add a note about hbar

src/4_sommerfeld_model.md

@@ -123,6 +123,12 @@ In the table below we summarize the properties of both phonons and electrons.
 | Degeneracy per $\mathbf{k}$ | 3 (polarization) | 2 (spin) |
 | Total particle number | temperature-dependent | constant |
 
+!!! note "About $\hbar$"
+
+    Within quantum mechanics energy and frequency are related by Planck's constant: $\varepsilon = \hbar\omega$.
+    Similarly, $p = \hbar k$ relates a particle's momentum with its wave vector.
+    This relation is so unambiguous that you may encounter these terms used synonymously in scientific literature.
+
 The last difference is important: warming a material up creates more thermally excited phonons.
 The number of electrons, on the other hand, stays the same: the electrons may not appear out of nowhere.[^1]
 
@@ -163,7 +169,7 @@ draw_classic_axes(ax, xlabeloffset = .8, ylabeloffset = 0.2);
 ```
 
 A good metaphor for describing this state of many electrons is a sea: electrons occupy a finite area in reciprocal space, starting from the "deepest" points with the lowest energy all the way up to the chemical potential—also called Fermi level.
-The border of the Fermi sea is called the Fermi surface (you should notice a pattern here), and in the free electron model it is a sphere with the radius equal to *Fermi momentum*.
+The border of the Fermi sea is called the Fermi surface (you should notice a pattern here), and in the free electron model it is a sphere with the radius equal to *Fermi wave vector*.
 To clarify the relation between these concepts let us take a look at the dispersion relation in 1D:
 
 ```python
@@ -204,9 +210,6 @@ $$
 \varepsilon_F = \frac{\hbar^2 \mathbf{k}_F^2}{2m}. 
 $$
 
-We mentioned earlier that the occupied states at $T = 0$ form a circle in the reciprocal space (or sphere or line depending on the dimensionality of the system). 
-This circle is called the _Fermi surface_.
-The shape of the Fermi surface is determined by the dispersion relation.
 The Fermi wavevector $\mathbf{k}_F$ also defines the _Fermi momentum_ $\mathbf{p}_F = \hbar \mathbf{k}_F$ and the _Fermi velocity_:
 
 $$
@@ -279,7 +282,6 @@ ax.set_ylabel(r"$g(\varepsilon)$")
 ax.set_xlabel(r"$\varepsilon$")
 ax.legend()
 draw_classic_axes(ax, xlabeloffset=.2)
-
 ```
 
 ## Relationship between the Fermi energy and the system parameters