diff --git a/src/1_einstein_model.md b/src/1_einstein_model.md index 5e249c00fa127f8d41ebd3c7dfebb0d092ba8655..bbb18d6fa5b39bc3bed76d777a37dbb43514771c 100644 --- a/src/1_einstein_model.md +++ b/src/1_einstein_model.md @@ -226,7 +226,7 @@ This oscillator has an energy spectrum given by $$\varepsilon_n=\left(n+\frac{1}{2}\right)\hbar\omega_0$$ where $\omega_0$ is the eigenfrequency of the oscillator. -What is the thermal occupation and corresponding energy of this oscillator? A harmonic oscillator is a bosonic mode $\Rightarrow$ its occupation is described by the Bose-Einstein distribution: +What is the thermal occupation and corresponding thermal energy stored in this oscillator? A harmonic oscillator is a bosonic mode $\Rightarrow$ its occupation is described by the Bose-Einstein distribution: $$ n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1} $$ @@ -234,8 +234,9 @@ The Bose-Einstein distribution describes the occupation probability of a state a $$ \bar{\varepsilon}=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1} $$ -The plot on the left shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The plot on the right shows the increasing thermal energy in the oscillator as a function of temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$, which is a consequence of the uncertainty principle. -. + +The left plot shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The right plot shows the increasing thermal energy in the oscillator for increasing temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$ - a consequence of the uncertainty principle. + ```python fig, (ax, ax2) = pyplot.subplots(ncols=2, figsize=(10, 5)) omega = np.linspace(0.1, 2)