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Posts Tagged ‘Dirac’s Equation’

Yesterday I introduced Paul Dirac, number 10 in “The Guardian’s” list of the 10 best physicists. I mentioned that his main contributions to physics were (i) predicting antimatter, which he did in 1928, and (ii) producing an equation (now called the Dirac equation) which describes the behaviour of a sub-atomic particle such as an electron travelling at close to the speed of light (a so-called relativistic theory). This equation was also published in 1928.

The Dirac Equation

In 1928 Dirac wrote a paper in which he published what we now call the Dirac Equation.

The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light. The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light.

This is a relativistic form of Schrödinger’s wave equation for an electron. The wave equation was published by Erwin Schrödinger two years earlier in 1926, and describes how the quantum state of a physical system changes with time.

The Schrödinger eqation

The time dependent Schrödinger equation which describes the motion of an electron The time dependent Schrödinger equation which describes the motion of an electron

The various terms in this equation need some explaining. Starting with the terms to the left of the equality, and going from left to right, we have i is the imaginary number, remember i = \sqrt{-1}. The next term \hbar is just Planck’s constant divided by two times pi, i.e. \hbar = h/2\pi. The next term \partial/\partial t \text{ } \psi(\vec{r},t) is the partial derivative with respect to time of the wave function \psi(\vec{r},t).

Now, moving to the right hand side of the equality, we have
m which is the mass of the particle, V is its potential energy, \nabla^{2} is the Laplacian. The Laplacian, \nabla^{2} \psi(\vec{r},t) is simply the divergence of the gradient of the wave function, \nabla \cdot \nabla \psi(\vec{r},t).

In plain language, what the Schrödinger equation means “total energy equals kinetic energy plus potential energy”, but the terms take unfamiliar forms for reasons explained below.

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