String Theory

Making the Schrödinger equation relativistic

The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle:

$-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi.$

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energy are the space and time parts of a space-time vector, the 4-momentum, and they are related by the relativistically invariant relation

$\frac{E^2}{c^2} - p^2 = m^2c^2$

which says that the length of this vector is proportional to the rest mass

m

. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,

$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$

with the wave function

ϕ

being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. The space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, then by the nature of solving differential equations, one must specify both the initial values of the wave function itself and of its first time derivative, in order to solve definite problems. Because both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression

$\rho=\phi^*\phi\,$

and this density is convected according to the probability current vector

$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$

with the conservation of probability current and density following from the Schrödinger equation:

$\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0.$

The fact that the density is positive definite and convected according to this continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace the probability density by the symmetrically formed expression

$\rho = \frac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*).$

which now becomes the 4th component of a space-time vector, and the entire probability 4-current density has the relativistically covariant expression

$J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*)$

The continuity equation is as before. Everything is compatible with relativity now, but we see immediately that the expression for the density is no longer positive definite - the initial values of both

ψ

and

\partial t ψ

may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson). Historically, Schrödinger himself arrived at this equation before the one that bears his name, but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the charge density, which can be positive or negative, and not the probability density.

String Theory

Friday, 7 June 2013

Making the Schrödinger equation relativistic

No comments:

Post a Comment