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Mathematics for Physics

92 Time Dependent Schrodinger Equation Generalized

The fundamental state-space equation that is used to describe how any quantum system’s state vector evolves over time under a Hamiltonian operator.

definition (Time Dependent Schrodinger Equation - Generalized) The coordinate-independent and dimension-independent form of the fundamental law of quantum mechanics. This abstract version represents the general principle that a quantum state changes continuously and unitarily over time, governing the evolution of any quantum system - including those with multiple particles or non-spatial degrees of freedom. The general time-dependent form is:

  • \(i\hbar \dfrac{\partial \Psi}{\partial t} = \hat{H}\Psi\)

where

  • \(i\) is the imaginary unit.
  • \(\hbar\) is the reduced Planck constant.
  • \(\Psi\) is the state-vector representing the state of the system.
  • \(t\) is time.
  • \(\hat{H}\) is the Hamiltonian operator representing the total energy of the system.

Note:

  • \(\Psi\) is also written \(|\Psi\rangle\).
  • \(\Psi\) is also called the wave function.
  • the total energy is kinetic energy plus potential energy.