FOUNDATIONS OF QUANTUM MECHANICS

VIII. 4. THE MEASUREMENT PROBLEM IN THE NARROW SENSE 171
iff its state is in a mixture of eigenstates of the operator A which, according to the observables
postulate, corresponds to A.
These postulates speak about the existence of properties, about physical quantities having values,
independent of a measurement or a measuring context.
EXERCISE 37. Discuss the link between the property postulates and the sufficient condition of
reality EPR(EPR) of Einstein, Podolsky and Rosen, section I. 2, p. 12, ff.
From this point of view it would be good to have a quantum mechanical description of the measurement
process in which, in any case, the measuring apparatus has a certain property after completion
of the measurement. This means that, instead of the superposition (VIII. 4), we require, as a final
state, the mixture
W ′ =
N∑
|c j | 2 |a j ⟩ ⊗ |r j ⟩ ⟨a j | ⊗ ⟨r j |. (VIII. 13)
j=1
Some authors, e.g. Landau and Lifshitz (1958, pp. 21 - 24), go still further and require as a final
state an eigenstate |r k ⟩ of the pointer quantity R, corresponding to the pointer position found after
measurement. According to them the measuring interaction finishes with an indeterministic jump,
with probability |c j | 2 , to one of the states |a j ⟩ ⊗ |r j ⟩.
Summarizing, we have the following options for the description of the measurement process. For
the initial state there is no comtroversy,
|ψ⟩ ⊗ |r 0 ⟩ =
N S ∑
j=1
For the final state there are three possibilities,
c j |a j ⟩ ⊗ |r 0 ⟩. (VIII. 14)
1.
N S ∑
j=1
c j |a j ⟩ ⊗ |r j ⟩, (VIII. 15)
2. W ′ = ∑ j
|c j | 2 |a j ⟩ ⊗ |r j ⟩ ⟨a j | ⊗ ⟨r j |, (VIII. 16)
3. |a j ⟩ ⊗ |r j ⟩ with probability |c j | 2 . (VIII. 17)
According to the foregoing line of reasoning we require that, at the end of a measuring interaction,
the pointer of the measuring apparatus, which is of course macroscopic, designates something.
The state (VIII. 15) does not satisfy this requirement, on the contrary, the quantum mechanical superposition
|ψ⟩ of eigenstates |a j ⟩ of the quantity that is measured and which prohibited us to ascribe,