Klein–Gordon equation

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The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation.

It is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.

Statement[edit]

The Klein–Gordon equation is

 \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0.

This is often abbreviated as

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

where  \mu = \dfrac{mc}{\hbar} and \Box is the d'Alembert operator, defined by

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

The equation is most often written in natural units:

 - \partial_t^2 \psi + \nabla^2 \psi = m^2 \psi

The form is determined by requiring that plane wave solutions of the equation:

\psi = e^{-i\omega t + i k\cdot x } = e^{i k_\mu x^\mu}

obey the energy momentum relation of special relativity:

 -p_\mu p^\mu = E^2 - P^2 = \omega^2 - k^2 = - k_\mu k^\mu = m^2

Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of ω for each k, one positive and one negative. Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction. For the time-independent case, the Klein–Gordon equation becomes

\left[ \nabla^2 - \frac {m^2 c^2}{\hbar^2} \right] \psi(\mathbf{r}) = 0

which is the homogeneous screened Poisson equation.

History[edit]

The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1926 proposed that it describes relativistic electrons. Other authors making similar claims in that same year were Vladimir Fock, Johann Kudar, Théophile de Donder and Frans-H. van den Dungen, and Louis de Broglie. Although it turned out that the Dirac equation describes the spinning electron, the Klein–Gordon equation correctly describes the spinless pion, a composite particle. On July 4th, 2012 CERN announced the discovery of the first spin-zero elementary particle, the Higgs boson. Further experimentation and analysis is required to discern whether the Higgs boson found is that of the Standard Model, or a more exotic form.

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, because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of \tfrac{4n}{2n-1} for the n-th energy level. The Dirac result is, however, easily recovered if the orbital momentum quantum number l is replaced by total angular momentum quantum number j.[1] 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 1927, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. The Klein–Gordon equation for a free particle has a simple plane wave solution.

Derivation[edit]

The non-relativistic equation for the energy of a free particle is

\frac{\mathbf{p}^2}{2 m} = E.

By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,

\frac{\mathbf{\hat{p}}^2}{2m} \psi = \hat{E}\psi

where

\mathbf{\hat{p}} =-i \hbar \mathbf{\nabla}

is the momentum operator (\mathbf\nabla being the del operator), and

\hat{E}=i \hbar \dfrac{\partial}{\partial t}

is the energy operator.

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special relativity.

It is natural to try to use the identity from special relativity describing the energy:

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

Then, just inserting the quantum mechanical operators for momentum and energy 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 the differential operator cannot be evaluated while under the square root sign. In addition, this equation, as it stands, is nonlocal (see also Introduction to nonlocal equations).

Klein and Gordon instead began with the square of the above identity, i.e.

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

which, when quantized, gives

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

which simplifies to

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

Rearranging terms yields

 \frac {1}{c^2} \frac{\partial^2}{(\partial t)^2} \psi - \mathbf{\nabla}^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0.

Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real valued as well as those that have complex values.

Using the inverse of the Minkowski metric diag(−c2, 1, 1, 1), we get

 - \eta^{\mu \nu} \partial_{\mu} \partial_{\nu} \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0

in covariant notation. This is often abbreviated as

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

where

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

and

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

This operator is called the d'Alembert operator. Today this form is interpreted as the relativistic field equation for a scalar (i.e. spin-0) particle. Furthermore, any solution to the Dirac equation (for a spin-one-half particle) is automatically a solution to the Klein–Gordon equation, though not all solutions of the Klein–Gordon equation are solutions of the Dirac equation. It is noteworthy that the Klein–Gordon equation is very similar to the Proca equation.

Klein-Gordon equation in a potential[edit]

The Klein–Gordon equation can be generalized to describe a field in some potential V(ψ) as:[2]

\Box \psi + \frac{\partial{}V}{\partial \psi} = 0

Relativistic free particle solution[edit]

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, known as the dispersion relation:

-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=\left\langle \psi \left|-i\hbar\mathbf{\nabla}\right|\psi\right\rangle = \hbar\mathbf{k},
\langle E\rangle=\left\langle \psi \left|i\hbar\frac{\partial}{\partial t}\right|\psi\right\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:

\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:

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

Action[edit]

The Klein–Gordon equation can also be derived via a variational method by considering the action:

\mathcal{S} = \int \left( - \frac{\hbar^2}{m} \eta^{\mu \nu} \partial_{\mu}\bar\psi \partial_{\nu}\psi - m c^2 \bar\psi \psi \right) \mathrm{d}^4 x

where \psi is the Klein–Gordon field and m is its mass. The complex conjugate of \psi is written \bar\psi . If the scalar field is taken to be real-valued, then \bar\psi = \psi .

