Physics Asked by Mikael Jensen on March 11, 2021
Laskar https://arxiv.org/abs/1209.5996 calculated the position of planets up to 5 billion years. The time period is chosen to cover a time scale before the Sun is thought to expand as a red giant.
In his calculations, Laskar found that there is a tenfold increase in an error of the earth position along its orbit over 10 million years. This leads to complete uncertainty of the Earth’s whereabouts in about 1200 million years for an initial uncertainty of 1 m. The planetary positions can only be described in statistical terms.
The quantum mechanical uncertainty of the earth’s position about 1E-66 m for a velocity determination of 1 mm per second would similarly lead to uncertainty of the orbit position within Laskar’s chosen time period.
Does this imply some connection between the first classical and the quantum mechanical views, both requiring a statistical description?
No, it does not. The underlying physics are completely different.
In the case of orbital dynamics, it is known that on long-enough timescales, a multiplanet system will exhibit chaotic behavior which arises from the effects of cumulative and nonlinear feedback. Quantum mechanical uncertainty does not enter into this picture at all because the gravitational effects on planetary motion are many, many orders of magnitude larger.
Answered by niels nielsen on March 11, 2021
When we refer to quantum mechanics being statistical in nature, we refer to something else.
In your case, the statistical nature of classical prediction is due to the fact we do not have full knowledge of initial state and therefore we need to propagate the error during computation of evolution. Error is statistical in nature. But if we had full knowledge of initial state, there would be no statistics and we could compute the evolution fully deterministically. In this regard, quantum mechanics is the same. The evolution in quantum mechanics, given exact initial quantum state, is just as deterministic as in classical physics.
When we talk about quantum statistical nature, we are not talking about evolution. We are talking about measurement. The difference between classical physics and QM is that if you have full knowledge of state of the system in classical physics, we can predict measurements on the system with infinite precision. Not so in QM. There, even if we had full knowledge of the quantum state of the system, there would be measurements we would not be able to predict with infinite precision. For example if we knew that Earth is in certain position, then we know that it is in position eigenstate, which encompasses full knowledge of quantum state of the Earth. But Earth being in position eigenstate means, we have no idea what velocity measurement would give us. Even though we know everything that is to know about Earths quantum state with infinite precision, we cannot predict result of velocity measurement.
Answered by Umaxo on March 11, 2021
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