3.2: Operators
( \newcommand{\kernel}{\mathrm{null}\,}\)
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
ˆHψ(x,t)=−ℏ22m∂2∂x2ψ(x,t)+V(x)ψ(x,t).
This appearance of operators (often denoted by hats) where we were used to see numbers is one of the key features of quantum mechanics.