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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?

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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.

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    $\begingroup$ @agaminon This is fine, so long as one as clear that the one particle wavefunctions $\psi_i(x)$ appearing in this equation are not the full quantum state $\Psi$. The Schrodinger equation must always be linear in $\Psi$. However, as GiorgioP says, you can approximate $\Psi$ using the $\psi_i$, so this is absolutely a reasonable thing to do. $\endgroup$
    – Andrew
    Commented Jul 12, 2022 at 21:20
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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

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    $\begingroup$ "A major obstruction to such a model is that it generically leads to faster than light propagation of information" I haven't read the two papers you linked, so maybe my comment is addressed there, but doesn't non-relativistic quantum mechanics also have this possibility? If you use the delta function as the initial condition for the linear Schrodinger equation for a non-relativistic particle, then the wavefunction in areas arbitrarily far away from the delta function will be affected instantly (if I'm not mistaken). $\endgroup$
    – Maximal Ideal
    Commented Jul 12, 2022 at 21:15
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    $\begingroup$ @MaximalIdeal Perhaps, but faster than light travel is not a fundamental issue in non-relativistic quantum mechanics. You could argue that anything that appears instantaneous in a non-relativistic theory would receive corrections in a relativistic theory that would make it causal. For example, information propagates instantaneously in Newton's "action at a distance" description of the gravitational field, but not in general relativity. $\endgroup$
    – Andrew
    Commented Jul 12, 2022 at 21:18
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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.

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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.

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  • $\begingroup$ I did find this wikipedia article when googling, but I figured it wasn't answering my question as you say. $\endgroup$
    – agaminon
    Commented Jul 12, 2022 at 20:23
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    $\begingroup$ I believe it does answer your question since you really ultimately asked if there was such an equation, the first few lines even detail exactly what physical situations one might find this equation. The article even gives you a form of the potential as $|\Psi|^2$ $\endgroup$
    – Triatticus
    Commented Jul 13, 2022 at 2:05