The Schrödinger Equation

In the last chapter the wavefunction ψ(x) gave us the full statistical description that quantum mechanics offers of a particle at one instant: square its magnitude and you have a probability density, whose area over any region is the chance of finding the particle there. But a description of one instant is only half of a physical theory. Classical mechanics earns its keep because Newton's second law, F = ma (force equals mass times acceleration), tells you how a state changes: supply a position and a velocity now and it hands you the whole future trajectory. Quantum mechanics needs the same thing for ψ, an equation of motion that turns the wavefunction now into the wavefunction later. That law is the Schrödinger equation, and nearly everything in the rest of this course, from the particle in a box to the qubit, runs on it.

By 1925 the ingredients lay scattered across the bench. Louis de Broglie had proposed in 1924 that every particle carries a wave of wavelength λ = h/p, and Werner Heisenberg, soon joined by Max Born and Pascual Jordan, had just built matrix mechanics, an abstract scheme that predicted spectra without offering any picture. Erwin Schrödinger, a 38-year-old professor at the University of Zurich, wanted the picture. According to Felix Bloch's later recollection, Peter Debye remarked at a Zurich colloquium that if electrons are waves, they ought to obey a wave equation. Over the Christmas holidays of 1925, in the Alpine resort of Arosa, guided by William Rowan Hamilton's analogy between light rays and particle paths, Schrödinger found one, and within weeks he had solved it for hydrogen.

Here is the equation for one particle of mass m moving along a line, first in words: iħ times the rate at which the wavefunction changes in time equals the Hamiltonian acting on the wavefunction. Three pieces of notation need unpacking. The rate of change is a partial derivative, written with a curly d: the rate at which ψ changes with time while you stand still at one position x, as a weather station records how fast the temperature changes at its own location. The constant ħ is the reduced Planck constant, h divided by 2π, about 1.055×10⁻³⁴ J·s: the exchange rate between angular frequency and energy, E = ħω. And H-hat, a capital H wearing a small hat, is the Hamiltonian operator, named after Hamilton: an operator is a rule that turns one function into another, and this one encodes the particle's total energy.

The Hamiltonian has exactly two parts, mirroring the classical bookkeeping that total energy is kinetic plus potential. The kinetic part takes the second derivative of ψ with respect to position and multiplies it by −ħ²/2m. The potential part multiplies ψ at each point by V(x), the potential energy the particle would have there. Altogether: iħ times the time-rate of change of ψ equals −(ħ²/2m) times its second space-derivative, plus V(x)ψ. Why the imaginary unit i, the square root of −1? Swap the i for a plain minus sign and the equation becomes the diffusion equation that governs heat soaking through a metal bar: ψ would smear out and fade, and total probability would leak away. The i converts that fading into rotation of the phase of ψ in the complex plane, which is why ψ must be complex (Schrödinger himself resisted this at first) and why the total area under |ψ|² stays exactly 1 for all time, a consequence of what physicists call unitarity.

The second derivative of a function measures its curvature: how sharply its graph bends at each point. A gently undulating wavefunction has small curvature; one crammed with tight wiggles has large curvature. Because the kinetic term is built from that second derivative, the rule of thumb is wonderfully visual: more bending means more kinetic energy. This is de Broglie's relation in disguise, since many wiggles per metre means a short wavelength, a large momentum p = h/λ, and therefore a large kinetic energy p²/2m. It also explains the uncertainty-principle lesson: squeeze a particle into a small region and its wavefunction is forced to bend sharply, so confinement always costs kinetic energy.

The potential V(x) is the landscape the wave travels across: valleys where a classical ball would settle, hills it must climb, cliffs it bounces off. For a particle of definite energy E, the equation says the curvature of ψ is proportional to (V − E) times ψ. Where the energy exceeds the local potential, the curvature points back toward the axis, so ψ oscillates, with an approximate local wavelength λ = h/√(2m(E − V)) that stretches where the particle is slow and tightens where it is fast. Where V exceeds E, a region forbidden to a classical particle, the curvature points away from the axis and ψ grows or decays exponentially. In a forbidden region that extends forever only the decaying tail is allowed, since a growing one could never be normalized; that tail is the seed of quantum tunnelling, which awaits you in Chapter 12.

