### Exercise 3: Total heat capacity of a diatomic material
One of the assumptions of the Einstein model states that every atom in a solid oscillates with the same frequency $\omega_0$. However, if the solid contains different types of atoms, it is unreasonable to assume that the atoms oscillate with the same frequency. One example of such a solid is a lithium crystal, which consists of the [two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium) $^6$Li (7.5%) and $^7$Li (92.5%) in their natural abundance. Let us extend the Einstein model to take into account the different masses of these different isotopes.
One of the assumptions of the Einstein model states that every atom in a solid oscillates with the same frequency $\omega_0$. However, if the solid contains different types of atoms, it is unreasonable to assume that the atoms oscillate with the same frequency. One example of such a solid is a lithium crystal, which consists of the [two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium) $^6$Li (7.5%) and $^7$Li (92.5%) in their natural abundance. Let us extend the Einstein model to take into account the different masses of these different isotopes. Assume that the solid is 1D (1D quantum harmonic oscillator).
1. Assume that the strength of the returning force $k$ experienced by each atom is the same. What is the difference in the oscillation frequencies of the two different isotopes in the lithium crystal?
2. Write down the total energy stored in the vibrations of each atom of the lithium crystal, assuming that all $^6$Li atoms are in $n=2$ vibrational mode and all $^7$Li atoms are in $n=4$ vibrational mode.
