What is Schrodinger’s equation? Is it deterministic or not? If it is, how can we prove that? And what conditions must be satisfied for it to be non-deterministic? 

Schrodinger’s equation was originally just the partial differential equation satisfied by the position-space wave function of a particle (or more general system) in non-relativistic quantum mechanics. The same name is sometimes also used for the equation [math]\frac{d}{dt}\Psi(t)=iH\Psi(t)[/math] satisfied by the state vector in any NRQM system regardless whether or not a position-space representation is being used (or is even available).

It is deterministic (in the sense of determining [math]\Psi(t)[/math] uniquely for all [math]t[/math] if given an initial condition [math]\Psi(0))[/math], so long as the Hamiltonian [math]H[/math] is self-adjoint (symmetry is NOT enough!).

The proof of this involves more analysis than I could fit into a Quora answer, but in the general case it follows from the fact that for any self-adjoint operator [math]H[/math] on a Hilbert Space, the equation [math]\frac{d}{dt}\Psi(t)=iH\Psi(t)[/math] is uniquely satisfied by [math]\Psi(t)=e^{iHt}\Psi(0)[/math] where the complex exponential of [math]H[/math] is defined in terms of its spectral resolution; and for the PDE special cases it might be done by various theorems involving greens functions or Fourier analysis and convergence properties of improper integrals.

It may be non-deterministic if [math]H[/math] has not been specified on a large enough domain to be essentially self-adjoint (as sometimes happens if boundary conditions are omitted from the specification of a problem in which the particle is confined somehow – either by an infinite potential or by living in a single cell of a crystal lattice for example). But such cases are normally just due to inadequate specification of the problem rather than to a real physical indeterminacy.

So I would say that in a properly defined quantum theory model the Schrodinger equation is indeed almost always deterministic.

[N.B. It wasn’t part of the actual question, but I should perhaps add that the reason this does not make quantum mechanics deterministic is because even complete knowledge of the quantum state of a system is not sufficient to predict the outcomes of all possible experimental measurements. For any a state which happens to produce a predictable value for one observable there will be other observables for which the outcome is uncertain.]

Source: (1000) Alan Cooper’s answer to What is Schrodinger’s equation? Is it deterministic or not? If it is, how can we prove that it is deterministic? If it isn’t, what conditions must be satisfied for Schrodinger’s equation to be non-deterministic? – Quora

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