FOUNDATIONS OF QUANTUM MECHANICS

IV. 5. THE UNCERTAINTY RELATIONS 105
Now there are, however, two parameters playing a role. The distance of the slits A is a measure for
the total width of |ψ ds (q)| 2 , the ‘enveloping’ cosine factor in (IV. 54), while the width of the slits a is a
measure for the ‘fine structure’ of this probability density. For | ˜ψ ds (p)| 2 the roles have reversed, A −1
is a measure for the width of the interference lines, while a −1 is a measure for the total width of the
interference pattern. This shows the well - known fact that the width of the interference lines and the
distance between the slits are inversely proportional. In a moment we will see that Bohr’s discussion
of the double slit experiment exactly rests on this fact.
◃ Remark
Consider the measures
∆ ψds Q ≃ A and ∆ ψds P = ∞, (IV. 55)
W α (Q, ψ ds ) ≃ A and W α (P, ψ ds ) ≃ . (IV. 56)
a
None of these measures gives the fine structure. Therefore, Bohr’s Copenhagen reasoning, treated
in the next subsection, cannot be based on the Kennard inequality (IV. 26) nor on the inequality of
Landau and Pollak (IV. 51). ▹
EXERCISE 30. Verify the calculations (IV. 55) and (IV. 56).
IV. 5. 6
A NEW UNCERTAINTY MEASURE
Bohr’s reasoning concerning the double slit experiment goes as follows. A way to determine
through which slit the particle has gone is measuring the recoil in the q - direction that the screen
experiences at the passage of this particle. To this end the screen must be able to move in the q - direction.
Instead of a fixed screen we take therefore a screen that is suspended from a spring, as can be
seen in figure IV. 10. The incoming momentum p is perpendicular to the screen.
We assume conservation of kinetic energy, i.e. a heavy screen, which means that only the direction
of the momentum changes. Consequently, a particle arriving at position q of the photographic
plate, gives a recoil to the screen of, assuming r ≫ A and therefore sin θ ≈ tan θ,
( q ± A
r
)
p, (IV. 57)
depending on which slit it has gone through. To be able to measure the difference in recoil, it must
hold for the inaccuracy δP with which the momentum of screen was known in advance, that
δP < 2 A p . (IV. 58)
r