Wave functions and
Wave functions and probabilities
If the configuration space X is continuous (something like the real line or Euclidean space, see above), then there are no valid quantum states corresponding to particular x ∈ X, and the probability that the system is "in the state x" will always be zero. An archetypical example of this is the L2(R) space constructed with 1-dimensional Lebesgue measure; it is used to study a motion in one dimension. This presentation of the infinite-dimensional Hilbert space corresponds to the spectral decomposition of the coordinate operator: ⟨x | Q | Ψ⟩ = x⋅⟨x | Ψ⟩, x ∈ R in this example. Although there are no such vectors as ⟨x |, strictly speaking, the expression ⟨x | Ψ⟩ can be made meaningful, for instance, with spectral theory.
Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function ψ is the wave function.
If the function ψ ∈ L2(X), ‖ψ‖ = 1 represents the quantum state vector |Ψ⟩, then the real expression |ψ(x)|2, that depends on x, forms a probability density function of the given state. The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in X to obtain probability values – as was stated above, the system can't be in some state x with a positive probability. It gives to both amplitude and density function a physical dimension, unlike a dimensionless probability. For example, for a 3-dimensional wave function the amplitude has a "bizarre" dimension [L−3/2].
Note that for both continuous and infinite discrete cases not every measurable, or even smooth function (i.e. a possible wave function) defines an element of L2(X); see #Normalisation below.
