Is the statistical nature of quantum mechanics due to uncertainty principle?

Question

Is the statistical nature of quantum mechanics due to position and momentum not defined for quantum particles?

Update: I mean the uncertainty of any conjugate pairs.

Answer 1

As pointed out in another answer, the measurement problem and Heisenberg uncertainty are not causally related.

Heisenberg uncertainty arises from the wave characteristics of propagation of quantum entities. This wave behavior is highly analogous to classical wave propagation (but not identical). For instance, most of the (interference) properties of light can be accounted for in terms of propagation of classical waves. (Well, in order to interpret propagation of light as classical wave propagation the existence of a medium that enables that propagation must be granted).

The key point:
Wave propagation, and interference effect of wave propagation, does not in itself lead to a statistical nature. Example: propagating sound waves do not have a statistical nature.

For context of how Heisenberg uncertainty arises from wave characteristics, see the video by Grant Sanderson titled The more general uncertainty principle, regarding Fourier transforms

The statistical nature, on the other hand, correlates with the fact that when a quantum entity is detected it is deteced as a whole unit, characteristic of detection of a single entire particle. This is referred to as the measurement problem.

Among the most vivid demonstrations of that: the electron double slit setup, 1989, Hitachi labs.

Uploaded by many to youtube, I have arbitrarily selected one of them: Single electron double slit wave experiment

Answer 2

No, the uncertainty principle is unrelated to the Born rule, which is what I guess you mean by the statistical nature of QM.

The uncertainty principle is simpler than you think. A wavefunction $\psi$ has a well defined momentum only if it is an eigenfunction of the momentum operator i.e.

$$ i\hbar\frac{d}{dx} \psi = p \psi $$

where $p$ is the momentum. Likewise it has a well defined position only if it is an eigenfunction of the position operator. The uncertainty principle simply says that there are no wave functions that are eigenfunctions of both the momentum and position operators¹, so no wavefunction can have a well defined momentum and a well defined position at the same time. This is just a mathematical property of wave functions and unrelated to the measurement problem.

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¹ the HUP says more than this of course, but this is the important point here

Answer 3

Some of the most fundamental principles of quantum mechanics are

• Particles can be in a superposition of many different positions/momentum states
• Observables like position and momentum behave in a statistical way when measured

Based on the more mathematically precise version of these statements, along with the relationship between position and momentum as described by their commutation relations, we can derive the uncertainty principle. So to answer the question in the title of your question, I would say that the statistical nature of quantum mechanics is not due to the uncertainty principle. It's the other way around. The uncertainty principle is due to the statistical nature of quantum mechanics. Of course, it's a little bit arbitrary to start with one set of "fundamentals" and then to say everything else is derived, but you'd need to start with more than just the uncertainty principle to derive all the statistical properties of quantum mechanics.

Answer 4

Of course not. Sequential Stern-Gerlach experiments, which are about spin and measurements, have nothing to do with position and momentum, yet the results are probabilistic, as is the passing of a polarized photon through a misaligned polarization filter.