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 finding the particle will be either greater or less than one, which violates the fundamental principle of probability conservation. Normalizing the wave function ensures that the probabilities calculated from it are consistent with the laws of probability theory and can be interpreted in a meaningful way. This normalization condition is fundamental to the interpretation of quantum mechanics and is a necessary requirement for any valid wave function. In summary, the normalization of the wave function ensures that the total probability of finding the particle in any state is equal to one. This condition is essential for the interpretation of quantum mechanics and guarantees that the probabilities calculated from the wave function are consistent with the principles of probability theory. By satisfying the normalization condition, the wave function provides a complete description of the system and allows for meaningful predictions to be made about the behavior of the particle in quantum mechanics.
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