Why do people say the dynamics of quantum mechanics is always linear?

Question

This statement seems false. An example of a non-linear equation governing the dynamics of a quantum system is the Gross-Pitaevskii equation.

Answer 1

The Gross-Pitaevskii equation is a very good approximate (or "effective") theory for an interacting Bose Einstein Condensate, and invokes a strong assumption that all particles occupy the same wave function in the lowest energy state. This assumption is well-motivated, but it isn't 100% accurate: in the actual ground state, bosonic particles might occupy strongly overlapping but ultimately non-identical orbitals, and positing a single monolithic state inhabited by all particles ignores this very real possibility. Quantum mechanics is referred to as linear (in spite of the complexity of solving many-body systems) because the starting point in non-relativistic mechanics is always a Schroedinger equation of some sort (even in density functional theory), and the Schroedinger equation is always linear.

Answer 2

The linearity of quantum comes from The Principle of Superposition and is one of the most fundamental ideas in Quantum Theory.

In the book "The Principles of Quantum Theory" chapter I, Paul Dirac argue about this principle. After giving some examples of experimental situations that can't be explained by Classical Mechanics, he says the following (ch I, section 4 pg 11)

There remains an overall criticism that one may make to the whole scheme, namely, that in departing from determinacy of the classical theory a great complication is introduced into the description of Nature, which is a undesirable feature. This complication is undeniable, but it is offset by a great simplification, provided by The general Principle of Superposition of states, which we shall now go on to consider.

We could conclude two things:

• As a principle, it can't be "derived" or concluded as a consequence of previous ideas. We accept it as a feature of the new scheme, in the same basis that we accept to depart from the determinacy of the classical theory.
• it comes as a simplification. We are not searching for the most complicated description of Nature, but the most accurate one. There would be no reason, for a matter of mathematical generalization, to consider a more complicated scheme, if this simple one already explain the natural phenomena.

Dirac mentions the Principle of Superposition before to define the mathematical structure of the theory and even before the definition of state.

Answer 3

Physicists believe that quantum mechanics is perfectly linear. But when you apply assumptions, or leave out parts of a system it might appear nonlinear. So if you were to describe the entire universe using the ket $|\Psi\rangle$, the universe would evolve according to $$i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle,$$ where $\hat H$ is a linear operator.

You might remember the time energy was first introduced to you in school. Energy is always conserved? Why does everything seem to slow down to a halt when left alone? Why is it possible to discharge a battery without doing any useful work? Although energy is always conserved$^\dagger$, in many systems it appears to be lost because it enteres the microscopic degrees of freedom. In those cases it is often simpler to act as if the energy is lost. Similarly, it is sometimes simpler to treat quantum mechanics as nonlinear.

$\dagger$ (except sometimes in general relativity)

Answer 4

The linear nature of quantum mechanics enters in the actual formulation of the theory.

It isn't so much that this is an arbtrary decision, but is really enshrined in postulates of the theory itself. There are different presentations of those postulates but the key point is:

Postulates are hypotheses that cannot be proven. If no discrepencies are found in nature then the postulate becomes an axiom...

These postulates and axioms are being challenged and tested regularly with results of experimentation regarding the possibility of nonlinearities in electromagnetism being updated as recently as this month November 2024.

Nonlinearity does in fact enter into the mix when dealing with practical open quantum systems where unitary evolution is not preserved. Most commonly one sees this in the application of a version of the Linblad Master Equation that is sometimes called the NonLinear Linblad Equation

So in short, quantum mechanics is by definition linear. Whether all of nature is quantum is a long standing argument. For practical applications the rigidness of linearity as a constraint is relaxed particularly when it is not practical to fully define a closed qauntum system.

Answer 5

People(even high credentialed ones) say false things all the time. Gross-Pitaevskii is an approximate replacement of the Schr. equation.The statement you ask about is implicitly about QT in general (the fundamental laws). It means time evolution of $\psi$ is given always by linear equation in $\psi$. It is true the Schroedinger equation (or similar fundamental equation giving evolution in time) is linear in $\psi$. But of course, time evolution in quantum theory is more complicated. Consistent account of single system before and after measurement has to involve also reduction of system's $\psi$ (projection, collapse), which isn't a linear process. This means $\psi$ across measurement can't be described by the linear Schroedinger equation. So measurement is special process, and we have no good mathematical description for it (the measurement problem), only the projection postulate, defining the appropriate final $\psi$ based on the result obtained from observation. So the statement that the whole theory is linear is not really true.

So you can tell, if the person is claiming whole QT is perfectly linear, they either aren't aware of the measurement problem, or they are, but believe it is solved by doubling down on the linear evolution, usually given by the Schr. equation. This really isn't possible, but some people claim it is, if we believe there are many worlds and $\psi$ and Schr. equation describe all these worlds at once. But they do not seem to be able to explain (so that other agree with them) why experiments have definite results implying non-linear evolution of $\psi$. E.g. those obtained in a measurement of spin component of silver atom in the SG-type experiment. There isn't a good explanation for how linear evolution of $\Psi$ of the whole super-system makes the effective $\psi$ evolve as if projection of $\psi$ happens (which is a nonlinear operation).

Answer 6

In general, this statement is incorrect. Quantum mechanics predicts the time evolution of states in a probabilistic way, and as such it has to be unitary. However, on infinite dimensional spaces unitary operators are non linear unless the Hamiltonian is bounded a self-adjoint operator. A more precise statement is to say that the Schrodinger equation predicts a linear regime. Mathematically speaking, the property of linearity at a given time $t$ holds if

$$\frac{d}{dt}\left(\left|\psi(t)\right\rangle +\left|\phi(t)\right\rangle \right)\,=\,\frac{d}{dt}\left|\psi(t)\right\rangle +\frac{d}{dt}\left|\phi(t)\right\rangle ,$$ for states $\left|\psi(t)\right\rangle ,\left|\phi(t)\right\rangle \in\mathcal{H}$. The linearity rule above works only when dealing with states that satisfy

$$\left\Vert \frac{d}{dt}\left|\psi(t)\right\rangle \right\Vert \,<\,M\left\Vert \left|\psi(t)\right\rangle \right\Vert ,$$ with $M\in\mathbb{R}$. The states $\left|\psi\right\rangle $ not satisfying the boundness condition are outside of the scope of the linear regime.

In order to make things more concrete and address the comment below, one can look at the Kronig–Penney potential for a particle in a one-dimensional lattice, $$V(x)\,=\,A\sum_{n=-\infty}^{\infty}\delta(x-na).$$ The corresponding time evolution operator, $e^{-iHt}$, has an indefinite meaning for states $\left|x\right\rangle $ with singular position eigenvalue of $x=na$, so the condition $\left\Vert \frac{d}{dt}e^{-iHt}\left|x_{0}\right\rangle \right\Vert <M\left\Vert \left|x_{0}\right\rangle \right\Vert $ is not satisfied. In other words, we are getting outside of Hilbert space formalism for any given time $t>0$. Since these measure-0 "holes" are outside of the domain of $H$ one cannot rely on Stone's theorem to guarantee unitarity there, and also linearity broken as well.