In quantum mechanics, a normalizable eigenfunction refers to a specific type of wavefunction that satisfies certain mathematical conditions. An eigenfunction represents a possible state of a quantum system, and it is associated with a particular observable quantity, such as energy or momentum. The corresponding eigenvalue is the value that the observable quantity will take when the system is in that particular state.
For an eigenfunction to be normalizable, it means that the integral of the absolute square of the wavefunction over all space must converge to a finite value. Mathematically, this condition can be expressed as:
∫ |ψ(x)|^2 dx < ∞
Here, ψ(x) represents the wavefunction, and the integral is taken over all possible values of the position variable x. The absolute square of the wavefunction, |ψ(x)|^2, represents the probability density of finding the particle described by the wavefunction at a given position x.
The normalizability condition ensures that the total probability of finding the particle somewhere in space is unity (or 100%). In other words, it guarantees that the particle is somewhere in the system and not spread out infinitely.
If an eigenfunction is not normalizable, it implies that the corresponding state is not physically realizable since its probability density does not integrate to a finite value. Normalizability is an essential requirement for eigenfunctions to be meaningful in quantum mechanics and to provide accurate predictions for observable quantities.