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Mathematics for Physics

151 Time Dependent Schrodinger Equation Generalized

A state-space equation that is used to describe how any quantum system’s state evolves in time under a Hamiltonian operator, where a Hamiltonian operator is the operator that represents the total energy.

The generalized Schrödinger equation. The time derivative of the state is proportional to the Hamiltonian acting on the state. This principle is used to evolve any quantum system, including systems with several particles or with spin.

The generalized time-dependent Schrödinger equation 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.
  • \(t\) is time.
  • \(\hat{H}\) is the Hamiltonian operator.

Unitary evolution. The evolution generated by a Hermitian Hamiltonian is unitary. Unitary evolution is evolution that preserves the norm of the state. This principle is used to keep the total probability equal to one at every time.

The time-evolution operator. If \(\hat{H}\) does not depend on time, the solution is a phase factor times the initial state in the energy basis. This principle is used to write the general solution as a superposition of stationary states.

The time-evolution operator for a time-independent Hamiltonian is

\[ \Psi(t) = e^{-i\hat{H}t/\hbar}\Psi(0) \]

where

  • \(\Psi(t)\) is the state at time \(t\).
  • \(\hat{H}\) is the Hamiltonian.
  • \(\hbar\) is the reduced Planck constant.

Note: \(\Psi\) is also written \(|\Psi\rangle\). Also called the wave function.

151.1 References

  1. Das, T. K. Quantum Mechanics: Axiomatic Approach and Understanding Through Mathematics. Springer, 2023. — Hilbert-space time-dependent Schrödinger equation.
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(i\hbar\partial_{t}|\Psi\rangle=\hat{H}|\Psi\rangle\).