Annihilation and creation operators

Over the complex numbers, u² + v² = (u − iv)(u + iv), and the two brackets may be written in either order because numbers commute. The dimensionless oscillator is exactly that shape: Ĥ = ½ℏω(X² + P²), with X = √(mω/ℏ) x̂ and P = p̂/√(mℏω) both Hermitian and obeying [X, P] = i. Try the factorisation anyway. Expanding (X − iP)(X + iP) gives X² + P² + i(XP − PX) = X² + P² + i[X, P], and i[X, P] = i⋅i = −1. The identity misses — but it misses by a number, not by an operator, which is the whole reason the method survives. Reversing the order flips the sign of that residue, so the two products differ by 2 before any scaling. Absorb a √2 into each bracket and the residue becomes ½ in each direction: a†a = ½(X² + P²) − ½ and aa† = ½(X² + P²) + ½. Nothing has been solved. One operator has been rewritten in two ways, and the difference between them is a number.

You cannot add x̂ to ip̂: one is a length, the other a momentum. The quadratures fix that by dividing out the two scales the Hamiltonian itself supplies, a length √(ℏ/mω) and a momentum √(mℏω), so X and P are pure numbers and X + iP is legal. In laboratory clothing the same operator reads a = √(mω/2ℏ)(x̂ + ip̂/mω). For ¹H³⁵Cl, with reduced mass 1.627 × 10⁻²⁷ kg and ω = 5.436 × 10¹⁴ rad s⁻¹, that length is 10.92 pm and √(ℏ/2mω) is 7.72 pm, set against a bond of 127 pm. The √2 is not decoration either. Define b = X + iP without it and you get [b, b†] = 2 and Ĥ = ½ℏω(b†b + 1): the same physics with every later factor doubled. Demanding [a, a†] = 1 is what fixes the normalisation once and for all.

Hermiticity is the licence to measure, and a fails it: a† is not a, so the value of a is not something an apparatus returns. What is Hermitian is built from the pair — a + a† = √2 X, i(a† − a) = √2 P, and both a†a and aa†, since (a†a)† = a†(a†)† = a†a. Worse than non-Hermitian, a is not normal: normality means [a, a†] = 0, and here that commutator is the identity, nonzero in every state. So the spectral theorem does not apply and a has no orthonormal eigenbasis; its eigenvectors, met later as coherent states, are non-orthogonal and over-complete. The pair is lopsided too. Take norms in aa† = a†a + 1 to get ‖a†ψ‖² = ‖aψ‖² + ‖ψ‖², so a† is bounded below and can never send a state to zero, while a can, and does, at the foot of the ladder. That single asymmetry is why the spectrum has a bottom and no top.

Suppose a and a† were d × d matrices. The trace is cyclic, so tr(aa†) = tr(a†a) and tr[a, a†] = 0, while tr 1_d = d, never zero. No finite representation of [a, a†] = 1 exists, in any dimension and any basis. Boundedness fails as well: induction on the commutator gives [a, (a†)ⁿ] = n(a†)ⁿ⁻¹, no power of a† can vanish, and taking operator norms leaves n ≤ 2‖a‖‖a†‖ for every integer n, which is the Wintner–Wielandt theorem. So a and a† are unbounded, defined not on the whole Hilbert space but on the dense domain of states of finite mean energy. In practice that rule governs every Fock-space computation: the d × d truncation obeys [a_d, a_d†] = 1_d − d |d−1⟩⟨d−1|, exact on every rung but the last and wrong by d on that one, so convergence is tested by raising d, never by trusting the top of the basis.

Inverting the definitions costs one line: X = (a + a†)/√2 and P = i(a† − a)/√2, or in laboratory units x̂ = √(ℏ/2mω)(a + a†) and p̂ = i√(mℏω/2)(a† − a). The payoff is structural. Because a and a† each shift the rung index by one, x̂ and p̂ are tridiagonal in the energy basis while x̂² and p̂² are pentadiagonal, so a moment that would have been a Gaussian integral becomes a count of which terms survive orthogonality. The same inversion is what makes the pair worth defining far from any spring: in a field theory the field is the observable, and a†, a are the operators that add and remove one quantum of each of its modes. The algebra travels; the picture of a mass on a spring does not have to.

Because ½(X² + P²) = a†a + ½ = aa† − ½, you may write the Hamiltonian in whichever ordering suits you, and normal ordering — pushing every a† to the left, then discarding the leftover number — is standard in field theory, where summing ½ℏω over infinitely many modes otherwise diverges. What you may not do is forget that a choice was made. Ĥ and ℏω a†a are different operators, differing by ℏω/2 times the identity, and they stop differing by a constant the moment ω changes. There is a numerical version of the same trap. In a d = 4 truncation, a†a + ½ returns diag(0.5, 1.5, 2.5, 3.5) exactly, while ½(X² + P²) built from the truncated X and P returns diag(0.5, 1.5, 2.5, 1.5). Identical operators, different matrices: a†a takes rung n down to n − 1 and back, while the aa† inside ½(X² + P²) must visit rung n + 1, which on the top rung is exactly the one the truncation removed.