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I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by [itex] i\in \mathbb{N} [/itex]. This set [itex] \left \{u_{i}(\overrightarrow{r}) \right \} [/itex] is orthonormal and canonical (can it be?). So my question is regarding the position vector of the basis: What is the explicit expression that accounts for the position in this case? The wavefunction has a value that depends on the position that can be expressed by the basis multiplied by a complex value, so the information of the variation of this constant by position must be in the basis function. What am I failing to see here?

Your help will be much appreciated!