Wavefunctions and position representation

A nonzero Hilbert-space vector specifies a pure state only up to multiplication by a nonzero complex number; after normalization the remaining freedom is an overall phase. The physical pure state is this ray. Choosing a position representation gives the amplitude density $\psi(x)=\langle x|\psi\rangle$. Multiplying the entire function by one constant phase leaves every prediction unchanged, while position-dependent phase changes generally alter the state and can change its momentum statistics.

The Hilbert space for a spinless particle on the full line is the space of square-integrable functions, with functions equal almost everywhere identified. Its inner product is an integral rather than a finite component sum. This means changing a function's value at one isolated point does not change its Hilbert-space vector. Internal spin would add component indices, and many-particle positions would label configuration space; neither extension turns the wavefunction into an ordinary probability distribution over classical trajectories.

The Born rule for an ideal position measurement assigns a detector region the integral of the squared amplitude over that region. A density can exceed one because its units are inverse length; the integral, not the height at a point, must lie between zero and one. For a continuous position distribution, the probability of any one exact position is zero even though intervals around it may have large probability. This does not mean that a detector cannot produce a finite-resolution reading.

Normalization fixes the magnitude of a multiplicative constant but leaves its global phase free. Change variables to a dimensionless coordinate before integrating when a length scale appears repeatedly. This exposes the units of the normalization constant and prevents a numerical answer from depending on whether length was entered in metres or nanometres. A probability calculated from an unnormalized trial function must be divided by the total norm, or the function must first be normalized.

The formal position ket corresponds to a delta distribution, not a square-integrable function. Its overlap with another position ket is delta normalized. These generalized eigenvectors are useful in integral resolutions of the identity and in defining representations, but they do not have ordinary unit norm. One must not square a delta distribution and announce a normalized position probability. Physical localized preparations use finite-width normalizable packets with finite-resolution measurement predictions.

Square integrability alone also does not place a function in every operator's domain. A state can be normalized while a derivative observable or an energy moment is undefined or divergent. Boundary regularity becomes essential when applying an unbounded momentum or Hamiltonian operator. This lesson uses normalization and position integrals only; additional dynamical assumptions must be checked before treating an arbitrary normalized shape as a finite-energy eigenstate of a particular potential.