The origin of Heisenberg uncertainty

Demystifying quantum mechanics II

Perhaps the one thing that everyone thinks about when they hear talk about quantum mechanics is Heisenberg’s uncertainty principle. It may even sometimes be considered as the essence of quantum mechanics. Now what would you say if I tell you that the Heisenberg uncertainty principle is not a fundamental principle and that the origin of this principle is not found in quantum mechanics? The fundamental origin of this uncertainty is a purely mathematical property and the reason that quantum mechanics inherited this principle is simply a result of the Planck relationship.

Werner Heisenberg

I have discussed this issue to some extent before. However, it forms an important part of the knowledge that would help to demystify quantum mechanics. Therefore, it deserves more attention.

Before the advent of quantum mechanics, the state of a particle was considered to be completely described by its position and velocity (momentum). The dynamics of a system could then be represented by a diagram showing position and velocity of the particle as a function of time. For historical reasons, the domain of such a diagram is called phase space. For a one-dimensional system (such as a harmonic oscillator), it would give a two-dimensional graph with position on one axis and velocity on the other. The state of the system is a point on the two-dimensional plane that moves along some trajectory as a function of time. For a harmonic oscillator, this trajectory is a circle.

A mathematical property (in Fourier analysis), which may have seemed to be complete unrelated at the time, is that the width of the spectrum of a function has a lower limit that is proportional to the inverse of the width of the function. This property has nothing to do with physical reality. It is a purely logical fact that can be proven with the aid of mathematics. If the function is, for instance, interpreted as the probability distribution of the position of a particle, the width of the function would represent the uncertainty in its location.

This mathematical uncertainty property was transferred to phase space by Planck’s relation, which links the independent variable of the spectrum (the wave number) with the momentum or velocity of the particle. The implication is that one cannot represent the state of a quantum particle with a single dimensionless point on phase space in quantum mechanics. Hence, the Heisenberg uncertainty principle.

So, the uncertainty associated with Heisenberg’s principle is inevitable due to Planck’s relation. And it is founded on pure logic in terms of which mathematics is based. Planck’s relation is the only physics that enters the picture. The Heisenberg uncertainty principle is therefore not a separate principle that is independent of Planck’s relationship as far as the physics is concerned.

Now, there are a few subtleties that we can address. There are also some interesting consequences based in this understanding, but I’ll leave these for later.

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