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Why the squared amplitude rule for probabilities?
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Quantum mechanics texts usually start off their formal development
with the Born rule or the equivalent squared amplitude rule, which
says that the squared absolute values of the wave function give the
probability density for observing particular results.
This rule usually is given without explanation, referring to the
success of quantum mechnaics to explain experiments rather than to a
rational insight why this rule should be used.
Here is the missing explanation:
A general state is described instead by a density matrix R - a positive
semidefinite Hermitian linear operator of trace 1 in the most general
case. Its diagonal elements are nonnegative and sum or integrate to 1 -
they define the probability density. If R is diagonal and remains so
in the course of time, the system behaves like a classical stochastic
system. Any off-diagonal elements account for quantum behavior.
Pure states are the special case when the density matrix has rank 1.
In this case, it can be written as R=psi psi^* with a vector psi, the
state vector or wave function of the system. (psi^* is the conjugate
transposed vector.) The wave function is not completely determined by
the state since multiplying psi by any number of absolute value one
(a ''phase'') gives another such vector. (This is the reason why psi
cannot be observable.)
By looking at the diagonal elements of R=psi psi^*, you get the
squared amplitude formula.