FOUNDATIONS OF QUANTUM MECHANICS

100 CHAPTER IV. THE COPENHAGEN INTERPRETATION
(v) An objection to the Robertson inequality (IV. 41) and the Schrödinger inequality (IV. 40) is that
the right - hand side depends on the state, therefore, it is no absolute lower limit for all states.
If |ψ⟩ is an eigenstate of A, the right - hand side of the Robertson inequality (IV. 41) is 0 and
does not provide any restriction on ∆B. Therefore, even if A and B are not both at the same
time sharp in any state, i.e., they do not have simultaneous eigenstates, this does not follow
from the inequality (IV. 41).
Only if the right - hand side of inequality (IV. 41) is unequal to zero for all states, the Robertson
inequality represents the uncertainty principle. This is the case if the commutator is a multiple
of unity, as in the case of P and Q, where [P, Q] = −i11, see p. 78, (IV. 1). It can, however, be
proved that this canonical commutation relation [P, Q] can only apply to unbounded operators
having no eigenstates in the, inevitably infinite dimensional, Hilbert space in which they act.
(vi) Already in 1929 E.U. Condon pointed out the following facts (Jammer 1974, p. 71). In certain
states, non - commuting operators can both be sharp. Take, for example, the ground state of the
H - atom, or any stationary state with total angular momentum l = 0. This is also an eigenstate
of L x , L y and L z with eigenvalue 0. Therefore, ∆L x ∆L y = 0, and likewise for L x and L z ,
and for L y and L z , although these operators do not mutually commute. Therefore, the fact that
operators do not commute does not guarantee an uncertainty relation. Furthermore, sometimes
an inequality holds for commuting operators. Take again a stationary state of the H - atom,
with l = 1 and m = 0. In that state ⟨[L x , L y ]⟩ = 0, whereas ∆L x ≠ 0 and ∆L y ≠ 0.
In conclusion, there are fundamental objections against accepting the Schrödinger inequality, and
by implication against the weaker inequalities which follow from it, to be the mathematical expression
of Heisenberg’s uncertainty principle.
And this is not everything yet.
IV. 5. 3
SINGLE SLIT EXPERIMENT
Relations (IV. 26) and (IV. 41) are considered to be the mathematical expression of the uncertainty
principle in the major part of textbooks on quantum mechanics. Next to the previous criticism, we
will show that this also is, remarkably enough, inconsistent with the experiments used as illustrations
of this principle (Uffink and Hilgevoord 1985, 1988 and Hilgevoord and Uffink 1988, 1990).
Consider the deflection of light, or of electrons, by a single slit in an absorbing screen, an example
Heisenberg also gives. Take for the wave function representing the particles passing through the
screen with the slit a simple square wave function, see figure IV. 6,
ψ ss (q) =
{
1 √
2 a
if |q| a
0 elsewhere
, (IV. 43)
where 2 a ∈ R + is the width of the slit, and q the Cartesian coordinate parallel to the screen and
perpendicular to the slit.