3.2: Operators

Notice that in deriving the wave equation we replaced the number $p$ or $k$ by a differential acting on the wavefunction. 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 wavefunction and function multiplying the wavefunction, symbolically written as

$$\hat{H} ψ(x,t) = − \dfrac{\hbar^2}{2 m} \dfrac{∂^2}{∂ x^2} ψ(x,t) + V(x) ψ ( x , t ) . \label{3.16}$$

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