In my syllabus about solid state physics they state that lattice vibration is quantized, analogous to the harmonic oscillator:
$$E = (n+\frac{1}{2})\hbar\omega$$
So the lattice vibration has zero-point energy $\frac{1}{2}\hbar\omega$.
I wonder what this actually means: are all possible vibration modes of the lattice quantized in this way? So do all the vibration modes (acoustical/optical and transversal/longitudinal) have nonzero energy for all possible wavevectors $\vec{k}$ in the Brillouin zone?
So If I interpret this quantization in this way, it means that the lattice is at all time vibrating in all possible vibration modes.
In the syllabus they state though (somewhat earlier (and translated to English)) :
"The vibration modes are purely longitudinal or transversal only in the case of sufficient symmetry, e.g. in some directions of a cubic crystal structure. Otherwise, the waves consist of a mixture of the two." (I thinks that this refers to f.i. the [100] direction in a cubic crystal)
This seems to be contradicting the idea that longitudinal and transversal modes should have zero point energy. I hope that someone can clarify this.
Best Answer
In classical mechanics, you can describe a crystal (in some approximation) by a Hamiltonian that is a quadratic form in coordinates and momenta of atoms. After you diagonalize this quadratic form, you obtain a Hamiltonian of a set of independent, rather than coupled, oscillators (modes). Then you can quantize this system and you do get zero-point energy for each independent mode (by the way, this energy is $\frac{1}{2}$ℏω, not ℏω). However, not all independent modes are purely longitudinal or transversal. In other words, longitudinal modes are often coupled with transversal modes and, therefore, they are not independent modes that you get as a result of diagonalization. In other words, longitudinal and transverse modes are some linear superpositions of independent modes (which are also called eigen modes, or characteristic modes, or normal modes:-) ) .