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By Berne B.J., Harp G.D.

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It is a convenient fiction to assert that every Hermitian operator symbolizes a physical quantity, and that every unit vector symbolizes a state. Then the expectation value of property in the state ¥ X is given by In particular, the probability of observing the values a1 in an A-measurement performed on sys- tems in the state 2,5 ¥ is UNITARY TRANSFORMATIONS The automophisms of the unitary geometry of states are produced by the unitary transformations THE GEOMETRY OF STATES 41 applied to every vector and operator, where the unitary operator U obeys All algebraic relations and adjoint connections among vectors and operators are preserved by this transformation.

2 RECONSTRUCTION OF THE MEASUREMENT ALGEBRA THE GEOMETRY OF STATES 33 The characteristics of the measurement operators M(a" of the *F f b') and * can now be derived from those symbols. 12). 6) show that the only significant products those not identically zero — ¥X — are of the form , in addition to XY , where the latin symbols are operators, elements of the physical measurement algebra. 4), all operators are linear combinations of products ff and the evaluation of the products XY Xf , $X , and reduces to the ones contained in (2,7), Hence, in any manipulation of operators leading to a product *¥ , the latter is effectively equal to a number, and in particular It should also be observed that, in any application of 1 as an operator we have, in effect, Accordingly, THE GEOMETRY OF STATES 35 which shows that The bracket symbols are designed to make this result an automatic consequence of the notation (Dirac).

70), we learn that or which expresses the fixed numerical values of the. transformation function Indeed, the latter is not an independent condition on transformation functions but is implied by the composition property and the requirement that transformation functions, as matrices, be nonsingular. 11 <§W are Hermitian. 33) to write the probability formula as the expectation value becomes 26 QUANTUM KINEMATICS AND DYNAMICS where the operator A is The correspondence thus obtained between operators and physical quantities is such that a function f(A) of the property A is assigned the operator £{A) , and the operators associated with a complete set of compatible physical quantities form a complete set of commuting Hermitian operators.

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