Pervasiveness of linear operators

[ralqs]

Obviously linear operators are ideal to work with. But is there a deeper reason explaining why they're ubiquitous in quantum mechanics? Or is it just because we've constructed operators to be linear to make life easier?

[D H]

That every observable q is associated with some linear operator Q is a key postulate of quantum mechanics. Experimental physicists don't quite trust those goofy ideas that theoreticians claim to be true (and that is exactly what a postulate is, a claimed rather than a derived truth), so those experimentalists test, test, test, and test again. As far as I know, linearity has so far withstood the test of time.

[arkajad]

Position and momentum operators are linear. In classical Hamiltonian mechanics every physical quantity is a function of positions and momenta. A function of linear operators (assuming some power series expansion) is a linear operator. In quantum mechanics we have to deal with noncommutativity of position and momenta, so association of operators to physical quantities is sometimes not quite unique - but in practice it occurs not so frequently.

Then we have a general theorem of Wigner associating every symmetry with a linear unitary (or antilinear antiunitary) operator from a very general assumptions. It follows that conserved quantities (generators of one-parameter groups of symmetries) are represented by linear operators.

Nonlinear operators may appear in more general formulations of quantum mechanics, when you start with a convex space of states which is not necessarily described by density matrices as for instance in Mielnik's Generalized Quantum Mechanics (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103859881) and Comments on Mielnik's generalized (non linear) quantum mechanics (http://www.springerlink.com/content/p1u40w867l10tm31/) by Haag and Bannier.

[Muthu]

Quantum Computers are not believed to be able to solve NP-Complete problems. But if quantum mechanics were non-linear, then it's a different story as shown here:
http://arxiv.org/abs/quant-ph/9801041
Probably adds to the evidence the QM is fundamentally linear.