Building the Schrödinger equation

Start with the trial wave ψ = A exp(i(kx − ωt)) and the two relations E = ℏω, p = ℏk. Differentiating once in time brings down a factor −iω; differentiating twice in position brings down −k². The classical energy of a slow free particle, E = p²/2m, becomes ℏω = ℏ²k²/2m in wave variables — one power of ω set against two powers of k. That single count fixes the shape of the equation: first order in time, second in position. Now assemble it. Since ∂ψ/∂t = −iωψ, multiplying by iℏ gives ℏωψ; since ∂²ψ/∂x² = −k²ψ, multiplying by −ℏ²/2m gives (ℏ²k²/2m)ψ. Setting those two equal is exactly ℏω = ℏ²k²/2m, for every k at once. Add potential energy and the classical relation becomes E = p²/2m + V, so a term Vψ joins the right-hand side. Put a number on it: an electron at λ = 0.10 nm has k = 6.28 × 10¹⁰ m⁻¹, so ω = ℏk²/2m = 2.29 × 10¹⁷ rad s⁻¹ and E = 150 eV — the working energy of a low-energy electron diffraction beam, recovered from the wave relations alone.

Write the free law in its most general first-order form, ∂ψ/∂t = C ∂²ψ/∂x², and put a mode ψ = exp(ikx) f(t) into it. Then f′ = −Ck²f, so f = exp(−Ck²t). Split C into modulus and phase, C = |C| exp(iθ): the amplitude falls as exp(−|C|k²t cos θ) while the phase turns at |C|k² sin θ. Real C, meaning θ = 0, is the diffusion equation. Its modes decay and never oscillate, so there is no frequency to identify with E/ℏ, no state of definite energy, and no conserved total probability. Only θ = 90°, a purely imaginary C, gives pure rotation: |ψ| is untouched and ω = |C|k², which matches ℏω = ℏ²k²/2m when |C| = ℏ/2m = 5.79 × 10⁻⁵ m² s⁻¹. The two behaviours are not subtly different. For an electron with k = 1.0 × 10¹⁰ m⁻¹, a real coefficient would drop the mode to 1/e of its height in 0.17 fs, while the imaginary one holds every bit of it and turns the phase through a full cycle every 1.09 fs.

Because only one time derivative appears, ψ(x, 0) alone determines ψ at every later instant. Compare what classical physics needs: Newton's law wants a position and a velocity, and the classical wave equation wants a displacement and its time derivative. Quantum mechanics gets away with one function because that function is complex and so is already carrying two real fields — |ψ| says where the particle is likely to be found and arg ψ says where it is going, since the local momentum is ℏ ∂(arg ψ)/∂x. That is what the complexity buys, and it is why 'the state' and 'the wave function at one instant' are the same object in this course. Go second order in time instead, as you must if you substitute into E² = p²c² + m²c⁴, and the bargain collapses: the Klein–Gordon equation needs ∂ψ/∂t at t = 0 as extra initial data, its plane waves come with ℏω of both signs, and the density it conserves is not |ψ|² and is not positive, so it cannot be read as a probability at all.

Every term contains exactly one power of ψ, so any combination aψ₁ + bψ₂ of solutions is a solution and the coefficients set at t = 0 are carried along untouched. Three things follow. An arbitrary initial state can be expanded on a convenient basis and evolved term by term, which is the method of the next four units. A two-path cross term survives the evolution, so interference is a consequence of the dynamics rather than an extra rule. And the map from ψ(0) to ψ(t) is a linear operator that, whenever V is real, also preserves the norm — the evolution is unitary. The cost is worth stating plainly: a linear, deterministic, norm-preserving law can never turn a spread-out superposition into one definite detector click, which is why the measurement postulate has to be a separate axiom rather than a theorem. Nor is a nonlinear patch free. A term like |ψ|²ψ makes the physics depend on how the state was normalised; such equations earn their place as mean-field approximations to many-body systems, never as the fundamental law.

Four assumptions sit on the face of it. It is non-relativistic, because E = p²/2m went in, and the mismatched derivative orders — one in time, two in space — mean it cannot be Lorentz covariant; a 1.5 eV electron has v/c = 2.4 × 10⁻³ and is safe, a 100 keV electron in a microscope has v/c = 0.55 and is not. It is spinless, because ψ is one complex number at each point with no internal index; an electron needs the two components of the Pauli equation or the four of Dirac's. It describes one particle in one dimension: three dimensions replace ∂²/∂x² with ∇², while N particles need a single wave function on 3N coordinates rather than N waves in space, which is where entanglement will come from. And V(x, t) is an external classical field, prescribed rather than solved for, with no back-reaction — and it must be real, since a complex V deliberately breaks norm conservation and is used precisely as a model of absorption.

The plane-wave construction above is a plausibility argument, not a proof. It leans on states that are not normalisable, so they are not states of the theory at all, and the substitution rule E → iℏ ∂/∂t is the physical content rather than a step in the algebra. The history was messier still: Schrödinger reached the equation through a Hamilton–Jacobi and optical analogy, tried the relativistic version first, and set it aside because it gave the wrong fine structure for hydrogen — the missing ingredient being spin, which nothing in the argument had suggested. What warrants the equation is the ledger of what it then predicted correctly: the hydrogen spectrum with the measured Rydberg constant, the evenly spaced vibrational ladder of molecular spectra, tunnelling currents falling exponentially with barrier width closely enough that a scanning tunnelling microscope resolves single atomic steps, and the band structure of solids. Say in an exam that it is assumed and tested, never that it is proved.