Schrödinger equation

It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time.

The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system.

Paul Dirac incorporated special relativity and quantum mechanics into a single formulation that simplifies to the Schrödinger equation in the non-relativistic limit.

Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only the concepts and notations of basic calculus, particularly derivatives with respect to space and time.

[5]: 10 Broadening beyond this simple case, the mathematical formulation of quantum mechanics developed by Paul Dirac,[6] David Hilbert,[7] John von Neumann,[8] and Hermann Weyl[9] defines the state of a quantum mechanical system to be a vector

This vector is postulated to be normalized under the Hilbert space's inner product, that is, in Dirac notation it obeys

[5]: 322 Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space.

The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:[12]: 143

[5]: 103–104  In solid-state physics, the Schrödinger equation is often written for functions of momentum, as Bloch's theorem ensures the periodic crystal lattice potential couples

Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts[19]

A solution of this type is called stationary, since the only time dependence is a phase factor that cancels when the probability density is calculated via the Born rule.

This is an example of the spectral theorem, and in a finite-dimensional state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.

Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.

[23] The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized.

For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories.

The set of all density matrices is convex, and the extreme points are the operators that project onto vectors in the Hilbert space.

For one reason, it is essentially invariant under Galilean transformations, which form the symmetry group of Newtonian dynamics.

Taking the "square root" of the left-hand side of the Klein–Gordon equation in this way required factorizing it into a product of two operators, which Dirac wrote using 4 × 4 matrices

Also, the solutions to a relativistic wave equation, for a massive particle of spin s, are complex-valued 2(2s + 1)-component spinor fields.

Heuristically, this complication can be motivated by noting that mass–energy equivalence implies material particles can be created from energy.

Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron.

He was guided by William Rowan Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system—the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.

Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl[45]: 3 ) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.

[47]: 220  Later, Schrödinger himself explained this interpretation as follows:[50] The already ... mentioned psi-function.... is now the means for predicting probability of measurement results.

In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog.The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time.

The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reality depends upon the interpretation of quantum mechanics that one adopts.

The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule.

[59] This interpretation removes the axiom of wave function collapse, leaving only continuous evolution under the Schrödinger equation, and so all possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition.

It attributes to each physical system not only a wave function but in addition a real position that evolves deterministically under a nonlocal guiding equation.