How time-dependent Schrödinger equation can describe a nonlinear phenomena where it is classified as linear equation?

Question

Many nonlinear phenomena are well described by QM (e.g. multiphoton ionization, high harmonic generation etc). The time-dependent Schrodinger equation is a linear equation, how a non-linear equation could describe such nonlinear phenomena ?.

Answer 1

In classical physics, we don't usually bother distinguishing between two things that must be distinguished in quantum physics. In classical physics, all observables commute with each other, so we can (and do) always take the state to be an eigenstate of all of the observables. For this reason, we don't really need to bother distinguishing between the state and the observables. But they are logically distinct: the state is what tells us the values of the observables. Observables represent the kinds of things we can measure, and the state tells us what the results of those measurements will be. In quantum physics, this distinction is essential, because most observables do not commute with each other, so they cannot all have predictable measurement outcomes. The state tells us what we will get, statistically, when we measure any given observable.

Quantum physics is linear in the state and (typically) nonlinear in the equations that govern relationships between observables.

Here's an example. In relativistic quantum electrodynamics (which implicitly accounts for the phenomena listed in the question), the Schrödinger equation is $$ \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\opsi}{{\overline\psi}} \newcommand{\pl}{{\partial}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfB}{\mathbf{B}} i\frac{d}{dt}|\Psi\ra = H|\Psi\ra \tag{1} $$ and the Hamiltonian is $$ H \sim \int d^3x\ H(x) \tag{2} $$ with \begin{align} H(x) &= \opsi(x)\gamma^ki\nabla_k\psi(x) + m\opsi(x)\psi(x) \\ & +\frac{\bfE^2(x)+\bfB^2(x)}{2} +e\opsi(x)\gamma^\mu A_\mu\psi(x). \tag{3} \end{align} (schematically, without carefully checking the coefficients and without messing with gauge-fixing), where $\psi$ is the spinor field (should be one for each species of charged particle, but I only included one here) and where $\bfE,\bfB,A$ are different representations of (components of) the electromagnetic field.

The Schrödinger equation (1) is linear in the state-vector $|\Psi\ra$, but the Hamiltonian is a moderately complicated combination of the field operators. To see how this relates to non-linear field equations in classical electrodynamics, we need to work in the Heisenberg picture instead of the Schrödinger picture. In the Heisenberg picture, the state $|\Psi\ra$ is independent of time, and all time-dependence is carried by the field operators (which are used to construct observables associated with regions of spacetime). The Heisenberg equations of motion describe the time-dependence of the field operators. In this case, these equations are (schematically again) $$ \gamma^\mu(i\pl_\mu + eA_\mu)\psi=0 \hskip2cm \pl_\mu F^{\mu\nu}=e\opsi\gamma^\mu\psi. \tag{4} $$ The first equation is the (operator version of the) Dirac equation, and the second equation is the (operator version of) Maxwell's equations. Both are nonlinear in the field operators: the Dirac equation is nonlinear because of the $A\psi$ term, and Maxwell's equations are nonlinear because of the $\opsi\psi$ term.

With allowance for my carelessness with the coefficients, equations (1)-(3) are equivalent to equations (4). The former are linear in the state-vector, and the latter are nonliner in the field operators. The nonlinear terms in equation (4) come from the non-quadratic term in the Hamiltonian (2)-(3).