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Quantum mechanics
\Delta x \, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction to...

Mathematical formulation of...

Equations
Schrödinger equation
Pauli equation
Klein–Gordon equation
Dirac equation
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The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system The Pauli Equation, also known as the Schrödinger-Pauli equation is the formulation of the Schrödinger equation for spin one-half particles which takes into account In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system It was named after Oskar Klein and Walter Gordon. Oskar Benjamin Klein ( September 15 1894 - February 5 1977) was a Swedish theoretical Physicist.

Contents

Details

The Schrödinger equation for a free particle is

\frac{\mathbf{p}^2}{2m} \psi = i \hbar \frac{\partial}{\partial t}\psi

where

\mathbf{p} = -i \hbar \mathbf{\nabla} is the momentum operator (\nabla being the del operator). In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product &nablaDel

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special relativity. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

It is natural to try to use the identity from special relativity

E = \sqrt{\mathbf{p}^2 c^2 + m^2 c^4}

for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

\sqrt{(-i\hbar\mathbf{\nabla})^2 c^2 + m^2 c^4} \psi= i \hbar \frac{\partial}{\partial t}\psi.

This, however, is a cumbersome expression to work with because of the square root. In addition, this equation, as it stands, is nonlocal. In Physics, nonlocality is a direct influence of one object on another distant object in violation of Principle of locality.

Klein and Gordon instead worked with the more general square of this equation (the Klein–Gordon equation for a free particle), which in covariant notation reads

(\Box^2 + \mu^2) \psi = 0,

where

\mu = \frac{mc}{\hbar} \,

and

\Box^2 = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.\,

This operator is called the d'Alembert operator. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally In Special relativity, Electromagnetism and wave theory, the d'Alembert operator \Box also called the d'Alembertian or the Today this form is interpreted as the relativistic field equation for a scalar (i. A field equation is an equation in a Physical theory that describes how a Fundamental force (or a combination of such forces interacts with Matter e. spin-0) particle. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin

The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, without taking into account the electron's spin, the Klein-Gordon equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of 4n/(2n-1) for the n-th energy level. In January 1926, Schrödinger submitted for publication instead his equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.

In 1926, soon after the Schrödinger equation was introduced, Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Year 1926 ( MCMXXVI) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar. Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фoк ( December 22 1898 &ndash December 27 1974 In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics, velocity is defined as the rate of change of Position. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves The Klein-Gordon equation for a free particle has a simple plane wave solution. In Physics, a free particle is a particle that in some sense is not bound In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose

Relativistic free particle solution

The Klein–Gordon equation for a free particle can be written as

\mathbf{\nabla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi= \frac{m^2c^2}{\hbar^2}\psi

with the same solution as in the non-relativistic case:

\psi(\mathbf{r}, t) = e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}

except with the constraint

-k^2+\frac{\omega^2}{c^2}=\frac{m^2c^2}{\hbar^2}.

Just as with the non-relativistic particle, we have for energy and momentum:

\langle\mathbf{p}\rangle=\langle \psi |-i\hbar\mathbf{\nabla}|\psi\rangle = \hbar\mathbf{k},
\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \hbar\omega.

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:

\left.\right.\langle E \rangle^2=m^2c^4+\langle \mathbf{p} \rangle^2c^2.

For massless particles, we may set m = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:

\left.\right.\langle E \rangle=\langle |\mathbf{p}| \rangle c.

Action

The Klein–Gordon equation can be derived from the following action

\mathcal{S}=\int \mathrm{d}^4x \left(\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{2}\frac{m^2 c^2}{\hbar^2} \phi^2 \right)

where φ is the Klein-Gordon field and m is its mass.

See also

References

External links

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