---
title: "The Time-Independent Schrödinger Equation"
description: "Energy conservation written for a wavefunction gives a differential equation whose solutions are the allowed stationary states. Solving a quantum problem means solving that equation with boundary cond"
canonical: https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T
source: https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T.md
retrieved: 2026-09-12
---

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# The Time-Independent Schrödinger Equation

Energy conservation written for a wavefunction gives a differential equation whose solutions are the allowed stationary states. Solving a quantum problem means solving that equation with boundary conditions.

Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20
Page: https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T

## Ready when they can

- Identifies kinetic, potential and total energy terms in the equation
- States the conditions a physical solution has to satisfy at a boundary
- Explains what makes a state stationary when the particle is still described by a wave

## Lesson: Energy conservation as a wave equation

Classical energy conservation says total energy is momentum squared over 2 m plus potential U. The time-independent Schrodinger equation writes the same budget for a wave: a curvature term for kinetic energy, plus U times psi for potential, equals E times psi. Solving a quantum problem means finding the psi and the E values that satisfy this equation with the boundary conditions.

**Example.** Inside an infinite well U is zero and the walls are impenetrable, so the equation becomes a pure curvature equation. The walls demand psi equals zero at both ends, which only sine waves with whole half-waves between the walls can meet. Each fitting wave carries its own allowed energy.

A stationary state has one fixed energy, and its probability density never changes in time. The particle is not sitting still: it stays spread in a wave whose shape of chances is frozen. This separated form applies when the potential itself does not change with time.

**Tip.** Read the three terms before calculating: curvature means kinetic, U psi means potential, E psi means total. Check the walls first, since they pick the solutions. Reject n equals zero: it is zero everywhere and cannot be normalised.

**Recap.** Each term of the equation names an energy, the walls select the solutions, and stationary states freeze the chances in time.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Uncertainty Principle from Wave Packets](https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd)
- [Newton's Second Law as a Differential Equation](https://lightmysky.com/learn/science/newtons-second-law-as-a-differential-equation-mt_LzSNSkEovH)
- [What a Differential Equation Says and What a Solution Is](https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o)

## Opens up

- [Unitary Time Evolution and the Heisenberg Picture](https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8)
- [The Infinite Square Well and Energy Quantisation](https://lightmysky.com/learn/science/the-infinite-square-well-and-energy-quantisation-mt_VV9ItoW7ew)