Hi guys,
I'm studying a classical ideal gas trapped in a one-dimensional harmonic potential and I first want to write out the partition function for a single particle. This, I believe, requires two Gaussian integrations, like so:
[tex] Z=\int_{-\infty}^{\infty} d\dot{x} \int_{-\infty}^{\infty}dx \,\, e^{-\beta E(\dot{x},x)} [/tex]
However, we should like the partition function to be unitless. The above expression has units of (length)^2 divided by (time), as best as I can tell. Now, I know we want to divide by constant parameters of the problem to make it dimensionless. However, there is no characteristic length in this problem! The only constants we have are:
[tex] \omega = \sqrt{\frac{k}{m}} [/tex]
and I can't figure out how these can be combined to cancel out the (length)^2 units in Z. How does one figure this out. Thanks :)
I'm studying a classical ideal gas trapped in a one-dimensional harmonic potential and I first want to write out the partition function for a single particle. This, I believe, requires two Gaussian integrations, like so:
[tex] Z=\int_{-\infty}^{\infty} d\dot{x} \int_{-\infty}^{\infty}dx \,\, e^{-\beta E(\dot{x},x)} [/tex]
However, we should like the partition function to be unitless. The above expression has units of (length)^2 divided by (time), as best as I can tell. Now, I know we want to divide by constant parameters of the problem to make it dimensionless. However, there is no characteristic length in this problem! The only constants we have are:
[tex] \omega = \sqrt{\frac{k}{m}} [/tex]
and I can't figure out how these can be combined to cancel out the (length)^2 units in Z. How does one figure this out. Thanks :)
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