The equation becomes far friendlier when the particle has a definite energy. Guess that ψ(x, t) is a fixed spatial shape φ(x) multiplied by a factor depending only on time, a technique called separation of variables. Substituting the guess shows that the time factor must be a pure phase, a complex number of size 1 circling at angular frequency ω = E/ħ, while the shape must satisfy a simpler rule: the Hamiltonian acting on φ gives back φ multiplied by the energy E. This is the time-independent Schrödinger equation. It is an eigenvalue equation: the operator acting on the special function φ returns the very same function multiplied by a plain number. The function is an eigenfunction (from the German eigen, meaning own or characteristic), the number is its eigenvalue, and here the eigenvalue is the energy.

Now the payoff. The probability density of such a solution is |φ|² times the squared size of the phase factor, and a pure phase always has size 1. So |ψ(x, t)|² = |φ(x)|² at every instant: the probability cloud does not drift, swell or shrink, and every measurable average, such as ⟨x⟩ or the average momentum, is frozen too. These are stationary states, the honest replacement for Bohr's stationary orbits. (Real excited atoms do eventually emit a photon and decay, because they couple to the electromagnetic field, which this single-particle equation leaves out; the ground state is genuinely stable.)

Do not mistake stationary for motionless. Hydrogen's ground state is stationary, yet its electron has an average kinetic energy of 13.6 eV, a typical speed of about 2.2×10⁶ m/s, roughly 0.7 percent of the speed of light; measure the momentum in many identically prepared atoms and you get a broad spread of results. A guitar string sounding a pure note is a good picture: every point vibrates furiously, but the pattern holds its shape. What is stationary is the probability distribution, not the particle. Interpretations differ about what, if anything, the electron is doing between measurements (Chapter 22), but all of them predict that same nonzero spread of momenta.

Two properties of the equation deserve to be engraved over the laboratory door. The first is determinism. Give it the complete wavefunction at one instant and the potential, and it fixes the wavefunction at every later instant, uniquely, smoothly and continuously, with no randomness anywhere. The randomness of quantum mechanics enters somewhere else: when a measurement is made, the Born rule assigns probabilities to the possible outcomes. How to understand that step physically is the measurement problem, which different interpretations answer differently, as you will see in Chapters 15 and 22. On one point there is no dispute: the Schrödinger equation, as written, rolls no dice.

That puts the phrase quantum jump in perspective. Schrödinger loathed the idea of electrons leaping instantaneously and at random between orbits; Heisenberg later recalled him telling Bohr, in Copenhagen in the autumn of 1926, that if these damned quantum jumps were here to stay he regretted ever getting involved with quantum theory. Under the equation, an atom absorbing light changes state continuously, amplitude flowing smoothly from the lower level to the upper. What looks abrupt and random is the result of a measurement. In 2019 Zlatko Minev, Michel Devoret and colleagues at Yale caught a superconducting artificial atom mid-jump: when a jump starts is random, but once begun it unfolds continuously and coherently, enough to be anticipated and even reversed in flight.

The second property is linearity. Every term contains ψ exactly once, to the first power: no ψ², no square roots of ψ. The consequence is the superposition principle: if ψ₁ and ψ₂ are solutions, so is any combination aψ₁ + bψ₂ with complex constants a and b. Don't panic — superpose! Linearity is why the double-slit amplitudes of Chapter 7 simply add, and it is the master key to the subject: whenever the potential does not change in time, any solution can be assembled from stationary states, each turning its own phase clock, so solving the time-independent equation once solves the time-dependent one for every starting condition.

