The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of quantum systems, such as particles, atoms, and molecules. It governs the evolution of the wave function, which encodes the probabilities of different states of a system.
While the Schrödinger equation is a key tool for understanding quantum mechanics, it is formulated within the framework of non-relativistic quantum mechanics, which does not take into account the effects of gravity or the structure of spacetime. It describes how quantum systems evolve in time, but it does not address the nature of gravity itself or the behavior of spacetime on a quantum level.
The need for a theory of quantum gravity arises when we try to combine the principles of quantum mechanics with those of general relativity, which is Einstein's theory of gravity. General relativity successfully describes gravity as the curvature of spacetime due to matter and energy. However, it is a classical theory and does not incorporate quantum effects.
On the other hand, quantum mechanics has been very successful in describing the behavior of particles and other quantum systems on small scales. It introduces probabilistic behavior, wave-particle duality, and uncertainty. However, when one tries to apply the principles of quantum mechanics to gravity, several conceptual and mathematical challenges arise.
In particular, the need for a theory of quantum gravity arises in situations where gravity and spacetime cannot be treated as classical, continuous entities. This is the case in extreme conditions such as the early universe near the Big Bang, or in the vicinity of black holes where gravitational effects are strong. At such scales, the discrete and probabilistic nature of quantum mechanics and the smooth and continuous nature of general relativity are expected to interact and have significant consequences.
A theory of quantum gravity is sought after to provide a consistent framework that incorporates both quantum mechanics and general relativity, allowing us to understand the behavior of spacetime and gravity at the quantum level. It is a frontier of theoretical physics that aims to address the fundamental nature of the universe and reconcile the seemingly disparate frameworks of quantum mechanics and general relativity. Such a theory would help us understand the behavior of the universe in extreme conditions and potentially shed light on questions about the early universe, the nature of black holes, and the fundamental structure of spacetime itself.