Heisenberg: Start from Observables
In 1925, Werner Heisenberg took a radical methodological stance: he would build a theory using only quantities that could be directly measured — spectral line frequencies and intensities. He would not speculate about electron orbits that could not be observed.
Spectral lines come in pairs: a photon emitted in a transition from energy level m to level n has frequency ν(m,n). Heisenberg represented these frequencies as a two-dimensional array — a matrix. The equations governing how these arrays combine turned out to be the rules of matrix multiplication.
Result: matrix mechanics. Physical observables become matrices. States become vectors. The equation of motion is a matrix equation. The energy levels of the hydrogen atom emerge as eigenvalues of the Hamiltonian matrix.
Hamming's framing: Heisenberg's approach is a lesson in scientific method — if a concept cannot be measured, perhaps it should not appear in the theory.
Schrödinger: Start from Waves
Erwin Schrödinger approached from a completely different starting point. Louis de Broglie had proposed that particles have an associated wavelength λ = h/p (momentum p, Planck's constant h). Schrödinger asked: if electrons are waves, what is the wave equation?
He found the Schrödinger equation (time-independent form):
Ĥψ = Eψ
where Ĥ is the Hamiltonian operator, ψ is the wave function, and E is energy. Solutions ψ satisfying this equation at specific energy values E form standing waves — the electron 'orbitals.'
The quantization of energy levels — the discrete spectral lines — emerges from the boundary conditions on the wave function. Only wave functions that remain finite and continuous everywhere are physical. These constraints admit only specific E values: the eigenvalues.
The Mathematical Unification
Paul Dirac (and independently von Neumann) showed that both matrix mechanics and wave mechanics are representations of the same abstract mathematical structure: Hilbert space.
A Hilbert space H is an inner product space that is also complete (every Cauchy sequence converges). Quantum states are unit vectors in H. Observables are Hermitian operators on H — linear maps from H to H that equal their own adjoint.
Eigenvalues and eigenstates: if an observable  has eigenstate |a⟩ with eigenvalue a:
Â|a⟩ = a|a⟩
Measurement of observable A on a system in eigenstate |a⟩ always returns value a with certainty.
Superposition: a general state |ψ⟩ is a linear combination (superposition) of eigenstates:
|ψ⟩ = Σᵢ cᵢ|aᵢ⟩
where the cᵢ are complex amplitudes satisfying Σᵢ |cᵢ|² = 1 (normalization).
The Born Rule
Max Born proposed the probabilistic interpretation: when observable A is measured on a system in state |ψ⟩ = Σᵢ cᵢ|aᵢ⟩, the probability of obtaining eigenvalue aᵢ equals the squared modulus of its amplitude:
P(aᵢ) = |cᵢ|² = |⟨aᵢ|ψ⟩|²
After measurement, the state collapses to the corresponding eigenstate |aᵢ⟩. Subsequent measurements of A will return aᵢ with certainty until the system evolves again.
A qubit state in the computational basis: |ψ⟩ = α|0⟩ + β|1⟩, with |α|² + |β|² = 1.
Hamming as Mathematical Consultant
Hamming described his role when working with physicists: he would find the class of mathematical functions to use by asking the physicist what they felt was relevant, then fitting the mathematical problem to their beliefs.
> I generally find the class of functions to use by asking the person with the problem, and then use the facts they feel are relevant — all in the hopes I will thereby, someday, produce a significant insight on their part.
This is a deliberate pedagogical strategy. Hamming did not impose a mathematical framework — he elicited the physicist's intuitions and formalized them. The goal: the physicist makes the insight, not Hamming.
The deeper lesson: quantum mechanics is philosophically unsatisfying (what does wave function collapse mean? what is the quantum state really?) but computationally successful. The act-as-if principle: treat the formalism as real — use it as if state vectors, operators, and eigenvalues are actual features of the world — when it gives correct predictions, regardless of whether you can explain what it means.
When Act-As-If Is Justified
The act-as-if principle is not intellectual laziness. It is a specific epistemic choice: prioritize computational reliability over metaphysical clarity when the two come apart.
QM provides the clearest example: the Born rule has been verified experimentally to extraordinary precision. The philosophical question of why the Born rule holds, or what 'wave function collapse' corresponds to physically, remains genuinely unsettled. Hamming's prescription: use the Born rule, act as if collapse happens, build the technology, make the predictions.