Why not an uncertainty principle?

It may seem strange that there are no fundamental physical principle for quantum physics that is associated with uncertainty among those that I have proposed recently. What would be the reason? Since we refer to it as the Heisenberg uncertainty principle, it should qualify as one of the fundamental principles of quantum physics, right? Well, that is just it. Although it qualifies as being a principle, it is not fundamental.

It may help to consider how these concepts developed. At first, quantum mechanics was introduced as an improvement of classical mechanics. Therefore, quantities like position and momentum played an important role.

A prime example of a system in classical mechanics is the harmonic oscillator. Think of a metal ball hanging from a spring. Being pulled down and let go, the ball will start oscillating, moving periodically up and down. This behavior is classically described in terms of a Hamiltonian that contains the position and momentum of the metal ball.

In the quantum version of this system, the momentum is replaced by the wave vector times the Planck constant. But position and the wave vector are conjugate Fourier variables. That is the origin of the uncertainty. Moreover, it also leads to non-commutation when position and momentum are represented as operators. The sum and difference of these two operators behave as lowering and raising operators for quanta of energy in the system. The one reduces the energy in discrete steps and the other increases it in discrete steps.

It was then found that quantum mechanics can also be used to improve classical field theory. But there are several differences. Oscillations in fields are not represented as displacements in position that is exchange into momentum. Instead, their oscillations manifest in terms of the field strength. So, to develop a quantum theory of fields, one would start with the lowering and raising operators, which are now called creation and annihilation operators or ladder operators. Their sum and difference produce a pair of operators that are analogues to the position and momentum operators for the harmonic oscillator. In this context, these are called quadrature operators. They portray the same qualitative behavior as the momentum and position operators. They represent conjugate Fourier variables and therefore again produce an uncertainty and non-commutation. The full development of quantum field theory is far more involved then what I described here, but I only focused on the origin of the uncertainty in this context here.

So, in summary, uncertainty emerges as an inevitable consequence of the Fourier relationship between conjugate variables. In the case of mechanical systems, these conjugate variables come about because of the quantum relationship between momentum and wave vector. In the case of fields, these conjugate variables comes from the ladder operators, leading to analogues properties as found for the formal description of the harmonic oscillator. Hence, uncertainty is not a fundamental property in quantum physics.

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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