form of a Gaussian integral, leading to:
$$ Z_1 = \frac{1}{h^2} \cdot \left(\frac{2\pi}{\beta m}\right) \cdot \left(\frac{2\pi}{\beta k}\right) = \frac{2\pi}{h^2} \frac{2\pi}{\beta^2 \sqrt{mk}} $$
- Hence, for N non-interacting particles:
$$ Z_N = \frac{Z_1^N}{N!} $$
- Combining, the canonical partition function becomes:
$$ Z_N = \frac{1}{N!} \left(\frac{2\pi}{h^2 \beta^2 \sqrt{mk}}\right)^N $$
This encapsulates the partition function for a system of N non-interacting classical particles in a two-dimensional harmonic potential.

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