FOUNDATIONS OF QUANTUM MECHANICS

106 CHAPTER IV. THE COPENHAGEN INTERPRETATION
q
q = r tan θ 1 + A
= r tan θ 2 − A
2 A
1
2
a
θ 2
θ 1
r
Figure IV. 10: Moving screen
Because of the inequality
δP δQ , (IV. 59)
to the inaccuracy with which the position Q of the screen was known then applies
δQ >
r . (IV. 60)
2 A p
But the width of the interference lines on the photographic plate is
λ r
2 A = r , (IV. 61)
2 A p
where λ = p
is the de Broglie wavelength of the electron. Bohr therefore concludes that the uncertainty
in the position of the screen will result in the erasure of the interference pattern.
◃ Remarks
First, we see that Bohr applies the uncertainty principle to the screen which means that he treats this
macroscopic body quantum mechanically. Second, he uses the uncertainty principle in a qualitative
manner, in particular, he does not give a definition of the uncertainties δP and δQ. Third, the relevant
uncertainty in Q is of the order of magnitude of the width A −1 of the interference lines. Bohr
therefore has no use of the Kennard inequality (IV. 26) or the inequality of Landau and Pollak (IV. 51),
which do not contain this width. Finally, Bohr does not show how erasure of the interference pattern
exactly takes place, obviously, he considers it to be intuitively evident. ▹
From the previous it should be clear that something is still lacking in the mathematical formulation
of the uncertainty principle. One would hope that there may exist some direct relation between the