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8.1.6 Eigenfunctions of $ \hat{x}$

The operator $ \hat{x}$ multiplies with x. Solving the equation

$\displaystyle \hat{x}$$\displaystyle \phi$(x) = x0$\displaystyle \phi$(x) (8.11)

we find that the solution must be exactly localised at x = x0. The function that does that is called a Dirac $ \delta$ function $ \delta$(x - x0). This is defined through integration,

$\displaystyle \int_{-\infty}^{\infty}$$\displaystyle \delta$(x - x0)g(x) dx = g(x0), (8.12)

and is not normalisable,

$\displaystyle \int$$\displaystyle \delta$(x - x0)2dx = $\displaystyle \infty$. (8.13)



© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk