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8.1.3 Hermitean operators

Hermitean operators are those where the outcome of any measurement is always real, as they should be (complex position?). This means that both its eigenvalues are real, and that the average outcome of any experiment is real. The mathematical definition of a Hermitean operator can be given as

$\displaystyle \int_{\mathrm{all\ space}}^{}$$\displaystyle \psi_{1}^{*}$(x)$\displaystyle \left[\vphantom{\hat O \psi_2(x)}\right.$$\displaystyle \hat{O}$$\displaystyle \psi_{2}^{}$(x)$\displaystyle \left.\vphantom{\hat O \psi_2(x)}\right]$dx = $\displaystyle \int_{\mathrm{all\ space}}^{}$$\displaystyle \left[\vphantom{\hat O \psi_1(x)}\right.$$\displaystyle \hat{O}$$\displaystyle \psi_{1}^{}$(x)$\displaystyle \left.\vphantom{\hat O \psi_1(x)}\right]^{*}$$\displaystyle \psi_{2}^{}$(x)dx. (8.4)

Quiz show that $ \hat{x}$ and $ \hat{p}$ (in 1 dimension) are Hermitean.



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