structural:expansion
More on wave functions and abstract state space

As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general infinite-dimensional Hilbert space. Due to the multiple possible choices of representation basis, these Hilbert spaces are not unique. One therefore talks about an abstract Hilbert space, state space, where the choice of representation and basis is left undetermined. Specifically, each state is represented as an abstract vector in state space. A quantum state | Ψ ⟩ in any representation is generally expressed as a vector | Ψ ⟩ = ∑ α ∫ d m ω Ψ (α, ω, t) | α, ω ⟩ {\displaystyle |\Psi \rangle =\sum _{\boldsymbol {\alpha }}\int d^{m}\!{\boldsymbol {\omega }}\,\,\Psi ({\boldsymbol {\alpha }},{\boldsymbol {\omega }},t)\,|{\boldsymbol {\alpha }},{\boldsymbol {\omega }}\rangle } where

These quantum numbers index the components of the state vector. More, all α are in an n -dimensional set A = A × A ×... × A where each A is the set of allowed values for α; all ω are in an m -dimensional "volume" Ω ⊆ ℝ where Ω = Ω × Ω ×... × Ω and each Ω ⊆ R is the set of allowed values for ω, a subset of the real numbers R. For generality n and m are not necessarily equal.

Example:

The probability density of finding the system at time t {\displaystyle t} at state | α, ω ⟩ is ρ α, ω (t) = | Ψ (α, ω, t) | 2 {\displaystyle \rho _{\alpha,\omega }(t)=|\Psi ({\boldsymbol {\alpha }},{\boldsymbol {\omega }},t)|^{2}}

The probability of finding system with α in some or all possible discrete-variable configurations, D ⊆ A, and ω in some or all possible continuous-variable configurations, C ⊆ Ω, is the sum and integral over the density, P (t) = ∑ α ∈ D ∫ C ρ α, ω (t) d m ω {\displaystyle P(t)=\sum _{{\boldsymbol {\alpha }}\in D}\int _{C}\rho _{\alpha,\omega }(t)\,\,d^{m}\!{\boldsymbol {\omega }}}

Since the sum of all probabilities must be 1, the normalization condition 1 = ∑ α ∈ A ∫ Ω ρ α, ω (t) d m ω {\displaystyle 1=\sum _{{\boldsymbol {\alpha }}\in A}\int _{\Omega }\rho _{\alpha,\omega }(t)\,d^{m}\!{\boldsymbol {\omega }}} must hold at all times during the evolution of the system.

The normalization condition requires ρ d ω to be dimensionless, by dimensional analysis Ψ must have the same units as (ω ω... ω).