I'm working in gravitational physics, unfortunately my solid state physics lecture is already a while a go. My question will probably sound quite trivial to solid state physicists, but I couldn't find a fully satisfactory answer, neither in textbooks nor on Google (which is probably just because I don't really no what to search for).
If I have a crystal (say Si, for example) at low temperature (1K or lower). What is the spatial probability distribution for the single atoms in the crystal?
My first guess was something like Einstein's model with independent harmonic oscillators, but if the temperature is low, this doesn't seem to make sense. Even when all oscillators are in the ground state, their energy will be much larger than kT.
So, if one can assume that the wave-function separates in the wave-functions of the single atoms, I guess, one would have to know the actual N-particle wave-function and calculate the marginal distribution for one atom?
I'm sure this is just a textbook problem and I can find the answer somewhere? I would be very thankful if someone could help me, even if it's just giving me the title of a book where I can find this.
To be a bit more precise, what I need is this:
Say we have a crystal consisting of N atoms at coordinates $\vec x_1$, ... $\vec x_N$. The crystal is in thermodynamic equilibrium at some small temperature T. The crystal will then be in some state with the spatial wave-function $\psi(\vec x_1,\vec x_2,...,\vec x_N)$. Once this wave-function is given, I could simply calculate the density
$\rho(x_1) = \int d^3 x_2 \,d^3 x_2 \, ... d^3 x_N \, \vert\psi(\vec x_1,\vec x_2,...,\vec x_N)\vert^2$
However, getting the wave-function is probably not so easy? So the question then is, if there is a good approximation/effective model to obtain $\rho(x_1)$, $\rho(x_2)$, ... that works well for small temperatures?