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For the one-dimensional quantum harmonic oscillator,This is the Time-dependent Schrodinger equation with harmonic potential.

The quantum harmonic oscillator Hamiltonian is
$H=\hslash \omega (N+\frac{1}{2})$. Thereforeso the evolved state remains an $A$-eigenstate, with eigenvalue
$\boxed{z{e}^{-i\omega t}}$.

The coherent-state overlap givesFor the classical period $T=2\pi /\omega$, this probability equals one whenever $t=kT$.

In units with $\hslash =1$, the quantum harmonic oscillator Hamiltonian and number operator areThe nontrivial commutators among the creation and annihilation operators and $N$ areTheir reversed commutators have the opposite signs, while the commutator of any operator with itself is zero.

The operator $N$ is self-adjoint becauseIts eigenvalues are therefore real. Moreover, for every state $|\psi ⟩$,so every eigenvalue is nonnegative.

If $N|\nu ⟩=\nu |\nu ⟩$, the commutator with $A$ givesandRepeated application of $A$ lowers the eigenvalue by one. It must terminate before producing a negative eigenvalue. If $m$ is the last occupied rung, then $A|\nu -m⟩=0$, and the norm identity forces $\nu -m=0$. HenceThis is the integer spectrum of the number operator.

Take $B>0$ and use the Landau gauge $\mathbf{A}=(0,Bx,0)$. With zero $z$-momentum, the Hamiltonian becomesFor a state $e^{i{k}_{y}y}\chi (x)$, this is a quantum harmonic oscillator in $x$, centred atwith cyclotron frequencyThe Landau levels are therefore

Periodicity in $y$ quantizes $k_y=2\pi \ell /R_y$. Requiring the corresponding oscillator centre to lie in an $x$ interval of length $R_x$ leavesallowed centres for each $n$. This is the degeneracy of a Landau level, equal to the magnetic flux through the rectangle divided by the magnetic flux quantum $h/|e|$. Consistent magnetic periodic boundary conditions make $D$ an integer.

Expanding the Hamiltonian about a static classical solution removes the linear term by its Euler-Lagrange field equation. For a canonical real scalar field with potential $U$, the quadratic fluctuation operator is $\mathcal{H}=-{\partial }_X^2+U^{\prime \prime }({\phi }_{\mathrm{cl}})$. Positive-frequency normal modes become quantum harmonic oscillators under canonical quantization, yielding the Gaussian contribution to a one-loop soliton mass correction. A negative eigenvalue signals an instability, while a physical zero mode in field theory must be replaced by a collective coordinate.

For a quantum harmonic oscillator with $\hslash =1$, $\omega >0$ and inverse temperature $\beta >0$, the canonical partition function is ${\sum }_{r\ge 0}{e}^{-\beta \omega (r+1/2)}$. The geometric series gives the displayed formula. Keeping the zero-point energy is essential to its prefactor.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.