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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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A system of N non interacting classical particles , each of mass m is ...
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.
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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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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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? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about 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? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about 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? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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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