FOUNDATIONS OF QUANTUM MECHANICS

IV. 5. THE UNCERTAINTY RELATIONS 107
total width of a distribution in the p - language (q - language), and the fine structure of this distribution
in the q - language (p - language) as exhibited by the wave function for the double slit, assuming that
this relation has general validity. Indeed, such a relation has been found (Uffink and Hilgevoord 1985),
w α (Q, ψ) W α (P, ψ) C α and w α (P, ψ) W α (Q, ψ) C α , (IV. 62)
where w α ( · , ψ) ∈ R + is a measure for the width of the fine structure of ψ, W α ( · , ψ) ∈ R + is
the measure for the total width of ψ as introduced earlier, and C α > 0 is a constant depending
on α ∈ (0, 1], but not on the state ψ.
Illustratively, if W is taken as a measure of the size of the objective of a microscope and w as a
measure of the fine structure of the image, the inequalities express the fact that the resolving power
must decrease if the aperture is reduced. Likewise, the direction of incoming radiation can better
determined by using a long array of radio telescopes than by using a short one, etc. These inequalities
thus express, among other things, the well-known fact in optics that the resolving power of an
apparatus improves as the apparatus is larger.
The inequalities (IV. 62) seem to solve the problem for Bohr. A closer consideration however
tells us that W α (P, ψ) is not the suitable measure to express whether the difference in recoil can or
cannot be observed. More precise, W α (P, ψ) > 2Ap
r
does not guarantee that this difference cannot be
observed. W α (P, ψ) can be large in this experiment, which makes the inequality (IV. 62) ineffective.
Actually, it is the question if Bohr’s argument can in fact be based on an uncertainty relation.
Nevertheless, his conclusion is correct! The fact is that a direct calculation of the double slit
experiment by D. Hauschildt, unpublished, shows that the intensity of the interference, in case the
screen is movable, is proportional to the factor
∣ ⟨χ| e
i 2 A p Q r sc
|χ⟩ ∣ . (IV. 63)
Here |χ⟩ is the state of the screen and Q sc is the position operator of the screen. The state
|χ⟩ ′
:= e i 2 A p
r Q sc
|χ⟩ (IV. 64)
is the state of which the momentum spectrum is shifted by 2Ap
r
with respect to the momentum spectrum
of the state |χ⟩,
⟨p | χ ′ ⟩ = ⟨ p − 2 A p
r
∣ χ
⟩
. (IV. 65)
The factor (IV. 63) is, therefore, exactly the quantum mechanical expression describing to what extent
the state of the screen after the recoil can be distinguished from the state of the screen before the
recoil.
If the momentum spectrum of |χ⟩ is broad with respect to 2Ap
r
, the overlap (IV. 63) will be large,
namely almost 1. In that case |χ⟩ and |χ⟩ ′ are difficult to distinguish and interference is large. If the
momentum spectrum of |χ⟩ only contains peaks which are narrow with respect to 2Ap
r
, then (IV. 63)
is small. The states |χ⟩ and |χ⟩ ′ are well distinguishable then and interference is small. The essence
of Bohr’s reasoning is therefore correct; to the extent in which the screen can serve as a measuring apparatus
to determine the slit a particle goes through, interference disappears. Whether this reasoning
can be based on an uncertainty relation, is unknown to this very day.