What Is the Schrödinger Equation?

Late in 1925, at a physics colloquium in Zurich, Erwin Schrödinger explained Louis de Broglie’s strange idea that electrons behave like waves. Peter Debye, who ran the colloquium, remarked that this way of talking was rather childish. A real wave, he said, needs a wave equation. A few weeks later Schrödinger opened his next talk by announcing that he had found one. He sent it for publication on 26 January 1926, which makes the Schrödinger equation a hundred years old this year. It governs every atom, every molecule and every transistor. In January 2026, physicists reported a clump of more than 7,000 sodium atoms obeying it.

The equation, written out

In its most general form the Schrödinger equation fits on one line:

iħ ∂Ψ/∂t = ĤΨ

For a single particle of mass m, moving along one direction through a potential energy V(x), that line becomes:

iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x)Ψ

Ψ, the Greek letter psi, is the wavefunction: a complex number attached to every point in space at every moment. i is the square root of minus one. ħ, read “h-bar”, is Planck’s constant divided by 2π, about 1.055 × 10⁻³⁴ joule-seconds. Since the 2019 redefinition of the SI units, Planck’s constant has an exact defined value, so ħ does too.

∂Ψ/∂t is how fast the wavefunction changes in time at a fixed place. Ĥ, the Hamiltonian, is the energy operator: a recipe that acts on Ψ and extracts the system’s energy. For one particle it has two parts. The term with the second derivative, ∂²Ψ/∂x², is kinetic energy. The term V(x)Ψ is potential energy. In three dimensions the second derivative becomes ∇²Ψ, the Laplacian, which adds up the curvature along all three directions.

What the equation actually says

Read it as a sentence. The left side is the rate at which the wavefunction changes. The right side is the energy operator acting on the wavefunction. So the equation says that energy drives change in time. Newton’s second law does the same job in classical mechanics, where force drives change in motion.

The kinetic term has a concrete meaning. A second derivative measures curvature, how sharply Ψ bends from one point to the next. A wavefunction that wiggles tightly, with a short wavelength, carries more kinetic energy than one that undulates gently. That is de Broglie’s relation hiding inside the mathematics: short wavelength means high momentum.

Two structural features matter more than any single term. The equation is first order in time. Give it Ψ now, everywhere, and it fixes Ψ at every later moment, with no randomness at all. Newton needs a position and a velocity; Schrödinger needs only the present wavefunction.

The equation is also linear. If Ψ₁ and Ψ₂ both solve it, so does any combination aΨ₁ + bΨ₂. That single property is the superposition principle, and most of what makes quantum mechanics strange follows from it.

Why the i matters

Delete the i and the equation turns into one physicists already knew. The heat equation, which describes warmth spreading along a metal bar, reads:

∂u/∂t = D ∂²u/∂x²

The Schrödinger equation for a free particle, with no potential, rearranges into exactly that shape:

∂Ψ/∂t = (iħ/2m) ∂²Ψ/∂x²

It is a diffusion equation with an imaginary diffusion coefficient. Richard Feynman described it as the diffusion of probability amplitude from point to point. That one factor of i changes the behaviour completely. Heat only spreads and smooths. Its peaks sink and never return, and nothing ever oscillates.

With the i in place, solutions rotate through the complex plane instead. They oscillate like waves, they interfere, and the total probability stays fixed at exactly one forever. The i is what turns diffusion into wave mechanics. It is also why the wavefunction must be complex. A purely real Ψ would make the left side imaginary and the right side real, so it could only satisfy the equation trivially.

The textbook derivation, step by step

Nearly every textbook gives the same short argument. It starts from a free particle, one feeling no force, described by a plane wave:

Ψ(x, t) = A e^{i(kx − ωt)}

Two relations from the early quantum era tie this wave to a particle. Max Planck and Albert Einstein linked energy to frequency. Louis de Broglie linked momentum to wavelength:

E = ħω and p = ħk

Now differentiate the plane wave. One time derivative brings down a factor of −iω. Two space derivatives bring down −k². Multiply through by the right constants:

iħ ∂Ψ/∂t = ħωΨ = EΨ

−ħ² ∂²Ψ/∂x² = ħ²k²Ψ = p²Ψ

On a plane wave, then, a time derivative measures energy and a space derivative measures momentum. The last step borrows classical bookkeeping. Total energy is kinetic plus potential, E = p²/2m + V. Replace each quantity with its derivative and the result is:

iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x)Ψ

That is the Schrödinger equation. It looks like a derivation. It is not quite one.

Why that is not a proof

Look hard at the final step. Every check in the argument ran on a plane wave, which describes a particle feeling no force. Then the potential V(x) went in, and the result was declared true for particles that do feel forces, whose solutions are not plane waves at all. Nothing forced that extension. It was a guess, and a bold one.

Richard Feynman was blunt about it in his lectures: “It’s not possible to derive it from anything you know.” The equation came out of Schrödinger’s head. Feynman added that some of Schrödinger’s original arguments were even wrong, and that this did not matter, because the equation describes nature correctly.

