Exploring the ψ(r,t) Wave Function: Probability & Position

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The wave function ψ(r,t) describes the quantum state of a particle, but the probability of locating the particle at an exact position r at an exact time t is zero. Instead, the square of the wave function represents the probability density, which allows for the calculation of probabilities over ranges of position and time. To determine the likelihood of finding the particle between two positions (r1 and r2) and two times (t1 and t2), integration of the probability density is necessary. This highlights the fundamental nature of quantum mechanics, where exact outcomes are replaced by probabilities. Understanding this relationship is crucial for interpreting quantum behavior.
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What is the relationship between the wave function ψ(r,t) of a particle and the probability of finding the particle at position r at time t?
 
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The probability of finding the particle at EXACTLY r and at EXACTLY time t is zero, just like how the probability of choosing the number 2 when given an infinite number of numbers to choose from is 0. However, the square of the wavefunction is the probability density. Given the probability density function, you find the probability of finding the particle between t1 and t2 and between r1 and r2 by integration.
 

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