{"id":1817,"date":"2026-08-29T20:36:43","date_gmt":"2026-08-29T20:36:43","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=1817"},"modified":"2026-08-29T20:36:44","modified_gmt":"2026-08-29T20:36:44","slug":"schrodinger-equation","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/modern-physics\/schrodinger-equation\/","title":{"rendered":"The Schrodinger Equation Explained"},"content":{"rendered":"\n
Definition<\/div>

\n\nThe Schrodinger equation is the central equation of quantum mechanics: it describes how a particle’s wave function evolves in time and which energies that particle is allowed to have. In words, the energy operator acting on the wave function equals the total energy multiplied by the wave function. Solving it for a given potential returns every permitted energy level.\n\n<\/p><\/div>\n\n

Look at the screen you are reading this on. The colour of every pixel, the switching of every transistor behind it, and the chemistry of the battery keeping it alive were all designed using solutions to one equation written down in 1926.<\/p>\n\n

That equation does something Newton’s laws cannot. It explains why an atom does not collapse, why the periodic table has the shape it does, and why an electron can pass through a wall it has no business crossing.<\/p>\n\n

What Is the Schrodinger Equation?<\/h2>\n\n

The Schrodinger equation is the equation of motion for a quantum particle \u2014 it plays the same role in quantum mechanics<\/a> that Newton’s second law plays in classical physics. Feed it a force landscape, and it tells you how the particle behaves.<\/p>\n\n

But the thing it evolves is not a position. Newton hands you x(t)<\/em>, a dot moving along a line. Schrodinger hands you \u03c8(x, t)<\/em>, a spread-out mathematical object called the wave function that encodes everything knowable about the particle at once.<\/p>\n\n

Erwin Schr\u00f6dinger \u2014 usually written “Schrodinger” in English \u2014 arrived at it over the Christmas holiday of 1925, chasing an idea of Louis de Broglie’s: that matter has a wavelength. If electrons are waves, Schrodinger reasoned, there must be a wave equation governing them. He shared the 1933 Nobel Prize in Physics<\/a> with Paul Dirac for the result.<\/p>\n\n

\n \"Erwin\n
Erwin Schr\u00f6dinger, who wrote down his wave equation in 1926 and shared the 1933 Nobel Prize in Physics.<\/figcaption>\n<\/figure>\n\n

Here is the payoff, and it is worth pausing on. Nobody puts quantisation into<\/em> the equation. You supply a potential, demand that the solution behave sensibly at the edges, and discrete energy levels fall out on their own \u2014 the way a guitar string clamped at both ends can only sound certain notes.<\/p>\n\n

The Schrodinger Equation Formula<\/h2>\n\n

The Schrodinger equation has two standard forms, and choosing between them is the first decision in any problem. The full time-dependent version describes how a wave function changes from moment to moment:<\/p>\n\n

i\u0127 \u2202\u03c8\/\u2202t = \u0124\u03c8<\/div>\n\n

For a single particle moving in one dimension, the Hamiltonian operator \u0124 is kinetic energy plus potential energy, which makes the time-independent form concrete:<\/p>\n\n

-(\u0127\u00b2\/2m)\u00b7d\u00b2\u03c8\/dx\u00b2 + V(x)\u00b7\u03c8 = E\u00b7\u03c8<\/div>\n\n

Read the second one aloud and it is just energy bookkeeping: kinetic term plus potential term equals total energy. Every variable, with its SI unit:<\/p>\n\n