FOUNDATIONS OF QUANTUM MECHANICS

VIII. 5. INCOMPATIBLE QUANTITIES 179
The step from the pure state (VIII. 22) to the mixture (VIII. 24) is therefore justified by limiting
ourselves to practically realizable states.
At first sight, this reasoning is in every way reasonable. Of course, the reasoning only refers to
a particular class of quantities; a physical quantity for a composite system is certainly not always a
direct product or a summation thereof. But it can be maintained that quantities which are not direct
products are even harder to measure in practice. It is, however, beyond doubt that experimentally
distinguishing the pure state (VIII. 22) from the mixed state (VIII. 24) using macroscopic quantities
will be extremely difficult.
Bell considers this FAPP solution as a pitfall, he speaks of the FAPP - trap. He emphasizes that the
measurement problem is not a practical but a fundamental problem. The core of the problem is if,
after the measurement process, certain properties are present in the measuring apparatus. The FAPP
reasoning shows that, generally, in practice the system behaves as if it had those properties, but it
leaves untouched the fact that ‘in reality’ the system does not have those properties, and that, if our
experimental possibilities would be more ample, this is also experimentally provable.
EXERCISE 39. Show that, using the physical quantity corresponding to the operator |Ψ⟩ ⟨Ψ|, in
which |Ψ⟩ is the right - hand side of (VIII. 22), experimental distinction can be made between the
pure state (VIII. 22) and the mixed state (VIII. 24).
VIII. 5
INCOMPATIBLE QUANTITIES
So far we considered measuring a single physical quantity or two compatible, or commeasurable,
physical quantities of the object system, where compatible quantities are quantities corresponding
to commutating operators. The simple measurement theory (VIII. 2) however, enables us to discuss
also the measurement of incompatible quantities.
Let A and B be two arbitrary, incompatible quantities of the object system S corresponding
to the maximal operators A and B. Measuring apparatus M 1 measures A and apparatus M 2 measures
B. The pointer observables of the apparatuses are R and T , corresponding to the operators
R and T , the eigenstates are |a j ⟩, |b j ⟩, |r j ⟩, |t j ⟩, respectively. The initial state is |ψ⟩ ⊗ |r 0 ⟩ ⊗ |t 0 ⟩
in H = H S ⊗ H 1 ⊗ H 2 , and with dim H S = N S ,
|ψ⟩ =
N S
∑
⟨a j | ψ⟩ |a j ⟩ =
j=1
N S
∑
⟨b k | ψ⟩ |b k ⟩. (VIII. 25)
k=1