Applying the formula for the Hilbert stress–energy tensor to the Lagrangian (the quantity inside the integral), we can derive the stress–energy tensor of the scalar field. It is

T^{\mu\nu} = \frac{\hbar^2}{m} \left (\eta^{\mu \alpha} \eta^{\nu \beta} + \eta^{\mu \beta} \eta^{\nu \alpha} - \eta^{\mu\nu} \eta^{\alpha \beta} \right ) \partial_{\alpha}\bar\psi \partial_{\beta}\psi - \eta^{\mu\nu} m c^2 \bar\psi \psi .

Electromagnetic interaction[edit]

There is a simple way to make any field interact with electromagnetism in a gauge invariant way: replace the derivative operators with the gauge covariant derivative operators. The Klein Gordon equation becomes:

D_\mu D^\mu \phi = -(\partial_t - ie A_0)^2 \phi + (\partial_i - ie A_i)^2 \phi = m^2 \phi

in natural units, where A is the vector potential. While it is possible to add many higher order terms, for example,

D_\mu D^\mu\phi + A F^{\mu\nu} D_\mu \phi D_\nu (D_\alpha D^\alpha \phi) =0

these terms are not renormalizable in 3+1 dimensions.

The field equation for a charged scalar field multiplies by i, which means the field must be complex. In order for a field to be charged, it must have two components that can rotate into each other, the real and imaginary parts.

The action for a charged scalar is the covariant version of the uncharged action:

S= \int_x \left (\partial_\mu \phi^* + ie A_\mu \phi^* \right ) \left (\partial_\nu \phi - ie A_\nu\phi \right )\eta^{\mu\nu} = \int_x |D \phi|^2

Gravitational interaction[edit]

In general relativity, we include the effect of gravity and the Klein–Gordon equation becomes[3]

\begin{align}
0 & = - g^{\mu \nu} \nabla_{\mu} \nabla_{\nu} \psi + \dfrac {m^2 c^2}{\hbar^2} \psi = - g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \psi) + \dfrac {m^2 c^2}{\hbar^2} \psi \\
& = - g^{\mu \nu} \partial_{\mu} \partial_{\nu} \psi + g^{\mu \nu} \Gamma^{\sigma}{}_{\mu \nu} \partial_{\sigma} \psi + \dfrac {m^2 c^2}{\hbar^2} \psi
\end{align}

or equivalently

\frac{-1}{\sqrt{-g}} \partial_{\mu} \left ( g^{\mu \nu} \sqrt{-g} \partial_{\nu} \psi \right ) + \frac {m^2 c^2}{\hbar^2} \psi = 0

where g^{\alpha \beta} is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, \nabla_{\mu} is the covariant derivative and \Gamma^{\sigma}{}_{\mu \nu} is the Christoffel symbol that is the gravitational force field.

Traveling wave solution[edit]

There exists traveling wave solutions in the form of solitons for Klein–Gordon equation [4]

Klein–Gordon equation traveling wave plot4 
Klein–Gordon equation traveling wave plot5 
Klein–Gordon equation traveling wave plot6 

See also[edit]

Notes[edit]

  1. ^ See C Itzykson and J-B Zuber, Quantum Field Theory, McGraw-Hill Co., 1985, pp. 73-74. Eq. 2.87 is identical to eq. 2.86 except that it features j instead of l.
  2. ^ David Tong, Lectures on Quantum Field Theory, Lecture 1, Section 1.1.1
  3. ^ S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press, 1996, p. 117
  4. ^ Inna Shingareva, Carlos Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple p64-72 Springer

References[edit]

  • Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2. 
  • Davydov, A.S. (1976). Quantum Mechanics, 2nd Edition. Pergamon. ISBN 0-08-020437-6. 

External links[edit]