The real history is messier than the textbook route. Schrödinger’s first attempt, at the end of 1925, was a relativistic equation. Applied to hydrogen, it gave the wrong fine structure, because electrons carry spin and that equation has no room for it. He retreated to the slower, non-relativistic case. His first paper reached the time-independent equation through a variational principle. In his second paper he conceded that a key step of the first made no sense on its own. The famous time-dependent equation, the one with the i, arrived only in the fourth paper, months later.

The modern answer to “why this form?”

There is a deeper account of why the equation looks the way it does, and it never mentions plane waves.

Start from two requirements. First, whatever the wavefunction does, the total probability of finding the particle somewhere must stay exactly one. Second, time evolution should be smooth, and it should not depend on when you start the clock. In 1932 the mathematician Marshall Stone proved what any evolution meeting those conditions must look like. A single operator generates it, an operator of the kind whose measured values are always real numbers. The rate of change of the state equals that operator acting on the state, times −i.

Identify that operator with energy, divide by ħ to fix the units, and the equation appears:

iħ ∂Ψ/∂t = ĤΨ

This version shows which part is forced and which part is physics. The overall shape, first order in time with an i on the left, follows from conserving probability. What goes inside Ĥ follows from nothing. Kinetic energy as a second derivative, the particular potential, the particle’s mass: those come from experiment. The mathematics fixes the grammar. Nature supplies the vocabulary.

Where quantisation comes from

Most of the time physicists do not solve the full equation. They hunt for stationary states, solutions whose shape never changes and whose complex phase simply rotates at a steady rate:

Ψ(x, t) = ψ(x) e^−iEt/ħ

Substitute that into the full equation and the time dependence cancels, leaving an equation in space alone:

−(ħ²/2m) d²ψ/dx² + V(x)ψ = Eψ

Written compactly, Ĥψ = Eψ. This is the equation Schrödinger published first, and it explains the title of his papers, Quantisation as an Eigenvalue Problem. For most values of E the equation has solutions, but they blow up at infinity or refuse to vanish where they must. No real particle can have them. Only special values of E, the eigenvalues, give well-behaved solutions, and those are the allowed energies.

This is the conceptual heart of the whole enterprise. Before 1926, quantisation was a rule added by hand; Niels Bohr simply declared that only certain orbits were allowed. Schrödinger’s first paper opened by promising a version with no whole-number rules at all. In it, nobody declares anything. Discreteness falls out of the equation plus sensible boundary conditions, just as a guitar string rings only at certain notes because its ends are pinned.

The simplest case: a particle in a box

Trap a particle between two impenetrable walls a distance L apart, with no force between them. Inside, V is zero, and the walls force ψ to vanish at both ends. The only waves that fit are those with a whole number of half-wavelengths across the box, so the allowed energies are:

E_n = n²h² / 8mL², n = 1, 2, 3, …

Three consequences come straight out of that line. Energy comes in steps, spreading apart as n². The lowest energy is not zero: at n = 1 the particle still has energy, so it can never sit perfectly still. That zero-point energy is the uncertainty principle expressed in the language of waves. And shrinking the box raises every level, in proportion to 1/L².

A particle in a box: the first three states

That last feature shows up in a laboratory flask. A semiconductor crystal a few nanometres across acts, to a first approximation, as a box for its electrons. Smaller crystals have wider-spaced levels, so they glow bluer, and a row of vials of these quantum dots can span the rainbow by size alone. Their discovery and synthesis won the 2023 Nobel Prize in Chemistry.

The first triumph: the hydrogen atom

The first real test was hydrogen: one electron held to one proton by the electric force. Schrödinger solved it in his first paper, and the allowed energies came out as:

E_n = −13.6 eV / n²

Those are the levels Bohr had found in 1913, and they reproduce the spectral lines hydrogen emits when its electron drops between levels. The difference lay in how the answer arrived. Bohr needed an unexplained rule about orbits. Schrödinger needed only the equation and the requirement that the wavefunction stay finite.

The solutions also delivered something Bohr’s model never could: shapes. The orbital pictures in every chemistry textbook, the spherical s orbitals and the dumbbell-shaped p orbitals, are plots built from solutions to this equation. Add electron spin and the Pauli exclusion principle, and the structure of the periodic table follows. The whole of chemistry sits on top of this one calculation.

What Ψ is: Born’s footnote

The equation says how Ψ changes. It does not say what Ψ is, and Schrödinger got that part wrong.

He hoped Ψ described something physical, the electron smeared out into a wave of charge. That picture fails fast. A wavepacket for a free particle spreads without limit, yet every detector records electrons as single, sharp hits. And for two or more particles, Ψ is not a wave in ordinary space at all. It lives in a space with three dimensions for every particle.

The reading that stuck came from Max Born, in a paper on electron scattering received in June 1926. His main text tied the wavefunction directly to probability. Then, in a note added at the proof stage, he corrected himself: the probability goes with the square. In modern notation:

probability density = |Ψ(x, t)|²

This Born rule is a separate postulate. The Schrödinger equation neither contains it nor produces it. Born shared the 1954 Nobel Prize in Physics for this and related work. Schrödinger stayed uneasy with it for the rest of his life.

