In quantum mechanics, the energy of a particle is described by its energy spectrum, which consists of discrete energy levels or states. These energy levels are determined by the specific quantum mechanical system under consideration, such as an atom or a particle in a potential well.
In most cases, the energy spectrum of a quantum mechanical system is bounded from below, meaning there is a lowest energy level called the ground state. This ground state represents the lowest possible energy that the system can have. According to the principles of quantum mechanics, energy is quantized, meaning it can only take on certain discrete values. These discrete energy levels are typically positive or zero, but they cannot be negative.
The reason for this limitation lies in the mathematical formulation of quantum mechanics. The Schrödinger equation, which describes the behavior of quantum systems, is a differential equation that involves the energy of the system. Solving the Schrödinger equation yields a set of possible energy eigenvalues, and these eigenvalues are typically positive or zero.
From a physical perspective, negative energy states would have peculiar consequences. For example, particles with negative energy could lower their energy indefinitely by emitting energy and reaching even lower energy states, violating the conservation of energy. This would lead to instability and inconsistencies within the framework of quantum mechanics.
However, it's worth noting that in certain theoretical contexts, such as the Dirac equation for relativistic particles, negative energy states do arise. In these cases, the interpretation of negative energy states is different, and they are typically associated with antimatter. These theories incorporate the concept of antiparticles, which have negative energy compared to the corresponding positive-energy particles.
In summary, within the standard framework of non-relativistic quantum mechanics, particles cannot have negative or zero energy due to the mathematical formulation of the theory and the physical consequences that would arise from allowing such states. However, in certain relativistic theories, negative energy states are incorporated to describe antimatter.