Editing Probability amplitude (section)

==Normalization==

In the example above, the measurement must give either {{math|{{ket| ''H'' }}}} or {{math|{{ket| ''V'' }}}}, so the total probability of measuring {{math|{{ket| ''H'' }}}} or {{math|{{ket| ''V'' }}}} must be 1. This leads to a constraint that {{math|1=''α''<sup>2</sup> + ''β''<sup>2</sup> = 1}}; more generally '''the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one'''. If to understand "all the possible states" as an [[orthonormal basis]], that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained [[#Mathematical|above]].

One can always divide any non-zero element of a Hilbert space by its norm and obtain a ''normalized'' state vector. Not every wave function belongs to the Hilbert space {{math|''L''<sup>2</sup>(''X'')}}, though. Wave functions that fulfill this constraint are called [[normalizable wave function|normalizable]].

The [[Schrödinger equation]], describing states of quantum particles, has solutions that describe a system and determine precisely how the state [[time evolution operator|changes with time]]. Suppose a [[wave function]] {{math|''&psi;''('''x''', ''t'')}} gives a description of the particle (position {{math|'''x'''}} at a given time {{math|''t''}}). A wave function is [[square integrable]] if 
:<math>\int |\psi(\mathbf x, t)|^2\, \mathrm{d\mathbf x} = a^2 < \infty.</math>
After [[Wave function#Normalization condition|normalization]] the wave function still represents the same state and is therefore equal by definition to{{sfn|Bäuerle|de Kerf|1990|p=330}}<ref>See also [[Wigner's theorem]]</ref>
:<math>\psi(\mathbf{x},t):=\frac{\psi(\mathbf{x},t)}{a}.</math>
Under the standard [[Copenhagen interpretation]], the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, {{math|''ρ''('''x''') {{=}} {{abs|''&psi;''('''x''', ''t'')}}<sup>2</sup>}} is a [[probability density function]] and the probability that the particle is in the volume {{math|''V''}} at fixed time {{math|''t''}} is given by
:<math> P_{\mathbf{x}\in V}(t) = \int_V |\psi(\mathbf {x}, t)|^2\, \mathrm{d\mathbf {x}}=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}}.</math>
The probability density function does not vary with time as the evolution of the wave function is dictated by the Schrödinger equation and is therefore entirely deterministic.{{sfn | Zwiebach | 2022 | p=170}} This is key to understanding the importance of this interpretation: for a given particle constant [[mass]], initial {{math|''&psi;''('''x''', ''t''<sub>0</sub>)}} and [[potential energy|potential]], the Schrödinger equation fully determines subsequent wavefunctions. The above then gives probabilities of locations of the particle at all subsequent times.