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Next: 8.1.3 Hermitean operators Up: 8.1 Key postulates Previous: 8.1.1 Wavefunction

8.1.2 Observables

In classical mechanics ``observables'' (the technical term for anything that can be measured) are represented by numbers. Think e.g., of x, y, z, px, py, pz, E, .... In quantum mechanics ``observables'' are often quantised, they cannot take on all possible values: how to represent such quantities?

We have already seen that energy and momentum are represented by operators,

$\displaystyle \hat{\vec{p}}$ = - i$\displaystyle \hbar$$\displaystyle \nabla$ = - i$\displaystyle \hbar$($\displaystyle {\frac{\partial}{\partial x}}$,$\displaystyle {\frac{\partial}{\partial y}}$,$\displaystyle {\frac{\partial}{\partial z}}$). (8.1)

and

$\displaystyle \hat{H}$ = - $\displaystyle {\frac{\hbar^2 d^2}{2mdx^2}}$ + V(x) (8.2)

Let me look at the Hamiltonian, the energy operator. We know that its normalisable solutions (eigenvalues) are discrete.

$\displaystyle \hat{H}$$\displaystyle \phi_{n}^{}$(x) = En$\displaystyle \phi_{n}^{}$(x). (8.3)

The numbers En are called the eigenvalues, and the functions $ \phi_{n}^{}$(x) the eigenfunctions of the operator $ \hat{H}$. Our postulate says that the only possible outcomes of any experiment where we measure energy are the values En!



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