QUANTUM PHYSICS FUNDAMENTALS - Chapter 5, Exercise 1 Solution
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Comparing Probabilities with Born's Interpretation

PROBLEM
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At position C, a particle's wave function has |Psi|^2 = 0.09. At
position D, |Psi|^2 = 0.03. How many times more likely is the
particle to be found at C than at D?

SOLUTION
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Under Born's real 1926 interpretation, |Psi|^2 gives the relative
probability density of finding the particle at a given position. The
ratio of two |Psi|^2 values directly gives the ratio of probabilities.

    Ratio = 0.09 / 0.03 = 3

ANSWER: The particle is three times more likely to be found at
position C than at position D.

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WHY THIS WORKS AS AN ANSWER

This is a direct, real application of Born's own probability-density
interpretation, following the same method worked through in the
chapter's own example (0.04 vs. 0.01, a ratio of 4). Born's real
insight was that the wave function itself carries no direct physical
meaning as a literal density of matter - only its squared magnitude
does, and only as a probability. Comparing two |Psi|^2 values this way
is exactly how Born's interpretation is used in practice throughout
quantum mechanics, from simple textbook problems up through real
calculations of electron distributions in atoms.