Harmonic-oscillator ladder algebra

For mass m, positive angular frequency $\omega$, and the usual full-line position and momentum operators, define a dimensionless annihilation operator from a scaled combination of x and p. Its adjoint reverses the imaginary momentum sign. The canonical commutator gives $[a,a^\dagger]=1$. Multiplication then factors the oscillator Hamiltonian into the number operator plus one half. The algebra is valid on a common invariant dense domain, such as finite linear combinations of oscillator eigenstates; unbounded operators still require domains.

The number operator is nonnegative because $\langle\psi|a^\dagger a|\psi\rangle=\|a\psi\|^2$. Lowering a number eigenstate reduces its eigenvalue by one until a zero-norm lowering action terminates the ladder. A normalizable lowest state therefore satisfies $a|0\rangle=0$, and repeated raising gives integer labels beginning at zero. Allowing negative occupation would contradict positivity rather than create another branch of bound states. This is an algebraic spectral argument supported by the normalizable ground solution.

For a normalized number state, the squared norm of its lowered vector equals n. The squared norm of its raised vector equals n + 1 because $aa^\dagger=N+1$. Dividing by these norms gives normalized adjacent states and fixes all ladder matrix elements. A lowered superposition generally remains unnormalized, so its norm must be calculated before it is used as a state for a new expectation value. Dropping the square roots changes both spectra and matrix elements.

Position is proportional to $a+a^\dagger$, and momentum to $i(a^\dagger-a)$. Orthogonality of number states immediately makes their mean position and mean momentum zero. Applying position twice leaves only number-preserving terms in a diagonal expectation. The resulting nonzero position variance and nonzero ground energy are compatible with stationary probability density. They do not imply that the ground state follows a hidden classical sinusoid whose phase has merely been forgotten.

Applying a lowering operator is a mathematical operation. A physical photon-subtraction or transition experiment needs a coupling, an outcome model, and normalization conditioned on the outcome. A closed oscillator Hamiltonian does not lower its own number state with time: each energy eigenstate acquires only a phase. An expectation of a noninteger occupation can occur in a superposition or mixture even though a number measurement yields an integer. Mean and possible outcome are distinct concepts.

The algebra is exact for a quadratic potential with positive curvature. An anharmonic term changes both level spacing and the simple ladder relation with energy, even if oscillator functions remain a useful expansion basis. A free particle, inverted oscillator, or finite well cannot inherit this bound spectrum by replacing a parameter name. Recognizing the physical Hamiltonian and its domain is therefore as important as recognizing the commutator identity in a formal calculation.