--- title: "The Time-Independent Schrödinger Equation" description: "For a nonrelativistic particle in a time-independent potential, the stationary Schrödinger equation is an energy eigenvalue equation. Its solutions must satisfy the differential equation and the physi" 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Time-Independent Schrödinger Equation For a nonrelativistic particle in a time-independent potential, the stationary Schrödinger equation is an energy eigenvalue equation. Its solutions must satisfy the differential equation and the physical 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: Finding stationary quantum states The wavefunction predicts probabilities, as in the packet topic. For a nonrelativistic particle of constant mass m in one dimension, with time-independent potential energy U(x), the stationary equation is -(ħ²/2m)ψ'' + Uψ = Eψ. Here ψ'' means d²ψ/dx². This quantum equation does not follow from classical energy conservation alone. Solve for ψ and allowed energies E. The kinetic-energy operator acting on ψ gives -(ħ²/2m)ψ''. Uψ is the potential term and Eψ the total-energy term. These are functions, not three energy numbers. In constant U, ψ = A sin(kx) has ψ'' = -k²ψ. Its kinetic eigenvalue is T = ħ²k²/(2m), so tripling k multiplies T by nine. Absolute curvature alone does not give a local kinetic energy. *(drawing: The equation relates functions: the kinetic operator term plus Uψ equals Eψ.)* **Example.** An infinite well from 0 to L has U = 0 inside and ψ(0) = ψ(L) = 0. Its normalised solutions are ψ_n = √(2/L) sin(nπx/L), with n = 1, 2, 3, ... . Each satisfies the equation and the wall conditions. The index n counts half-waves across the well. E_n = n²π²ħ²/(2mL²), so n = 3 has three half-waves and nine times the ground-state energy. The bars show energies in units of E_1, not wave shapes. Check a candidate by substituting ψ'' into the equation: one constant E must work throughout the interval. Also check the wall conditions and confirm that the squared norm is finite and nonzero. The n = 0 sine has norm zero, so the well starts at n = 1. For constant mass at a finite, nonsingular potential boundary, ψ and ψ' are continuous. At an infinite well wall, ψ is zero; ψ' need not match the zero function outside. A stationary energy eigenstate has Ψ(x,t) = ψ(x) exp(-iEt/ħ). Its time factor has magnitude one, so |Ψ(x,t)|² = |ψ(x)|². The overall phase can change while the density stays fixed. Stationary does not mean fixed at one position. Every infinite-well eigenstate has positive kinetic energy. A static potential also permits superpositions of different energies whose position densities may change with time. **Recap.** Solve the energy eigenvalue equation and check its boundary conditions and norm. A stationary state's density stays fixed even when its overall phase changes. *Lesson, questions and flashcards written for this dataset using standard stationary-equation and infinite-well methods.* ## 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)