Wave function
Normalization condition

The probability that its position x will be in the interval a ≤ x ≤ b is the integral of the density over this interval: P a ≤ x ≤ b (t) = ∫ a b | Ψ (x, t) | 2 d x {\displaystyle P_{a\leq x\leq b}(t)=\int _{a}^{b}\,|\Psi (x,t)|^{2}dx} where t is the time at which the particle was measured. This leads to the normalization condition: ∫ − ∞ ∞ | Ψ (x, t) | 2 d x = 1, {\displaystyle \int _{-\infty }^{\infty }\,|\Psi (x,t)|^{2}dx=1\,,} because if the particle is measured, there is 100% probability that it will be somewhere.

For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers. Technically, wave functions form a ray in a projective Hilbert space rather than an ordinary vector space.
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