You are correct that the Schrödinger equation, which is the fundamental equation of quantum mechanics, is linear. Linearity implies that the superposition principle holds, meaning that if you have two solutions to the equation, their sum is also a valid solution. This linearity is one of the key aspects that differentiates quantum mechanics from classical mechanics.
However, chaos in quantum systems can still arise due to the interplay of several factors:
Many-particle systems: Quantum systems involving multiple particles can exhibit chaotic behavior, even though the underlying Schrödinger equation is linear. Interactions between particles can lead to complex and unpredictable dynamics, especially when the number of particles is large.
Quantum chaos: Quantum systems that are classically chaotic can still exhibit chaotic behavior in their quantum description. This field, known as quantum chaos, studies the behavior of quantum systems whose classical counterparts display chaos. Quantum chaos typically manifests as statistical properties of energy levels or wavefunctions rather than in the individual trajectories of particles.
Decoherence: When a quantum system interacts with its environment, a process called decoherence occurs. Decoherence leads to the loss of quantum coherence and the emergence of classical-like behavior. In complex systems with many degrees of freedom, the interactions with the environment can result in chaotic behavior.
It's important to note that the concept of chaos in quantum systems is distinct from classical chaos. In classical chaos, small perturbations in initial conditions can lead to drastically different outcomes. In quantum systems, on the other hand, the superposition principle restricts the growth of such sensitivity to initial conditions.
Quantum chaos is an active area of research, and understanding the interplay between quantum mechanics and chaos remains an intriguing and challenging topic within the field of theoretical physics.