Question
Why is it necessary to deal with probability densities rather than probabilities such as $P(X=a)$ when the variable under consideration is continuous?
Step 1
Continuous random variables can take on an infinite number of values within a given range. This means that the probability of the variable taking on any specific single value, such as \( P(X = a) \), is actually zero. Show more…
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Key Concepts
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Schroedinger'S Theory Of Quantum Mechanics
Questions
The probability density for finding a particle at position $x$ is $$P(x)=\left\{\begin{array}{lr} \frac{a}{(1-x)} & -1 \mathrm{mm} \leq x<0 \mathrm{mm} \\b(1-x) & 0 \mathrm{mm} \leq x \leq 1 \mathrm{mm}\end{array}\right.$$ and zero elsewhere. a. You will learn in Chapter 41 that the wave function must be a continuous function. Assuming that to be the case, what can you conclude about the relationship between $a$ and $b ?$ b. Draw a graph of the probability density over the interval $$-2 \mathrm{mm} \leq x \leq 2 \mathrm{mm}$$ c. Determine values for $a$ and $b$. d. What is the probability that the particle will be found to the left of the origin?
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