Schrödinger Equation (Math View)

Key idea

Wave equation of quantum mechanics.

In depth

iħψ_t = −(ħ²/2m)Δψ + Vψ. It is a linear PDE for a complex wave function, whose |ψ|² gives probability density. Stationary states are eigenfunctions of the Hamiltonian, with energies as eigenvalues.

Worked example

A particle in a box of length L: ψₙ = √(2/L) sin(nπx/L), Eₙ = n²π²ħ²/(2mL²).

Formula

Real-world uses

• Calculating energy levels of atoms and designing semiconductors and lasers

Common mistake

Interpreting ψ itself as a probability; it is |ψ|².

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Differential Equations, Abstract Algebra & Applied Math

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• Second-Order Homogeneous Equations – Use the characteristic equation.
• Undetermined Coefficients – Guess the particular solution form.
• Variation of Parameters – General method for particular solutions.