Schrödinger equation
In physics, especially quantum mechanics, the Schrödinger 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. Schrödinger'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 Schrödinger who proposed the equation in 1926.^{[1]}
The Schrödinger 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, Schrödinger'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 Schrödinger equation has been found is in a singleelectron 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 Schrödinger equation is of central importance in nonrelativistic 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 Schrödinger 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 quantumlevel reality, and both are correct. Whilst the Schrödinger equation separates space from time, and is thus best applied to nonrelativistic 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 nonrelativistic quantum states.
Historical background and development
Although it can't be derived from classical arguments, a heuristic derivation of Schrödinger's equation follows very naturally from earlier developments:
Assumptions:
 1 The total energy E of a particle is

 This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, , and the potential energy V. p is the momentum of the particle or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well.

Note that the energy E and momentum p appear in the following two relations:
 2 Einstein's light quanta hypothesis of 1905:

 where the frequency f of the quanta of radiation (photons) are related by Planck's constant h.

 3 The de Broglie hypothesis of 1924:

 where is the wavelength of the wave. This hypothesis also requires:

 4 The association of a wave (with wavefunction ) with any particle.
Combining the above assumptions yields Schrödinger's equation:
Expressing frequency f in terms of angular frequency and wavelength in terms of wavenumber , with we get:
and
where we have expressed p and k as vectors.
Schrödinger'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 Schrödinger's famed equation for a single particle in the 3dimensional case in the presence of a potential V:
Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, , 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 noncommuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a paper in 1926.^{[2]}
The Schrödinger equation defines the behaviour of , but does not interpret what is. Schrödinger tried unsuccessfully to interpret it as a charge density.^{[citation needed]} In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted as a probability amplitude,^{[citation needed]} although Schrödinger was never reconciled to this statistical or probabilistic approach.^{[citation needed]}
Mathematical forms
There are various ways of writing Schrödinger's equation, depending on the precise mathematical framework used and whether the wavefunction varies over time.
Timedependent Schrödinger equation
In operator form, timedependent Schrödinger equation for a system with total energy is,
where is the wavefunction, is Planck's constant and is the imaginary unit. As with the force occurring in Newton's second law, the form of the Hamiltonian is not provided by the Schrödinger equation, but must be independently determined from the physical properties of the system.
As a standard example, a nonrelativistic particle with no electric charge and zero spin has a Hamiltonian which is the sum of the kinetic (T) and potential (V) energies :
The Schrödinger equation can then be written explicitly as a partial differential equation
where the dependence of on the space and time coordinates has been suppressed for clarity.
Timeindependent Schrödinger equation
For many realworld problems the Hamiltonian does not depend on time. Denoting this constant energy by results in the timeindependent Schrödinger equation
Together with Schrödinger's timedependent equation in operator form, this gives,
which can be solved for as
where is the value of at . Also, when such a given solution  representing a stationary state  is substituted into the timedependent Schrödinger equation, the resulting equation is^{[3]} the timeindependent Schrödinger equation.
An example of a onedimensional timeindependent Schrödinger equation for a chargeless, spinless particle of mass m, moving in a potential V(x) is: [1]
The analogous 3dimensional timeindependent equation is, [2]:
where is the Laplace operator.
Braket 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 Schrödinger equation provides a quantitative description of the rate of change of the state vector.
The timedependent Schrödinger equation can be written using Dirac's braket notation as,
where is a ket, is the reduced Planck's constant and is the Hamiltonian (a selfadjoint 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 timeindependent Hamiltonian operator, , there exists a set of quantum states, , known as energy eigenstates, and corresponding real numbers satisfying the eigenvalue equation,
Alternatively, is said to be an eigenstate (eigenket) of with eigenvalue . Such a state possesses a definite total energy, whose value is the eigenvalue of the Hamiltonian. The corresponding eigenvector is normalizable to unity. This eigenvalue equation is referred to as the timeindependent Schrödinger equation. We purposely left out the variable(s) on which the wavefunction depends.
