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We have so many experiments on quantum systems; many of those regard superposition principle; tests of probabilities; entanglement; quantum communication protocols; and others are related to properties of the energy eigenstates (eg. the spectrum of hydrogen atom, particle in a box, etc.)

My question is an open one: how do we test the left hand side of schrodinger equation? Say, for instance, that you add some non linear time derivative, or maybe higher order time derivative of the wavefunction into a modified Schrödinger equation. Imagine for instance $$ \epsilon \partial_t^2 \psi + i \partial_t \psi = H \psi. $$

What experiment would help to put constraints or refute such possibilities? Have this been done? I'm aware of the Weinberg paper on non-linear QM, but it doesn't deal with the time derivative if I remember it correctly.

I'm thinking about: Larmor precession, nuclear magnetic resonance, maybe time-of-flights?

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It sounds like you are asking for confirmation/refutation of the time evolution of the system. There's a new manuscript on the arxiv that watches Gaussian wave functions evolve into broader Gaussian wave functions by modifying the intensity of an optical trap (https://arxiv.org/abs/2404.05699). They pretty explicitly state that it is textbook Schrödinger equation dynamics. This work would rule out some hypothetical non-linear terms. The details depend on the limits of experimental resolution and any proposed non-linear model.

As a weaker example, the Stern-Gerlach experiment also relies on the separation of the centers of mass of the spin-up and spin-down halves of the wave function.

Likely to be less of an interest to you, there are also steady-state cases of time evolution, i.e., stationary states. I would take take an x-ray or electron crystallography-based reconstruction of a solid's electron density to be some evidence for a wave function in a stationary state. Additionally, the group associated with this PhysicsWorld article has published many articles that image stationary states of atoms.

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There are many experiments trying to test various fundamental assumptions and properties of quantum mechanics.

I am not aware of any direct exclusion experiments of the extension proposed in the OP, but it goes a bit in the direction of nonlinear quantum mechanics, which has been well excluded. As an overview, I recommend the introduction of the following thesis: https://link.springer.com/book/10.1007/978-3-031-04454-0. Its main focus is the testing of Born’s rule as an axiom of quantum mechanics, but the introduction provides a very thorough background on other topics as well.

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