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S1q. How can quantum mechanics be stochastic while the Schrödinger
equation is not?
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The Schroedinger equation is a deterministic wave equation.
But when we set up an experiment to measure either position or
momentum, we get uncertain, stochastic outcomes.
So - is quantum mechanics deterministic or stochastic?
One has to be careful in the interpretation of the foundations...
Fortunately, the same apparent paradox already occurs in classical
physics; hence the paradox cannot have anything to do with the
peculiarities of quantum mechanics.
Indeed, a Focker-Planck equation is a deterministic partial
differential equation. But when measuring a process modelled by it
- such as the position of a grain of pollen in Brownian motion -,
we get only probabilistic results. Now Focker-Planck equations are
essentially equivalent to classical stochstic differential equations.
So - do they describe a deterministic or a stochastic process?
The point resolving the issue is that, both in stochstic differential
equations and in quantum mechanics, probabilities satisfy deterministic
equations, while the quantities observed to deduce the probabilities
do not.
Thus, in both cases, probabilities are deterministic ''observables''
while the position of a grain of pollen in classical mechanics, or
position and momentum in quantum mechanics, ar not.
Note that this does not preclude that quantum mechanics has certain
deterministic features. For example, the energy levels of hydrogen are
deterministic. They can be computed from the Schroedinger equation and
can be compared with experiment. The experiment consists in observing
spectral lines, and these are differences of computed energy levels.
In practically all comparisons of theory and experiment, the situation
is similar: Some quantities computed with the help of the Schroedinger
equation can be compared with some quantities measured or computed from
measurements. Many of these are stationary or slowly varying in time,
and then behave deterministically, even in a quantum context.
But this does not make quantum mechanics deterministic. As the term is
usually understood, determinism refers to the predictability of all
observables of a system at all times given the observation at a fixed
time and a dynamical law.
The indeterminism of quantum mechanics for hydrogen is seen in the
impossibility of predicting at all times the position of the electron
surrounding the nucleus given its position at a fixed time.
Instead, one can only predict the probability of being in a
certain orbital or jumping to a different orbital, and the prediction
of the position within an orbital is even more fuzzy.
Note that quantum jumps are observable; see, e.g., the frequently cited
paper
Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek,
Observation of Quantum Jumps,
Phys. Rev. Lett. 57, 1696 - 1698 (1986).
In the absence of external forces, the stable ground state is attained
after some time; and at this level of description we see determinism.
Thus eing in the ground state is a deterministic feature of the state,
but the electron's position in the ground state is not. The ground
state only predicts probabilities for being somewhere, no certainties.
Essentially the same holds for all quantum systems.
For a more in depth discussion of the deterministic and stochastic
features of quantum mechanics see the Chapter on ''Models, statistics,
and measurements'' in
Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
Cambridge University Press, to appear (2009?).
http://www.mat.univie.ac.at/~neum/papers/physpapers.html#QML
arXiv:0810.1019