Selfadjoint 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 . If the maximum number of linearly independent eigenvectors corresponding to equals k, we say that the energy level is kfold degenerate. When k=1 the energy level is called nondegenerate.
On inserting a solution of the timeindependent Schrödinger equation into the full Schrödinger equation, we get
It is relatively easy to solve this equation. One finds that the energy eigenstates (i.e., solutions of the timeindependent Schrödinger equation) change as a function of time only trivially, namely, only by a complex phase:
It immediately follows that the probability amplitude,
is timeindependent. Because of a similar cancellation of phase factors in bra and ket, all average (expectation) values of timeindependent observables (physical quantities) computed from are timeindependent.
Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors form a basis for the state space. We introduced here the shorthand notation . Then any state vector that is a solution of the timedependent Schrödinger equation (with a timeindependent ) can be written as a linear superposition of energy eigenstates:
(The last equation enforces the requirement that , like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the timedependent Schrödinger equation in the lefthand 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 into the energy basis at time , its value at any subsequent time is given simply by
Note that when some values are not equal to zero for differing energy values , the lefthand side is not an eigenvector of the energy operator . The lefthand is an eigenvector when the only values not equal to zero belong the same energy, so that can be factored out. In many realworld application this is the case and the state vector (containing time only in its phase factor) is then a solution of the timeindependent Schrödinger equation.
Let and be degenerate eigenstates of the timeindependent Hamiltonian :
Suppose a solution of the full (timedependent) Schrödinger equation of has the form at t = 0:
Hence, because of the discussion above, at t > 0 :
which shows that only depends on time in a trivial way (through its phase), also in the case of degeneracy.
Apply now :
Conclusion: The wavefunction with the given initial condition (its form at t = 0), remains a solution of the timeindependent Schrödinger equation for all times t > 0.
Properties
Linearity
The Schrödinger equation (in any form) is linear in the wavefunction, meaning that if and are solutions, then so is , where a and b are any complex numbers. This property of the Schrödinger equation has important consequences.
Assumptions:
 The Schrödinger equation:
 and are solutions of the Schrödinger equation.
 (as the Hamiltonian is a linear operator)
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 with moving at speed 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 is defined as:
and measured in units of (probability)/(area × time) = r^{−2}t^{−1}.
The probability flux satisfies the required continuity equation for a conserved quantity, i.e.:
where is the probability density and measured in units of (probability)/(volume) = r^{−3}. 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 Schrödinger equation satisfies the correspondence principle.
Solutions
Analytical solutions of the timeindependent 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 can be found in the list of quantum mechanical systems with analytical solutions.
For many systems, however, there is no analytic solution to the Schrödinger 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 HartreeFock method which is the basis of the post HartreeFock methods)
 Quantum Monte Carlo methods
 Density functional theory
 The WKB approximation
 Discrete deltapotential method
Free particle Schrödinger equation
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An important form of the Schrödinger equation results when the potential function for a single particle is zero:
The wave function can then be shown ^{[3]} to satisfy,
i.e. the particle is in a plane wave state.
Relativistic generalisations
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The Schrödinger equation does not take into account relativistic effects, meaning that the Schrödinger equation is invariant under a Galilean transformation, but not under a Lorentz transformation.
Relativistically valid generalisations incorporating ideas from special relativity include the KleinGordon equation and the Dirac equation.
Applications
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See also
 Basic quantum mechanics
 Dirac equation
 KleinGordon equation
 Pauli equation
 Quantum number
 Schrödinger's cat
 Schrödinger field
 Schrödinger picture
 Theoretical and experimental justification for the Schrödinger equation
References
 ↑ Schrödinger, Erwin (December 1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules" (PDF). Phys. Rev. 28 (6): 1049–1070.
 ↑ Erwin Schrödinger, Annalen der Physik, (Leipzig) (1926), Main paper
 ↑ An initial condition must be used here, namely, that at time zero the wavefunction must be an eigenstate of
Modern reviews
 David J. Griffiths (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.
External links
 Linear Schrödinger Equation at EqWorld: The World of Mathematical Equations.
 Nonlinear Schrödinger Equation at EqWorld: The World of Mathematical Equations.
 The Schrödinger Equation in One Dimension as well as the directory of the book.
 Mathematical aspects of Schrödinger equation's are discussed on the Dispersive PDE Wiki.
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