# Chemistry - Orthogonal Wavefunctions

The notion of *orthogonality* in the context of the question referrers to the very well-known general concept of *linear algebra*, the branch of mathematics that studies *vector spaces*. Instead of going deep into the mathematics (that requires at least 50 textbook pages) let's just clear some OP's doubts.

First, a small (but important) correction: two wave functions $\psi_1$ and $\psi_2$ are called *orthogonal* to each other if
$$
\int \overline{\psi}_1 \psi_2 \, \mathrm{d} \tau = 0 \, ,
$$
where the first function is complex-conjugated as indicated by a bar on top of it. Wave functions are complex-valued functions and complex-conjugation of the first argument is important.^{1}

So, yes, orthogonality is a not a property of a single wave function. It either refers to a pair of them being orthogonal to each other as described above, or, in general, to a set of them, being all mutually orthogonal to each other, i.e. to a set $\{ \psi_i \}_{i=1}^{n}$ such that for any $i \neq j$ $$ \int \overline{\psi}_i \psi_j \, \mathrm{d} \tau = 0 \, . $$ In the last case it is said that the whole set $\{ \psi_i \}_{i=1}^{n}$ is orthogonal.

1) More precisely, one of the two functions has to be complex-conjugated in this expression, where which one is the matter of *convention*: in physical literature it is often the first function, while in mathematically oriented literature it is usually the second one.