In physics, the Schrödinger equation, proposed 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. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... The word physicist should not be confused with physician, which means medical doctor. ... Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. ... 1925 was a common year starting on Thursday (link will take you to calendar). ... 8:17 am, August 6, 1945, Japanese time. ... Fig. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ...

In the mathematical formulation of quantum mechanics, each system is associated with a complex Hilbert space such that each instantaneous state of the 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. One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ...

Using Dirac's bra-ket notation, we denote that instantaneous state vector at time t by |ψ(t)〉. The Schrödinger equation is: Paul Adrien Maurice Dirac, (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

where i is the unit imaginary number, is Planck's constant divided by 2π, and the Hamiltonian H(t) is a self-adjoint operator acting on the state space. The Hamiltonian describes the total energy of the system. As with the force occurring in Newton's second law, its exact form is not provided by the Schrödinger equation, and must be independently determined based on the physical properties of the system. In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. ... Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In physics, a force acting on a body is that which causes the body to accelerate; that is, to change its velocity. ...

Time-independent Schrödinger equation

For every time-independent Hamiltonian H, there exist a set of quantum states, known as energy eigenstates, satisfying the eigenvalue equation

Such a state possesses a definite total energy, whose value E is the eigenvalue of the state vector with the Hamiltonian. This eigenvalue equation is referred to as the time-independent Schrödinger equation. 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. On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... The text or formatting below is generated by a template which has been proposed for deletion. ...

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

.

It is easy to solve this equation. One finds that the state vectors of the energy eigenstates change by only a complex phase: Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ...

Energy eigenstates are convenient to work with because their time-dependence is so simple; that is why the time-independent Schrödinger equation is so useful. We can always choose a set of instantaneous energy eigenstates whose state vectors {|n>} form a basis for the state space. Then any state vector |ψ(t)〉 can be written as a linear superposition of energy eigenstates: In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

(The last equation enforces the requirement that |ψ(t)〉, like all state vectors, must be a unit vector.) Applying the Schrödinger equation to each side of the first equation, and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

Therefore, if we know the decomposition of |ψ(t)〉 into the energy basis at time t = 0, its value at any subsequent time is given simply by

Schrödinger wave equation

The state space of certain 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 that depends on position as well as time. This form of the Schrödinger equation is referred to as the Schrödinger wave equation. In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics and physics, a scalar field associates a single number (or scalar) to every point in space. ...

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.

The wavefunction

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

Since the position eigenstates form a basis for the state space, the integral over all projection operators is the identity operator: An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

This statement is called the resolution of the identity. With this, and the fact that kets have unit norm, we can show that

where ψ(r, t)* denotes the complex conjugate of ψ(r, t). This important result tells us that the absolute square of the wavefunction, integrated over all space, must be equal to 1:

We can thus interpret the absolute square of the wavefunction as the probability density for the particle to be found at each point in space. In other words, |ψ(r, t)|² d³r is the probability, at time t, of finding the particle in the infinitesimal region of volume d³r surrounding the position r.

We have previously shown that energy eigenstates vary only by a complex phase as time progresses. Therefore, the absolute square of their wavefunctions do not change with time. Energy eigenstates thus correspond to static probability distributions.

Operators in the position basis

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

The operators A on the two sides of the equation are different things: the one on the right acts on kets, whereas the one of the left acts on scalar fields. It is common to use the same symbols to denote operators acting on kets and their projections onto a basis. Usually, the kind of operator to which one is referring is apparent from the context, but this is a possible source of confusion.

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

This form of the Schrödinger equation is the Schrödinger wave equation. It may appear that this is an ordinary differential equation, but in fact the Hamiltonian operator typically includes partial derivatives with respect to the position variable r. This usually leaves us with a difficult linear partial differential equation to solve. In mathematics, and in particular analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. ... The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...

Non-relativistic Schrödinger wave equation

In non-relativistic quantum mechanics, the Hamiltonian of a particle 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 operator is Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... Kinetic energy (also called vis viva, or living force) is energy possessed by a body by virtue of its motion. ... Potential energy (U, or Ep), a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Electric charge is a fundamental FATTY STASHEconserved property of some subatomic particles, which determines their electromagnetic interactions. ... In physics, spin is an intrinsic angular momentum associated with microscopic particles. ...

where p is the momentum operator, defined as In physics, momentum is a physical quantity related to the velocity and mass of an object. ...

