In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei. Macroscopic particles vary in mass from small dust particles to heavy planets.
In the mathematical formulation of quantum mechanics, a physical system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a ray (a one-dimensional subspace) in that space. The elements of a Hilbert space are by definition normalizable and it is convenient, although not necessary, to represent a state by an element of the ray which is normalized to unity. This vector is often somewhat loosely referred to as wave function, although in a more rigorous formulation of quantum mechanics a wavefunction is a special case of a state vector. (In fact, a wavefunction is a state in the position representation, see below). A state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. It contains all information of the system that is knowable in a quantum mechanical sense. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.

In Dirac's bra-ket notation at time t the state is given by the ket |\psi(t)\rangle. The time-dependent Schrödinger equation, giving the time evolution of the ket, is:

    H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{d}{d t} \left| \psi \left(t\right) \right\rangle

where i is the imaginary unit, t is time, d / dt is the derivative with respect to t, \hbar is the reduced Planck's constant (Planck's constant divided by 2π), ψ(t) is the time dependent state vector, and H(t) is the Hamiltonian (a self-adjoint operator acting on the state space). If one assumes a certain representation for ψ, for instance position or momentum representation, the state vector is assumed to depend on more variables than time alone, and the time derivative must be replaced by the partial derivative \partial / \partial t.

The Hamiltonian describes the total energy of the system. As with the force occurring in Newton's second law, its form is not provided by the Schrödinger equation, but must be independently determined from the physical properties of the system.
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