ladder operators
There is a purely algebraic way to solve the harmonic oscillator that never touches a differential equation. The trick is to build two operators that step you up and down the ladder of energy levels one rung at a time — raise a state to the next level, or lower it to the one below. This is one of the most beautiful shortcuts in quantum mechanics, and it generalizes to become the backbone of quantum field theory.
For the oscillator, define a = sqrt(m omega / 2 hbar) (x + i p / (m omega)) and its adjoint a-dagger, which factor the Hamiltonian. Their key property is the commutator [a, a-dagger] = 1, and in terms of them H = hbar omega (a-dagger a + 1/2). The operator N = a-dagger a is the number operator with eigenvalue n. The lowering operator acts as a|n> = sqrt(n)|n-1> and the raising operator as a-dagger|n> = sqrt(n+1)|n+1>. The whole spectrum is generated by finding the ground state from a|0> = 0 (there is no lower rung) and then repeatedly applying a-dagger, with energies (n + 1/2) hbar omega falling out automatically.
The reason this matters far beyond one toy problem: in quantum field theory a-dagger and a become creation and annihilation operators that add or remove one quantum — one photon, one phonon, one particle — from a field mode, so the number n is reinterpreted as a particle count. The same algebra reappears for angular momentum as the raising and lowering operators J_+ and J_-, which step between the magnetic quantum numbers m. Learn the ladder once and you have a master key.
Starting from the ground state |0> defined by a|0> = 0, one photon-like excitation is a-dagger|0> = |1>, two is (a-dagger)^2|0>/sqrt(2) = |2>, and the energy climbs in equal steps of hbar omega.
Build the whole tower from the vacuum by repeated raising: no differential equation required.
a and a-dagger are not Hermitian and so are not observables themselves; their Hermitian combinations (x, p, and N = a-dagger a) are the measurable quantities.