- #1
Narcol2000
- 25
- 0
I'm having problems understanding how
[tex]
\frac{e^{-\hbar \omega / 2k_BT}}{1-e^{-\hbar \omega / k_BT}}
[/tex]
approximates to
[tex]
k_BT/ \hbar\omega
[/tex]
when
[tex]
T >> \hbar\omega/k_B
[/tex]
Seems like it should be simple but don't quite see how to arrive at this result.
*update*
I have tried using taylor expansions of [tex]exp(-x)[/tex] and [tex]1-exp(-x)[/tex] and just using the first expansion term since if [tex]T>>\hbar\omega/k_B[/tex] then [tex]\hbar\omega/k_BT[/tex] should be small. This seems to give the right answer but i'd be interested in knowing if indeed my method is ok and if there are alternate methods.
[tex]
\frac{e^{-\hbar \omega / 2k_BT}}{1-e^{-\hbar \omega / k_BT}}
[/tex]
approximates to
[tex]
k_BT/ \hbar\omega
[/tex]
when
[tex]
T >> \hbar\omega/k_B
[/tex]
Seems like it should be simple but don't quite see how to arrive at this result.
*update*
I have tried using taylor expansions of [tex]exp(-x)[/tex] and [tex]1-exp(-x)[/tex] and just using the first expansion term since if [tex]T>>\hbar\omega/k_B[/tex] then [tex]\hbar\omega/k_BT[/tex] should be small. This seems to give the right answer but i'd be interested in knowing if indeed my method is ok and if there are alternate methods.
Last edited: