next up [*]
Next: 3.3 Analysis of the Up: 3. The Schrödinger equation Previous: 3.1 The state of

3.2 Operators

Notice that in deriving the wave equation we replaced the number p or k by a differential acting on the wave function. The energy (or rather the Hamiltonian) was replaced by an "operator", which when multiplied with the wave function gives a combination of derivatives of the wave function and function multiplying the wave function, symbolically written as

$\displaystyle \hat{H}$$\displaystyle \psi$(x, t) = - $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{\partial^2}{\partial x^2}}$$\displaystyle \psi$(x, t) + V(x)$\displaystyle \psi$(x, t). (3.16)

This appearance of operators (often denoted by hats) where we were used to see numbers is one of the key features of quantum mechanics.



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