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
|
\hat{H}ψ(x,t) = −{{ℏ}^{2}\over
2m} {{∂}^{2}\over
∂{x}^{2}}ψ(x,t) + V (x)ψ(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.