- #1
damarkk
- 8
- 2
- Homework Statement
- Statistical Mechanics
- Relevant Equations
- Canonical Ensemble, Thermodynamic generalized forces
Assume you have a microscopic pendulum you can suppose is like quantum harmonic oscillator. If the length of pendulum has variation of ##dl##, calculate the work on the pendulum and thermodynamic generalized force.
Find also the variation of mean number of extitations.
My Attempt
Firstly, I find the partition function: ##Z= \sum_n e^{-\beta\epsilon_n}##, with ##\epsilon_n = \hbar\omega(1/2 + n)## and this is the result:
##Z= \frac{1}{2sinh(\beta\hbar\omega/2)}##.
In this result ##\beta= \frac{1}{kT}##.
Then, I can write Helmholtz Free Energy: ##F = U -TS = -kT\ln{Z}##
And of course ## U = F-T\frac{\partial F}{\partial T}##
I know that ##-\frac{\partial U}{\partial x} = F_x## but if i don't know how thermodynamic variables depends on ##l## how I can compute the thermodynamic generalized force?
Find also the variation of mean number of extitations.
My Attempt
Firstly, I find the partition function: ##Z= \sum_n e^{-\beta\epsilon_n}##, with ##\epsilon_n = \hbar\omega(1/2 + n)## and this is the result:
##Z= \frac{1}{2sinh(\beta\hbar\omega/2)}##.
In this result ##\beta= \frac{1}{kT}##.
Then, I can write Helmholtz Free Energy: ##F = U -TS = -kT\ln{Z}##
And of course ## U = F-T\frac{\partial F}{\partial T}##
I know that ##-\frac{\partial U}{\partial x} = F_x## but if i don't know how thermodynamic variables depends on ##l## how I can compute the thermodynamic generalized force?