Heisenberg’s Measurement-Disturbance


Werner Heisenberg, one of the founding fathers of quantum mechanics is famous for his uncertainty principle. Initially he came to his result based on a rather heuristic argument that measurement of a system disturbs the system and this leads to an inherent uncertainty in all measurements. We have the “Heisenberg’s microscope” in which he imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.

Later on the uncertainly principle was formulated correctly in terms of quantum operators that do not commute. This is of course the true deep reason why we have quantum uncertainty.

It has now been shown that Heisenberg’s original argument is wrong. Aephraim Steinberg and other researchers the Centre for Quantum Information & Quantum Control and Institute for Optical Sciences at the University of Toronto have published work disproving Heisenberg [1].

I will stress the claim is not that the uncertainty principle is wrong, it is not, the claim is that Heisenberg’s original argument is wrong.


[1] Lee A. Rozema, Ardavan Darabi, Dylan H. Mahler, Alex Hayat, Yasaman Soudagar, and Aephraim M. Steinberg. Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements. Phys. Rev. Lett. 109, 100404 (2012)

Fractal from Binomial Coefficients


Above is a discrete fractal generated by creating a table of zeros and ones by deciding if the binomial coefficients are even or odd. The “key” here is paint black if odd, otherwise leave light blue.

The pattern is closely related to Pascal’s triangle.

The pattern clearly shows self-similarity as all fractals do.

As far as I know, this pattern was first noticed in [1]. Also note that we have a structure very similar to the Sierpinski Sieve. In the limit of infinite rows we recover the Sierpinski Sieve, up to a shift in the positions of the zeros and ones.

A slight variant


Just for fun I used the same algorithm to study the pattern associated with modified binomial coefficients of the form

\(\left( \begin{array}{c} (-1)^{k}n\\ k \end{array} \right)\)

Again the pattern shows lots of self-similarity.


[1] S. Wolfram: American Mathematical Monthly, 91 (November 1984) 566-571