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 $α_{1},α_{2},…,α_{n}$ and $m$ continuous variables $ω_{1},ω_{2},…,ω_{m}$, 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

$∑_{α∈A}∫_{Ω}∣Ψ(α,ω,t)∣_{2}dω=1.$

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

$(Φ,Ψ)=⟨Φ∣Ψ⟩=∑_{α∈A}∫_{Ω}Φ_{∗}(α,ω,t)Ψ(α,ω,t)dω.$

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 $Φ$:

$∣(Φ,Ψ)∣_{2}=∣⟨Φ∣Ψ⟩∣_{2}=∑_{α∈A}∫_{Ω}∣Φ_{∗}(α,ω,t)Ψ(α,ω,t)∣_{2}dω=P(Ψ→Φ).$

Made by Anton Vasetenkov.

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