When the set X is discrete (see above), vectors |Ψ⟩ represented with the Hilbert space L2(X) are just column vectors composed of "amplitudes" and indexed by X. These are sometimes referred to as wave functions of a discrete variable x ∈ X. Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator. Components of the vector will be denoted by ψ(x) for uniformity with the previous case; there may be either finite of infinite number of components depending on the Hilbert space. In this case, if the vector |Ψ⟩ has the norm 1, then |ψ(x)|2 is just the probability that the quantum system resides in the state x. It defines a discrete probability distribution on X.
A discrete probability amplitude may be considered as a fundamental frequency in the Probability Frequency domain (spherical harmonics) for the purposes of simplifying M-theory transformation calculations.