Superposition is also where motion comes from. Mix two stationary states whose energies differ by ΔE and their phase clocks drift in and out of step, so the probability cloud oscillates at frequency f = ΔE/h. For hydrogen's ground state mixed with its 2p excited state that is 2.47×10¹⁵ hertz, the very frequency of the ultraviolet photon the atom emits when it drops from 2p to the ground state. Bohr had to postulate his frequency rule; Schrödinger's equation produces it as an oscillating cloud of charge.

Now set V = 0 and let a particle fly free. The simplest solution, a plane wave of a single wavelength, has a perfectly definite momentum but is spread uniformly over all of space, so it cannot describe a particle that is here rather than there. A localized particle is a wave packet: a superposition of many plane waves whose wavelengths cluster around a central value, reinforcing one another in one region and cancelling everywhere else, the Fourier trade-off you met in Chapter 8. By linearity, the packet simply evolves as the sum of its ingredients.

Each ingredient wave advances at its own phase velocity, ω/k, which for a free particle is ħk/2m, only half the classical speed. The packet as a whole moves at the group velocity, the speed of the region of constructive interference, given by the derivative dω/dk. For a free particle that is ħk/m = p/m, exactly the classical velocity: the centre glides like a Newtonian particle, a special case of Ehrenfest's theorem (1927), which shows that quantum averages obey classical-looking equations of motion. Phase velocity has no physical meaning in quantum mechanics, since shifting the zero of energy changes it without changing any prediction. In the lab you will see the ripples inside a packet slide backward through its envelope, travelling at half its speed.

Nor does the packet keep its shape. Its ingredients carry a range of momenta, so the fast components pull ahead of the slow ones; this dispersion makes a free packet spread. For a Gaussian packet whose initial width, the standard deviation of |ψ|², is σ₀, the width grows as σ(t) = σ₀·√(1 + (t/τ)²), with a characteristic time τ = 2mσ₀²/ħ. Narrow packets spread fastest, because squeezing σ₀ widens the momentum spread. An electron localized to 0.1 nm, roughly the size of an atom, has a spreading time τ of about 1.7×10⁻¹⁶ seconds and is about 6 nm wide after 10 femtoseconds. A one-gram marble localized to a micrometre has a τ of about 1.9×10¹⁹ seconds, more than forty times the age of the universe, one reason the everyday world looks so reassuringly classical.

Taken alone, the time-independent equation has solutions for any energy you care to try. Physics enters through boundary conditions, the requirements a wavefunction must meet to describe a real particle. A well-behaved wavefunction is single-valued, with one amplitude at each point; finite everywhere; normalizable, meaning the total area under |ψ|² is finite and can be scaled to exactly 1, so a bound particle's ψ must die away far from the well; and continuous, with a continuous slope wherever the potential is finite. At an infinitely high wall, where the particle can never be found, ψ must fall to zero, although its slope may jump.

These modest-sounding demands are what produce quantization. Pick an energy, start a solution as a tiny decaying tail deep in the forbidden region on the left, and integrate across a potential well. For almost every energy the solution emerging on the far side curves away from the axis and races off toward plus or minus infinity, which no real particle can do. Only at special, discrete energies does the tail settle gracefully to zero on both sides; those are the allowed levels. No one has to postulate them, as Bohr did; they fall out of the requirement that ψ behave itself, just as a guitar string pinned at both ends can sound only particular notes.

This is exactly what made Schrödinger famous. In his first 1926 paper he inserted the Coulomb potential energy of an electron and a proton, V(r) = −e²/(4πε₀r), solved the time-independent equation in three dimensions, and demanded well-behaved solutions. Out came Eₙ = −13.6 eV/n², precisely Bohr's levels and therefore the Balmer lines, with more besides: orbitals instead of orbits, a ground state with zero orbital angular momentum where Bohr had insisted on ħ, and a method that extends to molecules and solids. The equation is non-relativistic (Dirac's equation of 1928 refines it), yet for chemistry, materials and most of this course it is the engine. You will dissect hydrogen properly in Chapter 16.