FOUNDATIONS OF QUANTUM MECHANICS

VIII. 4. THE MEASUREMENT PROBLEM IN THE NARROW SENSE 173
quantum mechanical chain. If he knows that he is in the state |b k ⟩ and sees the meter indicating something
which corresponds to the pointer state |r k ⟩, then from that moment on the state has immediately
become |a k ⟩ ⊗ |r k ⟩ ⊗ |b k ⟩. Conscious introspection of the observer therefore causes the collapse
of the wave packet. This strange situation is expressed in the thought experiment called ‘Wigner’s
friend’, in which the measuring device is replaced by a friend who communicates the outcome of
measurement to Wigner.
The aforementioned authors emphasize the role of consciousness in the interpretation of quantum
mechanics. It need hardly be emphasized that for the majority of physicists something like this is
unacceptable. They are of the opinion that a measurement is finished as soon as the result is registered
somewhere in the equipment. It is not necessary that it subsequently comes to attention of a conscious
being. But of course, then the question remains again which criterion can be given for a permanent
registration.
VIII. 4. 2
BOHMIAN MECHANICS
An important advantage of the theory of chapter VI is its avoidance of the projection postulate.
This has consequences for the treatment of measurements. ‘Measuring’ is not a primitive concept in
Bohmian mechanics, measurements are treated on an equal footing with all other physical interactions.
The measuring apparatus is treated in the same manner as the measured object system, namely
with the Bohmian equations, which are derived from the Schrödinger equations. As a consequence,
the interaction between an object system and a measuring apparatus can be given according to the
measurement scheme (VIII. 4).
If, for the sake of simplicity, we limit ourselves to two terms, the interaction is of the
form (VI. 24), p. 134, where ϕ B and ϕ D are the eigen - wave functions of the pointer quantity, corresponding
to the various pointer positions. It is plausible to assume that ϕ B and ϕ D have no overlap.
Consequently, the wave function of the object system and the measuring apparatus is effectively factorizable
and we can regard the superposition as a mixture. There is no measurement problem in
Bohmian mechanics.
◃ Remark
The requirement that ϕ B and ϕ D in (VI. 24) have no overlap is stronger than what is required in
Von Neumann’s model. There it suffices that the wave functions are orthogonal, i.e., ⟨ϕ B | ϕ D ⟩ = 0
instead of ϕ B (⃗q)ϕ D (⃗q) = 0 for all ⃗q ∈ R 3 . ▹
VIII. 4. 3
SPONTANEOUS COLLAPSE
The next option has been developed by G.C. Ghirardi, A. Rimini, and T. Weber (1986), a related
proposal comes from F.A. Bopp (1947). In this view the evolution from the Schrödinger postulate has
to be replaced by an indeterministic evolution. A stochastic term is added, making the Schrödinger
equation non - linear. This has as a consequence that every physical system from time to time spontaneously
makes a small jump, so that the wave function collapses to, almost, a position eigenstate.
The new constant of nature characterizing the relevant time scale is such that the probability of a
spontaneous collapse of the wave function for a single elementary particle is extremely small, in the