A deterministic equation for a random world

Put the two rules side by side and the central puzzle of quantum mechanics appears.

The Schrödinger equation is deterministic, linear, and conserves probability exactly. Left alone, a wavefunction evolves smoothly and predictably forever, superpositions and all. Yet every measurement returns one outcome, at random, with odds set by the Born rule. Afterwards the wavefunction seems to have jumped to match the result. Nothing in the equation describes that jump.

So textbook quantum mechanics runs on two rules that do not fit together: smooth evolution when nobody looks, and a sudden probabilistic jump when somebody does. What counts as looking is never defined. This is the measurement problem, and a century on it remains open. Decoherence explains why superpositions become practically invisible once a system tangles with its surroundings. On most readings, it does not by itself explain why one particular outcome occurs.

Schrödinger saw the problem clearly. In 1935 he invented his cat, alive and dead at once in a sealed box, to show how absurd the equation’s linearity becomes when pushed to everyday scales. The same year he coined the word entanglement for the correlations that make the absurdity possible.

Where the equation stops working

The Schrödinger equation is not the last word, and its limits are well mapped.

It is non-relativistic. The kinetic term p²/2m is the low-speed approximation to Einstein’s energy formula. The equation also treats time and space unequally, with one time derivative against two space derivatives. For fast particles it fails. Schrödinger’s abandoned relativistic equation, now named after Oskar Klein and Walter Gordon, handles particles without spin. Paul Dirac found the correct relativistic equation for the electron in 1928, and spin fell out of it naturally.

In the Schrödinger equation, spin has to be added by hand. Wolfgang Pauli did it in 1927 with a two-component wavefunction, which is good enough for most of atomic physics.

The deepest limit is that the equation keeps the number of particles fixed. It cannot describe an electron and a positron appearing out of pure energy, which happens routinely at high energies. That takes quantum field theory, where particles are excitations of fields that fill even empty space. The Schrödinger equation reappears there as the low-energy limit, valid when particles are neither created nor destroyed.

The equation behind chemistry and quantum computers

In 1929 Dirac gave a famous verdict. The laws behind most of physics and all of chemistry were now known, he wrote. The trouble was that applying them exactly produced equations “much too complicated to be soluble.”

The reason is structural. For N particles, Ψ depends on 3N coordinates at once. Give each coordinate a modest ten grid points, and a single water molecule, with its ten electrons, already needs 10³⁰ numbers. The cost grows exponentially with size, so exact solutions stop at a handful of particles. Everything bigger, from drug molecules to battery materials, relies on clever approximations, such as the density functional methods honoured by the 1998 Nobel Prize in Chemistry.

Feynman turned the problem inside out in a 1981 lecture. If quantum systems are exponentially hard to simulate on ordinary computers, build a computer out of quantum systems. That is what a quantum computer is: a machine whose qubits evolve under a Hamiltonian the programmer chooses. Every gate is a controlled slice of Schrödinger evolution. Whether such machines have yet beaten classical ones on any task is the question behind every quantum advantage claim.

Tested at every scale so far

If the Schrödinger equation holds without limit, then anything, however large, can in principle sit in a superposition. That claim is testable, and physicists keep testing it with bigger objects.

The method is matter-wave interference: send particles through a set of gratings and look for the stripes that only waves produce. Electrons showed the effect in the 1920s. By 2019 Markus Arndt’s group in Vienna had seen interference with molecules of up to 2,000 atoms, whose quantum wavelength was about five orders of magnitude smaller than the molecules themselves.

In January 2026 the same group reported a new record in Nature. Sodium nanoparticles, each holding more than 7,000 atoms and weighing over 170,000 atomic mass units, passed through gratings made of ultraviolet light and interfered. Each cluster was spread over a distance more than ten times its own diameter. The result sets the tightest bound yet on theories that modify the Schrödinger equation to make large objects behave classically. The equation turned a hundred, and it held.

Established, contested, unproven

Established. The Schrödinger equation describes non-relativistic quantum systems with extraordinary accuracy, from single atoms to molecules and solids. A century of energy levels, chemical bonds and interference patterns confirms it. Superposition has been seen directly in objects of more than 7,000 atoms. The equation is a postulate, not a derived result, and the Born rule is a separate postulate on top of it.

Contested. What the wavefunction is. Some physicists treat it as a real physical object, others as a summary of what an observer knows. Whether the equation applies to everything, observers and instruments included, as the many-worlds interpretation holds, or whether measurement involves something extra. The measurement problem has several proposed solutions and no consensus.

Unproven. Whether the equation fails at some scale. Objective-collapse theories add small terms that make large superpositions decay by themselves. Each larger interference experiment rules out more of those modifications, and none has yet seen a deviation. How the equation behaves when gravity matters, with a massive object itself in superposition, remains untested territory.