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Consider an ensemble of microscopic systems prepared in the
same initial state . Suppose a measurement of the observable is
made on each system. We know that each measurement yields the value with
probability . What is the mean value of the measurement? This quantity,
which is generally referred to as the expectation value of , is
given by
which reduces to

(59) 
with the aid of Eq. (54).
Consider the identity operator, 1. All states are eigenstates of this operator
with the eigenvalue unity. Thus, the expectation value of this operator
is always unity: i.e.,

(60) 
for all . Note that it is only possible to normalize a given
ket such that Eq. (60) is satisfied because of the more general
property (21) of the norm. This property depends on the particular correspondence
(16), that we adopted earlier, between the elements of a ket space and those of its
dual bra space.
Next: Degeneracy
Up: Fundamental concepts
Previous: Measurements
Richard Fitzpatrick
20060216