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— CH. 1 · INTRODUCTION —

Schrödinger equation

~6 min read · Ch. 1 of 6
6 sections

  • The Schrödinger equation sits at the heart of quantum mechanics, a partial differential equation that governs how quantum systems evolve through time. Erwin Schrödinger, an Austrian physicist, postulated it in 1925 and published it in 1926, and the work it anchored earned him the Nobel Prize in Physics in 1933. Before that equation existed, physicists had no reliable way to describe the behavior of matter at the smallest scales. What does it mean for a particle to have a wave? How does energy become discrete rather than continuous? And why does measuring a quantum system seem to change it? Those are the questions this documentary will pursue.

  • Louis de Broglie proposed that all matter carries an associated wave, and Schrödinger built his equation on that foundation. Before Schrödinger arrived at his formulation, physicist Peter Debye made an offhand remark that if particles behaved as waves, they ought to satisfy a wave equation. That casual comment, by Debye's own account, set Schrödinger in motion. He was also guided by William Rowan Hamilton's analogy between mechanics and optics, the observation that the zero-wavelength limit of optics resembles a mechanical system, where light rays trace sharp tracks obeying Fermat's principle.

    Schrödinger first attempted a relativistic version of the wave equation, using the relativistic energy-momentum relation to arrive at what is now called the Klein-Gordon equation in a Coulomb potential. The relativistic corrections that equation produced, however, disagreed with Arnold Sommerfeld's refined atomic formula. Discouraged, Schrödinger retreated in December 1925 to a mountain cabin with a mistress, setting the relativistic problem aside.

    While secluded, he decided his nonrelativistic calculations were novel enough to publish on their own. Despite difficulties solving the differential equation for hydrogen, Schrödinger sought help from his friend, the mathematician Hermann Weyl, and succeeded. His nonrelativistic wave equation produced the correct spectral energies of hydrogen, published in a paper in 1926. He treated the hydrogen atom's electron as a wave moving in a potential well created by the proton, and the result accurately reproduced the energy levels of the Bohr model.

  • Schrödinger himself was initially uncertain about the physical meaning of the wave function he had discovered. He first tried interpreting the real part of it as a charge density, then revised that proposal to say the modulus squared was a charge density. Neither interpretation held up. In 1926, just days after Schrödinger published that revised proposal, Max Born offered the interpretation that endured: the wave function is a probability amplitude, and its modulus squared gives a probability density.

    Schrödinger later described the wave function in his own words as "the means for predicting probability of measurement results," containing what he called "the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog." That catalog metaphor captures something important. The equation tells you how probabilities shift over time, not what a particle is doing at any given instant.

    In 1952, Schrödinger himself suggested that the different terms of a quantum superposition are "not alternatives but all really happen simultaneously." That remark has since been read as an early gesture toward what Hugh Everett formulated independently in 1956 as the many-worlds interpretation, which holds that all possibilities described by quantum theory occur simultaneously across a multiverse of mostly independent parallel universes.

  • Schrödinger's approach was not the only path through quantum mechanics. Werner Heisenberg introduced matrix mechanics as a separate formulation, and Richard Feynman later developed the path integral formulation. When these approaches are placed side by side, Schrödinger's version is sometimes called "wave mechanics" to distinguish it from the others. All three lead to the same predictions; they differ in the mathematical objects they work with.

    The Schrödinger equation is explicitly nonrelativistic. It contains a first derivative in time and a second derivative in space, which means time and space are not treated on equal footing, a problem for any theory meant to incorporate special relativity. Paul Dirac addressed this by seeking a differential equation that would be first-order in both space and time. Taking the "square root" of the Klein-Gordon operator required factorizing it using 4x4 matrices, now called Dirac matrices. The resulting Dirac equation describes spin-1/2 particles, while the Klein-Gordon equation, which preceded the nonrelativistic Schrödinger equation historically, describes spin-less particles.

