Nonlinear evolution of the quantum mechanical state vector

[modaniel]

Hi,

My understanding that one of the postulates of quantum mechanics is that the vector describing the quantum mechanical state of a system evolves in a linear fashion. My question is how this can be reconciled with systems where the system evolves in a non-linear fashion for example systems with a non-linear optical response?

 

[fzero]

Hi,

My understanding that one of the postulates of quantum mechanics is that the vector describing the quantum mechanical state of a system evolves in a linear fashion. My question is how this can be reconciled with systems where the system evolves in a non-linear fashion for example systems with a non-linear optical response?

When we say that quantum mechanics is linear, we mean that measurements, as well as time-evolution, of a system is described by linear operators acting on the state vector. In this context, linear means that the operators preserve vector addition and scalar multiplication: http://en.wikipedia.org/wiki/Linear_map.

When systems respond nonlinearly to an input, they do not do so because the microscopic linear QM description has broken down. Instead the nonlinearity is due to the effect of higher order, multi-step processes.

For instance, the first-order absorption of light by a crystal is described by a matrix that is linear in the amplitude of the incident light. A material that has a nonlinear optical response does so because the microscopic description of the relevant absorption process is actually a 2nd-order effect that is a quadratic function of the amplitude of the incident light. The absorption process actually involves absorbing two or more photons at the same time. Each part of the process (absorption of a single photon) is linear, so the total absorption process is linear in the operator sense. However, the observables are nonlinear functions of the inputs.