Partition Function for Classical Harmonic Oscillator for interview of Assistant Professor,Phd#physic

Partition Function of Classical Harmonic Oscillator (Short Notes) Definition The partition function is a central quantity in statistical mechanics that contains all thermodynamic information of a system. For a classical one-dimensional harmonic oscillator, E=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}x^{2} where: = mass of oscillator = angular frequency = displacement = momentum --- Classical Partition Function The canonical partition function is Z=\frac{1}{h}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-\beta E}\,dx\,dp where \beta=\frac{1}{k_B T} Substituting , Z=\frac{1}{h} \int_{-\infty}^{\infty} e^{-\beta \frac{p^2}{2m}}dp \int_{-\infty}^{\infty} e^{-\beta \frac{1}{2}m\omega^2x^2}dx Using the Gaussian integral \int_{-\infty}^{\infty} e^{-ax^2}dx = \sqrt{\frac{\pi}{a}} we get \int e^{-\beta \frac{p^2}{2m}}dp = \sqrt{\frac{2\pi m}{\beta}} and \int e^{-\beta \frac{1}{2}m\omega^2x^2}dx = \sqrt{\frac{2\pi}{\beta m\omega^2}} Therefore, Z = \frac{1}{h} \sqrt{\frac{2\pi m}{\beta}} \sqrt{\frac{2\pi}{\beta m\omega^2}} Z = \frac{2\pi}{h\beta\omega} Since \beta=\frac{1}{k_B T} \boxed{ Z=\frac{2\pi k_B T}{h\omega} } --- Average Energy U=-\frac{\partial}{\partial\beta}\ln Z Since \ln Z = \ln\left(\frac{2\pi}{h\omega}\right)-\ln\beta U=\frac{1}{\beta} \boxed{ U=k_B T } --- Heat Capacity C_V = \left(\frac{\partial U}{\partial T}\right)_V \boxed{ C_V=k_B } --- Physical Meaning A harmonic oscillator has two quadratic degrees of freedom: Kinetic energy Potential energy By the equipartition theorem, \frac{1}{2}k_B T energy is associated with each quadratic term. Hence, U= \frac{1}{2}k_B T + \frac{1}{2}k_B T = k_B T --- Final Results (Exam Point of View) \boxed{E=\frac{p^2}{2m}+\frac12 m\omega^2x^2} \boxed{Z=\frac{2\pi k_B T}{h\omega}} \boxed{U=k_B T} \boxed{C_V=k_B} One-Line Conclusion For a classical harmonic oscillator, the partition function is proportional to temperature , the mean energy is , and the heat capacity is . ‪@FunwithPhysics‬ ‪@D_PHYSICS‬ ‪@dhruvrathee‬ ‪@iitjam_cuetpg_physics‬ ‪@PravegaaEducation‬ ‪@potentialg‬ ‪@physicsbyfiziks‬ ‪@physicsbyfiziks‬ ‪@Fizica.‬ ‪@Fizicăăspirant‬ ‪@ajphysics1992‬ ‪@Bmsharmaphysics‬ ‪@FunwithPhysics‬ ‪@FortheLoveofPhysics‬ ‪@D_PHYSICS‬ ‪@PhysicsEliteby‬