# In QM, do we deal with basis or orthonormal sets?

Most textbooks say, that given a (countable) basis ${|\phi_n\rangle}$ of a Hilbert space, that every vector $|\psi\rangle$ of the space can be written as:

$$\psi\rangle=\sum_{n=1}^\infty a_n|\phi_n\rangle$$

But this an infinite linear combination, and every vector should be expressed by a finite linear combination.

So I think that they are an orthogonal set, which generates the vector space as an infinite linear combination (the difference between infinite and finite LC means that orthonormal sets are usually smaller than basis).

But for the rest of the properties, ${|\phi_n\rangle}$ looks like a basis to me.

So, is ${|\phi_n\rangle}$ a basis or just an orthogonal set?

• What is your question? Dec 5 '13 at 9:21
• @Niel de Beaudrap Updated. Dec 5 '13 at 9:26
• in qm what is being called "basis" is indeed a complete orthogonal set. I think it is called basis because it resembles of the concept of basis in linear algebra, not the concept of basis of a metric (or more generally, topological) space. Dec 5 '13 at 9:33

• That's right: the Hermite polynomials form an orthogonal Schauder basis if the inner product has an appropriate (Gaussian) weight function, similarly for Legendre with different caveats, but yes. Another example is the Fourier basis $\{1,\cos nx,\sin nx:\;n>0\}$ for (equivalence classes of) functions of period $2\pi$. Dec 5 '13 at 9:46
An orthonormal system $\{x_i\}_{i\in I}$ that meets the [proven equivalent] conditions of this theorem is called a Hilbert basis or just basis of $X$.