Time-Dependent and Time-Independent Schrodinger Equation

The time-dependent Schrodinger equation tells a wavefunction how to evolve, and separating out the time dependence reduces it to the time-independent equation, an eigenvalue problem whose solutions are the stationary states with definite energy.

Definition

The time-dependent Schrodinger equation states that the Hamiltonian generates the time evolution of the wavefunction, while the time-independent Schrodinger equation is the resulting eigenvalue equation whose solutions are the stationary states of definite energy.

Scope

The topic covers the time-dependent Schrodinger equation and conservation of probability, separation of variables for time-independent Hamiltonians, the time-independent equation as an energy eigenvalue problem, stationary states and their trivial phase evolution, the expansion of a general state in energy eigenstates, and the propagator that advances any state in time.

Core questions

• How does the Hamiltonian determine the evolution of any quantum state?
• Why does separating time from space yield an energy eigenvalue problem?
• What is special about stationary states under time evolution?
• How is the future state of an arbitrary superposition computed?

Key concepts

• Hamiltonian operator
• stationary state
• energy eigenvalue
• separation of variables
• probability conservation
• propagator

Key theories

Separation of variables

When the Hamiltonian has no explicit time dependence, solutions of the form spatial function times time phase reduce the full equation to the time-independent eigenvalue problem, with each energy eigenstate acquiring only an oscillating phase as time passes.

Spectral expansion and the propagator

Any initial state can be written as a superposition of energy eigenstates, each evolving by its own phase, so the full time evolution is captured by a propagator built from the energy spectrum that maps the state at one time to any later time.

Clinical relevance

This pair of equations is the starting point for nearly all quantum calculations: stationary states give the spectral lines measured in atomic and molecular spectroscopy, while the time-dependent form governs transitions, wave-packet dynamics, and the coherent control of qubits in quantum technology.

History

Schrodinger presented both forms of his equation in his 1926 series of papers, immediately applying the time-independent equation to the hydrogen atom; Dirac and von Neumann later recast the time evolution in the abstract operator language of unitary propagators.

Key figures

• Erwin Schrodinger
• Paul Dirac
• John von Neumann

Seminal works

• griffiths2018
• sakurai2017

Frequently asked questions

Why is it called a stationary state if it still evolves in time?

A stationary state acquires only an overall oscillating phase, which cancels in any measurement probability or expectation value, so all observable properties remain constant in time even though the wavefunction itself keeps rotating in the complex plane.

When can the time-independent Schrodinger equation be used?

It applies when the Hamiltonian does not depend explicitly on time, allowing separation of variables; for time-varying potentials one must solve the full time-dependent equation or use time-dependent perturbation theory.