QUANTUM METAPHYSICS - E-thesis

Geometrically, the components z1, z 2, z3 .. measure the lengths of the orthogonal projections of
the state vector at various axes I0>, I1>, I2>. These axes or base vectors correspond to
measurable properties, i.e. observables, and the lengths of the projections reveal the possible
values or test results. 476 The base vectors can be chosen in numerous different ways, which
means that a state vector can be shown in various systems of coordinates. In general a state can
be an eigenstate for many different observables, but if certain operators do not commute, then a
type of uncertainty relation connects the precision with which the two observables can be
determined. 477 Accordingly, it is not possible, for example, to measure simultaneously position
and momentum with arbitrary accuracy, because there are no available observables that could
simultaneously correspond to both properties. Moreover, when the operators that correspond to
position and momentum do not commute with each other, the sequence of measurement
influences the result. The state vector also rotates in the Hilbert space, and thus the values of the
projections are constantly changing, depending on the forces that influence the particles in a
given situation. This rotation equals the motion of the system.
If the system is left to develop without external interaction, time development occurs according
to the deterministic dynamics of Schrödinger. It is however impossible to observe this kind of
idealised system, since the observation in itself already means an interaction. This observation
interaction can be described by using the dynamics of von Neumann, which states that a system
"collapses" to one of the possible observed states with a probability that can be calculated from
quantum theory. This can be illustrated in the vector space with a projection of the system state
to the eigenstate of a given observable. Observables are represented by operators in Hilbert
space. 478 Even though many operators are employed in quantum theories only a certain class, the
self-adjoint operators, represent observables. Self-adjoint operators have spectra that consist only
of real numbers. For an observable A, the spectrum Λ(A) of its representing operator comprises
the set of all possible values obtainable in measurements of A. A spectrum can be discrete,
continuous, or a combination of both. If the observable A has a pure point spectrum so that
Λ(A)=(ai) the real numbers ai are called the eigenvalues of A. The eigenvalues can be the direct
476
Auyang 1995, 16-17, 86. The dynamical variables in quantum mechanics are called observables. In addition to
describing state - as in classical mechanics - they also provide the possible outcomes of measurements. Quantum
representations are conventions. The choice of a particular representation is arbitary and depends on what is most
convenient for the solution of a particular problem.
477
Gasiorowicz 1974, 119. For example, the transition from coordinate space to momentum space corresponds to
the complex rotation of Hilbert's space.
478
Operators are transformations or mapping-like functions, but unlike functions, the range of operators is the state
space itself. From a theoretical point of view, measuring simply means that one calculates the expected value of a
given operator.
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