Wave function normalization ensures total probability equals one from "summary" of The Principles of Quantum Mechanics by P. A. M. Dirac
The wave function Ψ of a particle in quantum mechanics contains all the information that can be known about the system. This wave function must satisfy the normalization condition, which ensures that the total probability of finding the particle in any state is equal to one. This condition is crucial for the interpretation of quantum mechanics, as it guarantees that the probabilities calculated from the wave function are meaningful and consistent with the principles of probability theory. The normalization condition for the wave function is given by the integral of the square of the absolute value of Ψ over all space, which must be equal to one. Mathematically, this condition can be written as ∫|Ψ|² dτ = 1, where dτ represents an infinitesimal volume element in space. This integral represents the total probability of finding the particle in any state, and the normalization condition ensures that this probability is conserved. If the wave function is not normalized, the total probability of f...
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