The potential energy operator is

where V is a real scalar function of the position operator r. Putting these together, we obtain The text or formatting below is generated by a template which has been proposed for deletion. ... The concept of a scalar is used in mathematics, physics, and computing. ...

where ² is the Laplacian. This is a commonly encountered form of the Schrödinger wave equation, though not the most general one. Del symbol (also known as nabla; used in mathematical physics). ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

The corresponding time-independent equation is

The relativistic generalisations of this wave equation are the Dirac equation, Klein-Gordon equation, Proca equation, Maxwell equations etc, depending on spin and mass of the particle. See Relativistic_wave_equations for details. The Dirac equation is a relativistic quantum mechanical wave equation invented by Paul Dirac in 1928. ... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is a relativistic version (describing scalar (or pseudoscalar) spinless particles) of the Schrödinger equation. ... In field theory, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... Spin has several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the periodical, see Spin Magazine Computer: For unproductive repetition in a computer program, see spin (software) For finding bugs in multi-threaded code, see SPIN model checker For the computer... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Before the creation of quantum field theory, physicists attempted to formulate versions of the Schrödinger equation which were compatible with special relativity. ...

Probability currents

In order to describe how probability density changes with time, it is acceptable to define probability current or probability flux. The probability flux represents a flowing of probability across space. In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ...

For example, consider a Gaussian probability curve centered around x₀, imagine that x₀ moving in a speed v toward the right. Then one may say that the probability is flowing toward right, i.e., there is a probability flux directed to the right. GAUSSIAN is a computational chemistry software program. ...

The probability flux j is defined as:

and measured in units of (probability)/(area × time) = r⁻²t⁻¹.

The probability flux satisfy a quantum continuity equation, i.e.: Note that all the examples given below express the same idea (i. ...

where P(x, t) is the probability density and measured in units of (probability)/(volume) = r⁻³. This equation is the mathematical equivalent of probability conservation law. In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... The word probability derives from the Latin probare (to prove, or to test). ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

It is easy to show that for a plane wave, In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ...

the probability flux is given by

.

Solutions of the Schrödinger equation

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: Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

• The free particle
• The particle in a box
• The particle in a ring
• The particle in a spherically symmetric potential
• The quantum harmonic oscillator
• The hydrogen atom
• The ring wave guide
• The particle in a one-dimensional lattice (periodic potential)

For many systems, however, there is no analytic solution to the Schrödinger equation. In these cases, one must resort to approximate solutions: In physics a free particle is a particle that is never under the influence of an external force Classical Free Particle The classical free particle is characterized simply by a fixed velocity. ... In physics, the particle in a box (or the square well) is a simple idealized system that can be completely solved within quantum mechanics. ... In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. ... The particle in a spherically symmetric potential describes the dynamics of a particle in a central force field, i. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... A hydrogen atom is an atom of the element hydrogen. ... In quantum mechanics, the ring wave guide starts from the one dimensional, time independent Schrödinger equation: This must be solved under the circularity condition. ... In quantum mechanics, the particle in a one-dimensional lattice is an idealised system that can be solved completely with some simplifications. ...

• Perturbation theory
• The variational principle underpin most approximate methods
• Hartree-Fock solutions are some of the simplest solutions to real situations
• Quantum Monte Carlo methods
• Vibronic coupling methods

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ... A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ... In computational physics, the Hartree-Fock calculation scheme is a self-consistent iterative procedure to calculate the so-called best possible single determinant solution to the time-independent Schrödinger equation of a many-electron system in a Coulombic potential of fixed nuclei. ... Monte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. ... Vibronic coupling is a branch of theoretical chemistry which deals with the interactions between electronic and nuclear motions of molecules. ...

References

• E. Schrödinger, Phys. Rev. 28 1049 (1926)

External links

• Linear Schrödinger Equation (http://eqworld.ipmnet.ru/en/solutions/lpde/lpde108.pdf) at EqWorld: The World of Mathematical Equations.

• Nonlinear Schrödinger Equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde1403.pdf) at EqWorld: The World of Mathematical Equations.