FOUNDATIONS OF QUANTUM MECHANICS

134 CHAPTER VI. BOHMIAN MECHANICS
◃ Remark
For Bell, this observation was a reason to examine if quantum mechanical HVT’s can, in fact, be local
at all. We will come back to this in chapter VII. ▹
An intermediate form occurs if A, B, C, D ⊂ R 3 are certain areas in space, such that A ∩ C = ∅
or B ∩ D = ∅, ψ A , ψ C , ϕ B , ϕ D are wave functions which are 0 outside these areas, and the wave
function is, analogously to (VI. 14), of the form
ψ(⃗q 1 , ⃗q 2 ) = a ψ A (⃗q 1 ) ϕ B (⃗q 2 ) + b ψ C (⃗q 1 ) ϕ D (⃗q 2 ), (VI. 24)
with a, b ∈ R. Since the pair ψ A and ψ C , or the pair ϕ B and ϕ D , or both, have no overlap, for
all ⃗q 1 , ⃗q 2 ∈ R 3 we have
ψ A (⃗q 1 ) ψ C (⃗q 1 ) = 0 or ϕ B (⃗q 2 ) ϕ D (⃗q 2 ) = 0. (VI. 25)
Therefore, the probability density belonging to (VI. 24) is
ρ(⃗q 1 , ⃗q 2 ) = R 2 (⃗q 1 , ⃗q 2 ) = |a ψ A (⃗q 1 ) ϕ B (⃗q 2 )| 2 + |b ψ C (⃗q 1 ) ϕ D (⃗q 2 )| 2 , (VI. 26)
without a cross - term, and we see that the ensemble, again analogously to (VI. 14), behaves like a
mixture. In this case we call the wave function ψ(⃗q 1 , ⃗q 2 ) effectively factorizable.
With
⎧
S ⎪⎨ A (⃗q 1 ) + S B (⃗q 2 ) for ⃗q 1 ∈ A, ⃗q 2 ∈ B
S tot (⃗q 1 , ⃗q 2 ) = S C (⃗q 1 ) + S D (⃗q 2 ) for ⃗q 1 ∈ C, ⃗q 2 ∈ D
(VI. 27)
⎪⎩
0 elsewhere,
and ψ A (⃗q 1 ) = R A (⃗q 1 )e i S A(⃗q 1 ) , etc., because of (VI. 25) it holds that
ψ(⃗q 1 , ⃗q 2 ) = a R A (⃗q 1 ) R B (⃗q 2 ) e i (S A(⃗q 1 ) + S B (⃗q 2 ))
+ b R C (⃗q 1 ) R D (⃗q 2 ) e i (S C (⃗q 1 ) + S D (⃗q 2 ))
(VI. 28)
= ( a R A (⃗q 1 ) R B (⃗q 2 ) + b R C (⃗q 1 ) R D (⃗q 2 ) ) e i Stot(⃗q 1, ⃗q 2 ) .
Therefore, also in case of composite systems, the quantum potential can be taken as a sum of terms
belonging to the separate particles, and the momentum of a particle does not depend on the other
particle.
Consequently, we can interpret the system as being composed of a pair of particles of which one
particle is in area A and the other in B, or, likewise, in area C and D. The pair of particles is not
influenced by the wave functions or the quantum potential in the other area. For this reason, these
pilot waves are also called empty waves. They have no dynamic influence on the particles, but they
do contain energy. If, at some time, the wave functions will have overlap again, they will of course
also regain influence.