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why does the wave function have to be normalizable, and why does it have to go to 0 and x approaches positive/negative infinity and y approaches positive/negative infinity ?
The wave function represents the probability amplitude of a particle in quantum mechanics. For this probability to make physical sense, it must be finite and have a total probability of 1. Normalization ensures that the probability of finding the particle anywhere in space is equal to 1, making the wave function physically meaningful.
If the wave function is not normalized, it means that the total probability of finding the particle in any location is not 1. This would violate the fundamental principle of quantum mechanics, which states that the total probability of all possible outcomes must equal 1. In other words, the wave function would not accurately describe the behavior of the particle.
Normalization of the wave function is achieved by dividing the wave function by its norm, which is the square root of the integral of its absolute square. This ensures that the total probability of finding the particle anywhere in space is 1.
No, the wave function must always be normalized to a value of 1. This is a fundamental principle of quantum mechanics and is necessary for the wave function to accurately represent the probability of finding a particle in a given location.
If the wave function is not normalized, it can lead to incorrect predictions and interpretations in quantum mechanics. For example, it can result in probabilities exceeding 1 or negative probabilities, which do not have physical meaning. Moreover, normalization is necessary for the wave function to be used in mathematical calculations and to make accurate predictions about the behavior of particles.