quantum mechanics - Do holes have wavefunctions?

Do holes (as in the absence of an electron) have wavefunctions?

In my understanding, when we talk about holes, we are implicitly invoking two multiparticle wavefunctions: $$\tag{1} \Psi(x_1,...,x_N)= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_N(x_1) \\ \vdots & & \vdots \\ \psi_1(x_N) & ... & \psi_N(x_N) \end{matrix} \right|$$ and $$\tag{1} \Phi(x_1,...,x_{N-1})= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_{N-1}(x_1) \\ \vdots & & \vdots \\ \psi_1(x_{N-1}) & ... & \psi_{N-1}(x_{N-1}) \end{matrix} \right|$$ Then we can say that $\Phi$ is $\Psi$ with an additional hole at a single-particle orbital $N$. (I am ignoring the fact that not all multiparticle states can be written as a Slater determinant.)

I think I've heard people talk about "hole wavefunctions." How do we define a hole wavefunction? In fact, can we even define a single particle wavefunction in a multiparticle system? If it exists, does the hole wavefunction need to be antisymmetrized with electron states (i.e., for an exciton, writing a Slater determinant for the hole and the electron)?

As a side note, I could also ask similar questions about positrons.

Answer

You have everything pretty much correct. If you have a piece of semiconductor with $10^{18}$ electrons, a full valence band would be $$\tag{3} \Psi(x_1,...,x_{10^{18}})= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_{10^{18}}(x_1) \\ \vdots & & \vdots \\ \psi_1(x_{10^{18}}) & ... & \psi_{10^{18}}(x_{10^{18}}) \end{matrix} \right|$$ and then a valence band with five holes in it would be $$\tag{4} \Phi(x_1,...,x_{10^{18}-5})= \left| \begin{matrix} \psi_1(x_1) & ... & \psi_{10^{18}-5}(x_1) \\ \vdots & & \vdots \\ \psi_1(x_{10^{18}-5}) & ... & \psi_{10^{18}-5}(x_{10^{18}-5}) \end{matrix} \right|$$

Those five holes would have the wavefunctions $\psi_{10^{18}-4}(x)$, $\psi_{10^{18}-3}(x)$, $\psi_{10^{18}-2}(x)$, $\psi_{10^{18}-1}(x)$, $\psi_{10^{18}}(x)$.

So a single-particle wavefunction in a multiparticle system is just one of the entries of the Slater determinant.

Most metals and semiconductors can be described in the "single-particle approximation", i.e. there is at least one way to write the Slater determinant so that to a very good approximation you can treat each entry $\psi_i$ as a separate particle, behaving like you would expect a typical particle to behave, and only weakly interacting with the other particles (described by the other $\psi_j$). That is the situation in which people are usually talking about single-electron or single-hole wavefunctions.