User:Thierry Dugnolle/Wave-particle duality and the superposition principle

Wave-particle duality, the quantum superposition principle and the Born rule
 One of the important consequences of the first postulate (the quantum superposition principle)'' when it is combined with the others, is the appearance of interference effects such as those which led us to wave-particle duality. 

The quantum superposition principle is:

 Physical states are represented by rays in Hilbert space. A Hilbert space is a kind of complex vector space; that is, if Φ and Ψ are vectors in the space (often called 'state vectors') then so is $$\xi$$Φ + $$\eta$$Ψ, for arbitrary complex numbers $$\xi,\eta$$. .

The vector Φ is also written $$ | \phi \rangle $$ in Dirac notation. Two vectors $$| u \rangle$$ and $$| v \rangle$$ (different from the null vector) belong to the same ray if and only if there is a complex number $$\alpha$$ such that $$| u \rangle = \alpha | v \rangle$$.

The existence of a wave function for a particle is a direct consequence of the quantum superposition principle: since any particle can be in any localized state $$|x\rangle$$ (a localized state centered on $$x$$ with width $$dx$$) it can also be in a superposition of these states $$\sum_x \psi(x) |x\rangle$$, or $$\int \psi(x) |x\rangle dx =  | \psi \rangle$$ (if $$dx$$ tends to zero) where $$\psi(x)$$ is the wave function (at a given time) of the particle:

 The wave functions $$\psi(x)$$ that we have been using to describe physical states in wave mechanics should be considered as the set of components of an abstract vector $$\psi$$ known as the state vector. 

$$\psi (x) = \langle x | \psi \rangle$$ in Dirac notation. $$x$$ is a real number for a one-dimensional wave function, and a three-dimensional vector, for a three-dimensional wave function. In the latter case $$dx$$ is a three-dimensional box. A wave function at a given time is a state vector. A wave function at all times is the evolution of a state vector.

The Born rule states that the squared modulus $$|\psi (x) | ^2$$of the normalized wave function $$\psi (x)$$ is the density probability of detection of the particle. (Normalized means that $$\int | \psi(x) | ^2 dx = 1$$). This leads to wave-particle duality : the motion of a particle (the evolution of its state) is represented by a wave function (at all times) but the particle is always detected at a single place.

A wave function can be a wave packet. Wave packets are waves which are localized. This means that their spreading in space, at a given time, is negligible (not measurable) at a distance. A wave packet can represent the evolution of the state of a quantum particle :