    A persistent problem with the Klein-Gordon equation was that it allowed probability densities to go negative, which is physically unviable. Dirac's construction of the square-root operator resolved that issue and, as a consequence, introduced the concept of the antiparticle: the four-component spinor fields that solve the Dirac equation include two components for the particle and two for its antimatter counterpart.

  • The particle in a one-dimensional box is the most mathematically simple example of the Schrödinger equation producing quantized energy levels. The box is defined as a region of zero potential energy bounded by walls of infinite potential energy. The boundary conditions at the walls force the wave function to zero at each end, and that constraint limits the allowed standing waves to discrete values, yielding a discrete set of allowed energies.

    The harmonic oscillator extends the lesson to systems like vibrating atoms, molecules, and ions in crystal lattices. Its lowest-energy solution, called the ground state, carries what is known as zero-point energy, and its wave function is a Gaussian. The eigenvalues are discretized, illustrating what the particle-in-a-box already showed: bound systems under the Schrödinger equation do not have a continuous range of allowed energies.

    The hydrogen atom is the one case for which the Schrödinger equation has been solved exactly. Using spherical polar coordinates and separating the equation into radial and angular parts, the solutions involve generalized Laguerre polynomials and spherical harmonics, characterized by three quantum numbers: the principal, azimuthal, and magnetic quantum numbers. Multi-electron atoms cannot be solved exactly and require approximate methods such as variational techniques, the WKB approximation, and perturbation theory, which treats a complex system as a small modification of one that can be solved precisely.

  • The Copenhagen interpretation holds that a system's wave function encodes statistical information, and that time evolution under the Schrödinger equation is continuous and deterministic, right up until a measurement occurs. At measurement, the wave function changes discontinuously and unpredictably, though the probabilities for different outcomes can be calculated using the Born rule. Relational quantum mechanics and QBism, both more recent frameworks, treat the equation in broadly similar terms.

    Bohmian mechanics takes a different approach by adding a "quantum potential" that exerts a nonlocal guiding force on particles. In this picture, a physical system carries both a wave function and a real definite position at all times; the position evolves deterministically under a guiding equation, and the Schrödinger equation governs the wave function alongside it. Determinism is restored, but at the cost of a nonlocal hidden variable.

    The many-worlds interpretation removes wave function collapse entirely. All possible outcomes of a measurement occur simultaneously in parallel universes, and only the Schrödinger equation governs the full multiverse. What remains contested in that framework is why observers should assign probabilities at all when every outcome is certain to occur somewhere, and why those probabilities should follow the Born rule. Several proposals have been advanced, but no consensus has formed.

Common questions

Who discovered the Schrödinger equation and when was it published?

Erwin Schrödinger, an Austrian physicist, postulated the equation in 1925 and published it in 1926. The work it anchored earned him the Nobel Prize in Physics in 1933.

What does the Schrödinger equation describe in quantum mechanics?

The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. It gives the evolution of the wave function over time, allowing calculation of the probability of different measurement outcomes via the Born rule.

What is the difference between the time-dependent and time-independent Schrödinger equation?

The time-dependent Schrödinger equation describes how a quantum system evolves over time in the most general case. The time-independent version applies when the Hamiltonian does not explicitly depend on time, and its solutions are stationary states, called atomic orbitals in chemistry and energy eigenstates in physics.

What is the physical meaning of the wave function in the Schrödinger equation?

Max Born established in 1926 that the wave function is a probability amplitude, and its modulus squared gives a probability density. Schrödinger himself later described the wave function as the means for predicting the probability of measurement results.

How does the Schrödinger equation differ from the Dirac equation?

The Schrödinger equation is nonrelativistic, with a first derivative in time and a second derivative in space, so time and space are not on equal footing. The Dirac equation, formulated by Paul Dirac, incorporates special relativity and is first-order in both space and time; it describes spin-1/2 particles and reduces to the Schrödinger equation in the nonrelativistic limit.

For which atom has the Schrödinger equation been solved exactly?

The hydrogen atom is the only atom for which the Schrödinger equation has been solved exactly. Multi-electron atoms require approximate methods such as perturbation theory and variational techniques.

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All sources

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