section_path: Mathematical formulation
section_name: Mathematical formulation
paragraph_text: Mathematical formulation
  
  In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a (separable) complex Hilbert space H {\displaystyle {\mathcal {H}}}. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ, ψ ⟩ = 1 {\displaystyle \langle \psi,\psi \rangle =1}, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 (C) {\displaystyle L^{2}(\mathbb {C})}, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product.
  
  Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ →, ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}, where λ → {\displaystyle {\vec {\lambda }}} is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ, P λ ψ ⟩ {\displaystyle \langle \psi,P_{\lambda }\psi \rangle }, where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the probability density.
  
  After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}}, in the non-degenerate case, or to P λ ψ / ⟨ ψ, P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi,P_{\lambda }\psi \rangle }}}, in the general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases to exist as an independent entity (see measurement in quantum mechanics).