Schrodinger equation
In physics, especially quantum mechanics, the Schrodinger equation is an equation that describes how the quantum state of a physical system varies. According to the Copenhagen interpretation of quantum mechanics, the state vector is used to calculate the probability that a physical system is in a given quantum state. Schrodinger's equation is primarily applied to atomic and subatomic systems, such as electrons and atoms, but is sometimes applied to macroscopic systems (such as the whole universe). The equation is named after physicist Erwin Schrodinger who proposed the equation in 1926.
The Schrodinger equation is commonly written as an operator equation describing how the state vector evolves over time. By specifying the total energy (Hamiltonian) of the quantum system, Schrodinger's equation can be solved, the solutions being quantum states. To date, the only instance where an exact analytical (purely mathematical) solution to the unconfined Schrodinger equation has been found is in a single-electron system, e.g. the hydrogen atom. Numerical approximations can be found for all other systems, assuming the equation is confined and, if large, that there are boundary conditions that allow computation within the timespan of the universe.
The Schrodinger equation is of central importance in non-relativistic quantum mechanics, playing a role for subatomic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Subatomic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei.
The Schrodinger equation is a differential equation that represents the temporal evolution of a wavefunction, where the state vector varies with time, whilst the operator remains static. Although this is opposite in fashion to the Heisenberg picture, the two are in fact alternative mathematical formulations of the same underlying physical quantum-level reality, and both are correct. Whilst the Schrodinger equation separates space from time, and is thus best applied to non-relativistic problems, the Heisenberg picture for solving Quantum Mechanical problems does not distinguish between the two, making it more general in relativistic scope, although less useful in the study of non-relativistic quantum states.

# Historical background and development
Although it can't be derived from classical arguments, a heuristic derivation of Schrodinger's equation follows very naturally from earlier developments:
Assumptions:
- 1- The total energy E of a particle is
Note that the energy E and momentum p appear in the following two relations:
- 2- Einstein's light quanta hypothesis of 1905:
- 3- The de Broglie hypothesis of 1924:
- 4- The association of a wave (with wavefunction \psi) with any particle.
Combining the above assumptions yields Schrodinger's equation:
Expressing frequency f in terms of angular frequency \omega = 2\pi f\; and wavelength \lambda\, in terms of wavenumber k = 2\pi / \lambda\;, with \hbar = h / 2 \pi\; we get:
and
where we have expressed p and k as vectors.
Schrodinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:
and to realize that since
then
and similarly since:
then
and hence:
so that, again for a plane wave, he obtained:
And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrodinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V:
Using this equation, Schrodinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, \psi\;, moving in a potential well, V, created by the positively charged proton.  This computation tallied with experiment for the Lyman, Balmer, Paschen and Brackett series, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrodinger published his wave equation and the spectral analysis of hydrogen in a paper in 1926.
The Schrodinger equation defines the behaviour of \psi\;, but does not interpret what \psi\; is.  Schrodinger tried unsuccessfully to interpret it as a charge density.  In 1926 Max Born, just a few days after Schrodinger's fourth and final paper was published, successfully interpreted \psi\; as a probability amplitude, although Schrodinger was never reconciled to this statistical or probabilistic approach.

# Mathematical forms
There are various ways of writing Schrodinger's equation, depending on the precise mathematical framework used and whether the wavefunction varies over time.

## Time-dependent Schrodinger equation
In operator form, time-dependent Schrodinger equation for a system with total energy \hat H is,
\hat H \psi\left(\mathbf{r}, t\right) = i \hbar \frac{\partial \psi}{\partial t} \left(\mathbf{r}, t\right)
where \psi is the wavefunction, \hbar is Planck's constant and i is the imaginary unit. As with the force occurring in Newton's second law, the form of the Hamiltonian is not provided by the Schrodinger equation, but must be independently determined from the physical properties of the system.
As a standard example, a non-relativistic particle with no electric charge and zero spin has a Hamiltonian which is the sum of the kinetic (T) and potential (V) energies :
The Schrodinger equation can then be written explicitly as a partial differential equation
where the dependence of \psi on the space and time coordinates has been suppressed for clarity.

