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S12d. Stochastic quantum mechanics
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For certain Hamiltonians, the Schroedinger equation can be interpreted
as a classical diffusion process. This leads to the stochastic
quantum mechanics of Nelson. For an overview, see, e.g.,
http://www-stud.uni-essen.de/~sb0264/stochastic.html
While it gives an interesting aspect to quantum mechanics and its
classical limit, Nelson's description has a severe deficiency
in that it cannot handle the situation when the wave function vanishes
at some point. At all such points, R has a singularity, and S is
entirely undefined. This happens, e.g., for excited states of hydrogen,
hence is an integral part of standard quantum mechanics.
Even if one argues that such states are idealized and cannot occur,
it seems not be possible to show that a state that is everywhere
nonzero will preserve this property under time evolution.
Thus Nelson's representations may develop spurious singularities
which are not in the observable part of quantum mechanics.
Also, it is awkward to do scattering calculations in Nelson's
framework. Moreover, Nelson, as quoted on p. 16 of the above paper,
says correctly,
''Quantum mechanics can treat much more general Hamiltonians
for which there is no stochastic theory.''
Thus it is unlikely to be useful as a 'fundamental' description
of nature.
Instead, natural stochastic forms of quantum mechanics are those of
quantum diffusion processes and quantum jump processes, in which the
wave function itself is regarded as a classical random object.
For their use in an experimental context, see, e.g., quant-ph/9805027.