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[5. Heisenberg indeterminacy principle.]
The effect of measurement on Ψ function.
The state of a quantum system is described by a function Ψ(q) that can be represented
as a sum of the eigenfunctions of a physical quantity u:
where the square of the module of the expansion coefficients σ_{i} determines the probability
that u has the value u_{i} .
Let u and v two non commutative physical quantities; they have two different sets of eigenfunctions.
Suppose that a measurement of u is performed, and the resulting value for u is u_{k} .
Thus the probability that u has the value u_{k} is equal to 1.
Hence σ_{k} = 1 and σ_{i} = 0 for every i≠k.
The equation (5.2) becomes
Ψ ^{u}(q) = φ_{k}^{u}(q)
(the superscript u in the equation marks the fact that the state of the system is determined immediately
after a measurement of u).
Immediately after the measurement of u, a measurement of v is performed; the result is
v_{h} and thus equation (5.2) becomes
Ψ ^{v}(q) = φ_{h}^{v}(q) .
Since u and v have different sets of eigenfunctions, we have
φ_{k}^{u}(q) ≠ φ_{h}^{v}(q)
and thus Ψ ^{u}(q) ≠ Ψ ^{v}(q) .
A measurement of v, performed after the measurement of a non commutative quantity u, changes
the function describing the state of the system.
The measurement of a quantity determines a function Ψ describing the quantum system.
If, immediately after this measurement, another one is performed on a non commutative quantity, a different function
Ψ is determined. Since function Ψ completely describes the status of the system, we must conclude that
the second measurement has altered the state of the system. The situation is different if we consider two
commutative entities, since they have the same set of eigenfunctions.
We can summarize the situation in the following way: In a quantum system, described by a function Ψ
defined as the sum of the eigenfunctions of an entity, the measurement of a commutative entity does not
alter the state of the system, while the measurement of a non commutative entity does alter the state
of the system.
A consequence of the fact that two non commutative entities have different sets of eigenfunctions is that
there is no functions Ψ which, as the same time, can be written as the sum of the eigenfunctions of two
non commutative entities. Thus there is no physical state corresponding to a quantum system in which two
commutative entities have, at the same time, definite unique values. If u and v are non commutative
entities, then there is no quantum system in which u(t) = u_{o} and
v(t) = v_{o} for the same time t. As an example, consider a quantum
particle moving along the xaxis; there is no physical state of this particle corresponding to the situation
in which the particle has a definite position and definite momentum in the same time. As an obvious consequence,
it is impossible to measure the exact position and momentum of a quantum particle, since there is no physical state
corresponding to such measurement: There is nothing such as an exact position and momentum of a quantum particle.
The impossibility of a such kind of measurement is not a human limitation, but it is a limitation of the applicability
of classical concepts, which assign a definite position and momentum to a particle. In quantum mechanics the concept
of the path of a particle is not applicable; it is only a classical notion, used as a first approximation, that
in some circumstances is inadequate.