## Time-independent Schrodinger equation
For many real-world problems the Hamiltonian does not depend on time. Denoting this constant energy by E results in the time-independent Schrodinger equation
Together with Schrodinger's time-dependent equation in operator form, this gives,
which can be solved for \psi as
where \phi\left(\mathbf{r}\right) is the value of \psi at t=0. Also, when such a given solution - representing a stationary state - is substituted into the  time-dependent Schrodinger equation, the resulting equation is the time-independent Schrodinger equation.
An example of a one-dimensional time-independent Schrodinger equation for a chargeless, spinless particle of mass m, moving in a potential V(x) is: 
The analogous 3-dimensional time-independent equation is, :
where \nabla^2 is the Laplace operator.

## Bra-ket versions
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 (ket) in that space. 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 Schrodinger equation provides a quantitative description of the rate of change of the state vector.
The time-dependent Schrodinger equation can be written using Dirac's bra-ket notation as,
where |\psi(t)\rangle is a ket, \hbar is the reduced Planck's constant and \hat H(t) is the Hamiltonian (a  self-adjoint operator acting on the state space).
The nonzero 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 a wavefunction, 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).
For every time-independent Hamiltonian operator, \hat H, there exists a set of quantum states, \left|\psi_n\right\rang, known as energy eigenstates, and corresponding real numbers E_n satisfying the eigenvalue equation,
Alternatively, \psi is said to be an eigenstate (eigenket) of \hat H with eigenvalue E. Such a state possesses a definite total energy, whose value E_n is the eigenvalue of  the Hamiltonian. The corresponding eigenvector  \psi_n\, is normalizable to unity. This eigenvalue equation is referred to as the time-independent Schrodinger equation. 
We purposely left out the variable(s) on which the wavefunction \psi_n\, depends.
Self-adjoint operators, such as the Hamiltonian, have the property that their eigenvalues are always real numbers, as we would expect, since the energy is a physically observable quantity. Sometimes more than one linearly independent state vector correspond to the same energy E_n. If the maximum number of linearly independent eigenvectors corresponding to E_n equals k, we say that the energy level E_n is k-fold degenerate. When k=1 the energy level is called non-degenerate.
On inserting a solution of the time-independent Schrodinger equation into the full  Schrodinger equation, we get
It is relatively easy to solve this equation. One finds that the energy eigenstates
(i.e., solutions of the time-independent Schrodinger equation) change as a function of time only trivially, namely, only by a complex phase:
It immediately follows that the probability amplitude,
\psi(0)^*\psi(0)  = |\psi(0)|^2,
is time-independent. Because of a similar cancellation of phase factors in bra and ket, all  average (expectation) values of time-independent observables (physical quantities) computed from \psi(t)\, are  time-independent.
Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors   \left\{\left|n\right\rang\right\}  form a basis for the state space. We introduced here the short-hand notation
|\,n\,\rang = \psi_n.
Then any state vector that is a solution of the time-dependent  Schrodinger equation (with a time-independent  \hat H)   
 \left|\psi\left(t\right)\right\rang  can be written as a linear superposition of energy eigenstates:
(The last equation enforces the requirement that  \left|\psi\left(t\right)\right\rang ,  
like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the time-dependent Schrodinger equation in the left-hand side and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain
Therefore, if we know the decomposition of  \left|\psi\left(t\right)\right\rang  into the energy basis at time t = 0, its value at any subsequent time is given simply by
Note that when  some values c_n(0)\, are not equal to zero for 
differing energy values E_n\,, the left-hand side is not an eigenvector of
the energy operator  \hat H. The left-hand is an eigenvector when
the only c_n(0)\,-values not equal to zero belong the same energy, so that
\mathrm{e}^{-\mathrm{i}E_nt/\hbar} can be factored out. In many real-world
application this is the case and the  state vector \psi(t)\, (containing time only in its phase factor) is then a solution of the time-independent Schrodinger equation.
Let |\,1\,\rangle and  |\,2\,\rangle be degenerate eigenstates of the time-independent Hamiltonian \hat H\,:
\hat H\,|\,1\,\rangle = E |\,1\,\rangle \quad \hbox{and} \quad \hat H\,|\,2\,\rangle = E |\,2\,\rangle.
Suppose  a solution \psi(t)\, of the full (time-dependent) Schrodinger equation of \hat H\,
has the form at t = 0:
c_1 |\,1\,\rangle +
c_2 |\,2\,\rangle.
Hence, because of the discussion above, at t > 0 :
\mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 |\,1\,\rangle +
\mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 |\,2\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar}
which shows that \psi(t)\, only depends  on time in a trivial way (through its phase),
also in the case of degeneracy.
Apply now \hat H\,:
\hat H\,|\,\psi(t)\,\rangle = 
\mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 E\,|\,1\,\rangle +
\mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 E\, |\,2\,\rangle =
E\mathrm{e}^{-\mathrm{i}Et/\hbar} \left( c_1 |\,1\,\rangle +
= E \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle = E\,|\,\psi(t)\,\rangle.
Conclusion: The wavefunction \psi(t)\, with the given initial condition (its form at t = 0),  remains a solution of the time-independent Schrodinger equation 
\hat H\psi(t) = E\psi(t) for all times t > 0.

