Incoming links: Position representation
One dimension in position representation:
In three dimensions In position representation, we define it by using the gradient, and so we see that
Position and Momentum from Wavefunctions by Faculty of Khan (2018)
Source. Proves in detail why the momentum operator is . The starting point is determining the time derivative of the expectation value of the position operator.Contains the full state of the quantum system.
This is in contrast to classical mechanics where e.g. the state of mechanical system is given by two real functions: position and speed.
The wave equation in position representation on the other hand encodes speed in "how fast does the complex phase spin around", and direction in "does it spin clockwise or counterclockwise", as described well at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)". Then once you understand that, it is more compact to just view those graphs with the phase color coded as in Video "Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019)".