First quantisation

First quantisation is a process for converting classical problems into their quantic counterparts.

1. Take the classical hamiltonian $H=\frac{1}{2m}(p_x^2+p_y^2+p_z^2)+V(x,y,z)$
2. Transform the Hamiltonian using Operators: $H=\frac{1}{2m}(P_x+P_y+P_z)+V(X,Y,Z)$, where $X,Y,Z$ are Position Operators and $P_x,P_y,P_z$ are Momentum Operators.
3. Plugin the resulting Hamiltonian into Schrodinger Equation: $i\hslash \dot{\psi }=H\psi$.