# Properties

## Linearity
The Schrodinger equation (in any form) is linear in the wavefunction, meaning that if \psi(x, t) and \phi(x,t) are solutions, then so is  a \psi +  b \phi, where a and b are any complex numbers. This property of the Schrodinger equation has important consequences.
Assumptions:
- The Schrodinger equation:
- |\psi\rangle and |\phi\rangle are solutions of the Schrodinger equation.

## Conservation of probability
In order to describe how probability density changes with time, we define a probability current or probability flux. The probability flux represents a flowing of probability across space.
For example, consider a Gaussian probability curve centered around x_0 with x_0 moving at speed v to the right. One may say that the probability is flowing towards the right, i.e., there is a probability flux directed to the right.
The probability flux \mathbf{j} is defined as:
and measured in units of (probability)/(area  time) = r2t1.
The probability flux satisfies the required continuity equation for a conserved quantity, i.e.:
where P\left(x, t\right) is the probability density and measured in units of (probability)/(volume) = r3.
This equation is the mathematical equivalent of the probability conservation law.
A standard calculation shows that for a plane wave described by the wavefunction,
the probability flux is given by
showing that not only is the probability of finding the particle in a plane wave state the same everywhere at all times, but also that it is moving at constant speed everywhere.

## Correspondence principle
The Schrodinger equation satisfies the correspondence principle.

# Solutions
Analytical solutions of the time-independent Schrodinger 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 can be found in the list of quantum mechanical systems with analytical solutions.
For many systems, however, there is no analytic solution to the Schrodinger equation.  In these cases, one must resort to approximate solutions.  Some of the common techniques are:
- Perturbation theory
- The variational principle underpins many approximate methods (like the popular Hartree-Fock method which is the basis of the post Hartree-Fock methods)
- Quantum Monte Carlo methods
- Density functional theory
- The WKB approximation
- Discrete delta-potential method

# Free particle Schrodinger equation
An important form of the Schrodinger equation results when the potential function for a single particle is zero:
The wave function can then be shown  to satisfy,
i.e. the particle is in a plane wave state.

# Relativistic generalisations
The Schrodinger equation does not take into account relativistic effects, meaning that the Schrodinger equation is invariant under a Galilean transformation, but not under a Lorentz transformation.
Relativistically valid generalisations incorporating ideas from special relativity include the Klein-Gordon equation and the Dirac equation.

# Applications