In physics, the **Schrödinger equation**, developed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to Newton's second law in classical mechanics.

In quantum mechanics, the set of all possible states of a system is described by a complex Hilbert space, and any instantaneous state of a system is described by a unit vector in that space. This state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. 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.

Using Dirac's bra-ket notation, we denote that instantaneous state vector at time *t* by |ψ(*t*)⟩. The Schrödinger equation is:

For more information on the role of operators in quantum mechanics, see mathematical formulation of quantum mechanics.

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2 The position basis 3 Solutions of the Schrödinger equation |

On inserting the time-independent Schrödinger equation into the full Schrödinger equation,

The state space of many (but not all) quantum systems can be spanned with a **position basis**. In this situation, the Schrödinger equation may be conveniently reformulated as a partial differential equation for a **wavefunction**, a complex scalar field depending on position as well as time. This form of the Schrödinger equation is referred to as the **Schrödinger wave equation**.

Elements of the position basis are called position eigenstates. We will consider only a single-particle system, for which each position eigenstate may be denoted by |**r**⟩, where the label **r** is a real vector. This is to be interpreted as a state in which the particle is localized at position **r**. In this case, the state space is the space of all square-integrable complex functions.

We define the wavefunction as the *projection* of the state vector |ψ(*t*)⟩ onto the position basis:

We have previously shown that energy eigenstates vary only by a complex phase as time progresses. Therefore, the absolute square of their wavefunctions do note change with time. Energy eigenstates thus correspond to static probability distributions.

Any operator *A* acting on the wavefunction is defined in the position basis by

Using the position-basis notation, the Schrödinger equation can be written in the position basis as:

This form of the Schrödinger equation is the
Often, the Hamiltonian can be expressed as the sum of two operators, one corresponding to kinetic energy and the other to potential energy. For a single particle of mass *m* with no electric charge and no spin, the kinetic energy term can be written as

Analytical solutions of the time-independent Schrödinger equation can be obtained for a variety of relatively simple conditions. These solutions provide insight into the nature of quantum phenomena and sometimes provide a reasonable approximation of the behavior of more complex systems (e.g., in statistical mechanics, molecular vibrations are often approximated as harmonic oscillators). Several of the more common analytical solutions include:

- The particle in a box
- The quantum harmonic oscillator
- The hydrogen atom
- The ring wave guide