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

Published on by Anton Vasetenkov

Topics: Quantum mechanics

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

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

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 $Φ$:

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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