Heisenberg's Microscope - Heisenberg's Argument

Heisenberg's Argument

Heisenberg's argument can be found in (Heisenberg 1930), and is summarized as follows. Heisenberg begins by supposing that an electron is like a classical particle, moving in the direction along a line below the microscope, as in the illustration to the right. Let the cone of light rays leaving the microscope lens and focusing on the electron makes an angle with the electron. Let be the wavelength of the light rays. Then, according to the laws of classical optics, the microscope can only resolve the position of the electron up to an accuracy of

When an observer perceives an image of the particle, it's because the light rays strike the particle and bounce back through the microscope to their eye. However, we know from experimental evidence that when a photon strikes an electron, the latter has a Compton recoil with momentum proportional to, where is Planck's constant. It is at this point that Heisenberg introduces objective indeterminacy into the thought experiment. He writes that "the recoil cannot be exactly known, since the direction of the scattered photon is undetermined within the bundle of rays entering the microscope" (p.21). In particular, the electron's momentum in the direction is only determined up to

Combining the relations for and, we thus have that

,

which is an approximate expression of Heisenberg's uncertainty principle.