# On understanding the relation between Heisenberg Uncertanity Principle, Indeterminism and classical chaos

I recently read Emperor's New Mind by Roger Penrose. In it he talks at length on the notion of determinism in science. In this context how does the Heisenberg Uncertanity Principle bring about the notion of so called indeterminism ,in the sense that how do we know it is a fundamental law of nature that doesn't allow us to make some observations and unlike the average Newtonian chaotic system (turbulent flow of fluids and the like) which is labelled deterministic chaos.

I would like to mention that I am not that well versed in these concepts.

In this context how does the Heisenberg Uncertanity Principle

Physics is the discipline that uses mathematical models, called Physics Theories, to describe observations and data, and, very important to predict future behavior. (A model that only fits existing data is a map, not a theory). So quantum mechanics, the theory developed from observations and measurements, has introduced indeterminacy because it is necessary to describe the data.

It was found that the differential equations that describe wave functions can be used to model the observations and data, if extra axioms pick up the correct solutions that can be descriptive and predictive of data. Principles, laws, postulates are the names used for these axiomatical statements. The Heisenberg Uncertainty Principle (HUP) was deduced from data that were incompatible with the microscopic world, where mostly the new quantum theory is needed, can be shown to emerge from the commutation relations of the complicated theory of Quantum Mechanics.

bring about the notion of so called indeterminism ,in the sense that how do we know it is a fundamental law of nature

In the final theory, the indeterminacy comes from the wavefunction postulate

$$Ψ(x,t)$$ = single valued probability amplitude at $$(x,t)$$

$$Ψ^*(x,t)Ψ(x,t)$$ = the probability of finding the particle at $$x$$ at time $$t$$ provided the wave function is normalized

This is what makes for the basic indeterminacy in quantum mechanics, and the theory was developed in order to explain the data of that time: photoelectric effect, black body radiation, spectra of atoms. It prevailed because it was predictive of new data.

that doesn't allow us to make some observations and unlike the average Newtonian chaotic system (turbulent flow of fluids and the like) which is labelled deterministic chaos.

The concept of probability in both classical mechanics and quantum mechanics is the same, the same with the simple probabilities of throwing a dice.

In classical deterministic chaos dealing with the many particle states, it is the complexity of the enormous number of particles that displays an emerging chaotic behavior, inability to exactly determine individual particle tracks. Quantum mechanics, by identifying particles with a probability distribution are inherently non deterministic. There are theoretical efforts to define an underlying deterministic layer of physical quantities from which the indeterminacy of quantum mechanics emerges, but they are not successful up to now, that is another long story.

• This is not correct: "deterministic chaos it is the complexity of the enormous number of particles that displays an emerging chaotic behavior". Three degrees of freedom is all one needs for chaos. – stafusa Nov 18 at 22:34
• @stafusa OK, I will insert "ffor many particle states", I was not defining deterministic chaos. – anna v Nov 19 at 5:13

The Heisenberg uncertainty principle reflects the fact that, according to quantum mechanics, certain combinations of the observable properties of particles are fundamentally incompatible (the technical term is 'non-commuting'), so that the particle cannot posses both properties at once. If a particle has a definite position say, its momentum is undefined. Conversely, if a particle has a definite momentum its position is undefined. There is no exact classical analogy, but you might consider, for example, the viscosity of liquid water and the shear strength of an ice crystal. They are both properties of water, but they are not properties that any given volume of water can possess simultaneously.

Quantum mechanics says that if you make a measurement of one property of a particle, and then make a measurement of another property that does not commute with the first, the result will be unpredictable, although there is a rule (the Born rule) which allows you to calculate the relative probabilities of one result compared to any other. It is the unpredictable nature of this effect which makes for an indeterministic outcome.

The effect is quite different in principle from classical chaos.

• It would be great to know more on how classical chaos differs from heisenberg's principle – Maan Nov 12 at 21:09
• Hi Maan, the difference is that in classical theory you could in principle pin down all of the positions and momenta of the particles that make up a turbulent liquid- what stops you is that there are just too many to deal with. In quantum theory you cannot in principle pin them down, because a particle cannot possess both a position and a momentum simultaneously- it either has one or the other. – Marco Ocram Nov 12 at 21:54
• QM does not say that a particle cannot have both momentum and position, but only that you cannot prepare a state where both properties are exactly known. This is a consequence of the fact that different properties require different experimental arrangements (like different orientation of a magnetic field in the case of spin preparation). – Andrei Nov 13 at 5:51

The HUP is fully compatible with determinism. Quantum mechanics without any special role for measurement (i.e. with the Schrodinger equation as the only law of motion) is deterministic, for example.

• I don't see anything to disagree with here, but this doesn't seem like an answer that will help the OP, nor is it a very complete answer even for someone who is at a higher level of knowledge. – Ben Crowell Nov 12 at 20:52