On the nature of the wave function

The modern theory of quantum mechanics is formulated using rigorous mathematical formalisms. In this modern formulation, the wave function, commonly denoted as $Ψ$ , is defined as a function of the degrees of freedom of a quantum system such as the positions or momenta of particles and their spin, which describes the state of the system. A complex-valued function, the wave function assigns a complex number to each element of its domain, i.e. every point in space or every possible spin value of each particle.

Given $n$ discrete degrees of freedom $α_{1}, α_{2}, \dots, α_{n}$ and $m$ continuous variables $ω_{1}, ω_{2}, \dots, ω_{m}$ , the wave function can be written as $Ψ (α, ω, t)$ . The wave function is an abstract mathematical construct and cannot be "measured" directly. The squared modulus of the wave function, $∣ Ψ ∣^{2} = Ψ^{*} Ψ$ , is interpreted as the probability density. In other words, $∣ Ψ ∣^{2}$ defines a probability distribution and therefore, for every $t$ , it satisfies

$\sum_{α \in A} \int_{Ω} ∣ Ψ (α, ω, t) ∣^{2} d ω = 1 .$

The mathematical formalism of quantum mechanics defines an inner product on the space of all wave functions. For any two wave functions $Φ$ and $Ψ$ , the inner product $(Φ, Ψ)$ is defined as

$(Φ, Ψ) = ⟨ Φ ∣ Ψ ⟩ = \sum_{α \in A} \int_{Ω} Φ^{*} (α, ω, t) Ψ (α, ω, t) d ω .$

Upon measurement of an observable, the wave function "collapses" to a new wave function. The modulus squared of the inner product of two wave functions $Φ$ and $Ψ$ is interpreted as the probability of the wave function $Ψ$ collapsing to the wave function $Φ$ :

$∣ (Φ, Ψ) ∣^{2} = ∣ ⟨ Φ ∣ Ψ ⟩ ∣^{2} = \sum_{α \in A} \int_{Ω} ∣ Φ^{*} (α, ω, t) Ψ (α, ω, t) ∣^{2} d ω = P (Ψ \to Φ) .$

On the nature of the wave function

See also