Schrödinger equation describes evolution of wave function from "summary" of The Principles of Quantum Mechanics by P. A. M. Dirac
The Schrödinger equation is the fundamental equation of quantum mechanics. It describes how the state of a system evolves over time. In particular, it specifies how the wave function of a system changes in response to the system's Hamiltonian operator. The Hamiltonian operator represents the total energy of the system, including both kinetic and potential energy. The Schrödinger equation is a partial differential equation that is typically written in the form of an eigenvalue equation. In this form, the equation relates the Hamiltonian operator to the wave function of the system. The eigenvalues of the Hamiltonian operator correspond to the energy levels of the system, and the eigenfunctions are the possible wave functions of the system. The time-dependent Schrödinger equation describes how the wave function evolves over time. It specifies how the wave function changes with time by relating the time derivative of the wave function to the Hamiltonian operator acting on the wave function. This equation allows us to predict the future state of the system based on its current state. The time-independent Schrödinger equation, on the other hand, describes the stationary states of the system. These states do not change with time and correspond to the energy eigenstates of the system. The solutions to the time-independent Schrödinger equation are the wave functions that represent these stationary states. In summary, the Schrödinger equation is a powerful tool in quantum mechanics that allows us to describe the evolution of the wave function of a system. By solving the Schrödinger equation, we can predict how a quantum system will behave over time and understand its stationary states. This equation forms the foundation of quantum mechanics and is essential for understanding the behavior of particles at the microscopic level.
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