General form of the Schrödinger equation for one particle in any dimension ↩

aliases: []
shorthands: {}
created: 2022-01-06 03:02:19
modified: 2022-01-10 04:13:04

The Hamiltonian operator is usually expressed as the sum of two operators corresponding to the kinetic and potential energies of a system:

Where the potential energy is generalized

Where is a vector.
And for a particle moving in , the kinetic energy is:

Where is the momentum operator, it is a vector of operators and is the mass of the particle, is the nabla operator.

The Hamiltonian operator takes the form:

So the Schrödinger equation of the system is:

Canonical commutation relations

The position and momentum operators commute if they operate in different dimensions:

See Commutator of position and momentum operators.
And of course they commute with themselves in different dimensions as well: