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The energy of harmonics oscillator, started of [tex]U=-\frac{\partial}{\partial \beta} \ln Z[/tex] is equal to [tex]\frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\beta \hbar \omega)-1}[/tex].
At high temperature, i could say that [tex]exp (\beta \hbar \omega ) \approx 1 + (\beta \hbar \omega )[/tex], and then [tex]U=\frac{\hbar \omega}{2} + kT[/tex], therefore at high temperature [tex]\frac{\hbar \omega}{2}[/tex] is negligible compared to [tex]kT[/tex], and then [tex]U \approx k T [/tex].
I need find arguments about why is incorrect say that [tex]\frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\beta \hbar \omega)-1}[/tex] at [tex]\beta \to 0[/tex] (high temperature) is equal to [tex]\infty[/tex]. This motivated by the fact that [tex]k T = k \cdot \infty = \infty[/tex]. I understand that at high temperature the energy has a asyntote equal to kT (http://www.av8n.com/physics/oscillator.htm#sec-e-vs-t ), but still need argumens.
Also ¿why the harmonics oscillators need a specific heat at high temperature?. In this case the specific heat is equal to k. But if the energy us infinity, then the specific heat would be zero.
The energy of harmonics oscillator, started of [tex]U=-\frac{\partial}{\partial \beta} \ln Z[/tex] is equal to [tex]\frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\beta \hbar \omega)-1}[/tex].
At high temperature, i could say that [tex]exp (\beta \hbar \omega ) \approx 1 + (\beta \hbar \omega )[/tex], and then [tex]U=\frac{\hbar \omega}{2} + kT[/tex], therefore at high temperature [tex]\frac{\hbar \omega}{2}[/tex] is negligible compared to [tex]kT[/tex], and then [tex]U \approx k T [/tex].
I need find arguments about why is incorrect say that [tex]\frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\beta \hbar \omega)-1}[/tex] at [tex]\beta \to 0[/tex] (high temperature) is equal to [tex]\infty[/tex]. This motivated by the fact that [tex]k T = k \cdot \infty = \infty[/tex]. I understand that at high temperature the energy has a asyntote equal to kT (http://www.av8n.com/physics/oscillator.htm#sec-e-vs-t ), but still need argumens.
Also ¿why the harmonics oscillators need a specific heat at high temperature?. In this case the specific heat is equal to k. But if the energy us infinity, then the specific heat would be zero.
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