Are there any nonlinear Schrödinger equations?

Question

The 1D Schrödinger equation reads:

$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2 \Psi}{\partial x^2}-\frac{i}{\hbar}V\Psi.$$

Now, generally we have $V=V(x)$ (or it dependending on any other number of real variables). But consider $V=V(\Psi)$, suddenly the Schrödinger equation presents a nonlinear term (except for specific cases like $V=0$ or $V=1/\Psi$).

Essentially my question, then, is: are there any systems where a potential depends partially or completely on the wavefunction, or is this simply a mathematical and non-physical curiosity?

Answer 1

Previous answers focus on the fundamental approach to Quantum Mechanics where the Hamiltonian operator is always a linear operator. However, they miss an extremely important situation where a non-linear Schrödinger-like equation appears in a natural way. That's the case with the so-called self-consistent one-particle approximations to the quantum many-body problem. They are approximations, but still well inside Quantum Mechanics.

For example, the Hartree approximation for the electronic problems introduces an effective interaction including the electrostatic interaction in a Schrödinger-like equation for the state $\psi_i({\bf r})$, due to the charge density (which depends quadratically on the one-particle wavefunctions):

$$ \left(-\frac12 \nabla^2 + U_{ion}({\bf r})+ \sum_{j \neq i} \int d {\bf r'} \frac{|\psi_j ({\bf r'})|^2}{| {\bf r}- {\bf r'}|} \right) \psi_i( {\bf r})=\varepsilon_i \psi_i( {\bf r}). $$ Similar equations appear in the Hartree-Fock and Kohn-Sham approaches to Density Functional Theory.

In conclusion, although a fundamental quantum Hamiltonian must be a linear operator, important approximate schemes introduce non-linear Schrödinger-like equations. Taking into account the basic importance of Kohn-Sham approximations for applications, the whole issue cannot be considered a curiosity, but it is a pillar of modern computational methods for electronic properties.

Answer 2

Quantum mechanics depends on the Schrodinger equation being linear. It is perfectly possible to invent a model with a nonlinear Schrodinger equation describing the wavefunction and ask if it describes experiments well. However, such a model would not be quantum mechanics, but rather a modification of quantum mechanics.

A major obstruction to such a model is that it generically leads to faster than light propagation of information when relativity is taken into account, indicating that the theory is not causal. See, for example:

1. https://www.sciencedirect.com/science/article/abs/pii/037596019090786N

2. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.66.397

Answer 3

To add to the cases mentioned in other answers: Ginzbourg-Landau theory of superconductivity leads to a non-linear Schrödinger-like equation for the order parameter: $$ i\hbar\partial_t \Psi=\frac{1}{2m}\left(-i\hbar\nabla -2e\mathbf{A}\right)^2\Psi + \alpha\Psi +\beta|\Psi|^2\Psi $$ This equation is literally solved, e.g., when studying superconducting transition in specific geometries.

Answer 4

There is something called the nonlinear Schrödinger equation, but it does not describe the dynamics of a quantum wavefunction, which is always linear. Rather, the NLSE is a classical field equation.