Dec 11, 2020

The wave function is an abstract mathematical concept and cannot be "measured" directly. So what is it then?

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 $Φ$:

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

See also

The Hadamard gate

The definition of the Hadamard gate and some of its properties.

Dirac notation for quantum states

How to read the bra–ket notation?

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