FOUNDATIONS OF QUANTUM MECHANICS

IV. 1. HEISENBERG AND THE UNCERTAINTY PRINCIPLE 79
He starts (1927, Eng. tr. p. 64) with linking measuring and defining operationally,
When one wants to be clear about what is to be understood by the words “position of the
object”, for example of the electron, relative to a given frame of reference, then one must
specify definite experiments with whose help one plans to measure the “position of the
electron”, otherwise this word has no meaning.
We will call this the measuring = defining principle.
One could, for example, determine the position of an electron by examining it under a microscope.
According to classical optics a microscope has a limited resolution. The Abbe criterion gives the
smallest distinguishable details as
δq ∼
λ , (IV. 2)
sin ε
where λ is the wavelength of light and ε is the aperture, the opening angle of the lens. For a precise
measurement we must therefore use a very short wavelength, i.e. gamma radiation. But in that case
the Compton effect cannot be neglected. The radiation behaves as a flow of particles, with momentum
p 0 = h λ
, which collides with the electron and causes it to recoil.
Figure IV. 1: Heisenberg’s γ - microscope
To allow for an observation at least one photon has to collide with the electron, which will bring
about a change of momentum. But as we do not know anything more about the direction of the
photon after the collision than that it has gone through the lens, we cannot indicate the size of the
recoil exactly. As can be seen in figure IV. 1, the transfer of momentum remains unknown to an
amount
δp ∼ p 0 sin ε = h λ
sin ε (IV. 3)
and therefore
δq δp ∼ h. (IV. 4)