Quantum harmonic oscillator

Intuition [Beginner]

A mass on a spring is the simplest oscillating system. Pull the mass, release it, and it bounces back and forth at a frequency set by the spring stiffness and the mass. In classical mechanics the total energy can be any positive number — a gentle pull gives a slow oscillation, a hard pull gives a fast one.

In quantum mechanics the same system has discrete energy levels. The allowed energies form a ladder with equally spaced rungs: $E_n=\hslash \omega (n+1/2)$ where $n=0,1,2,3,\dots$ and $\omega$ is the classical oscillation frequency. The gap between any two adjacent levels is always the same amount $\hslash \omega$. This uniform spacing contrasts with the particle in a box (unit 12.04.01), where the spacing grows as $n$ increases. Equal spacing is the signature of the harmonic oscillator and the reason it appears in virtually every corner of physics.

The bottom rung, $n=0$, is the ground state. Its energy $E_0=\hslash \omega /2$ is nonzero. A classical mass sitting at the equilibrium point has zero energy. The quantum oscillator can never be perfectly at rest. This minimum energy is the zero-point energy, a direct consequence of the Heisenberg uncertainty principle: confining the particle near the origin forces a minimum spread in momentum, and the resulting kinetic energy is $\hslash \omega /2$.

The ground-state wave function is a bell-shaped Gaussian curve centred at the origin, falling off smoothly on both sides. Excited-state wave functions are the same Gaussian multiplied by polynomials (Hermite polynomials). The $n$-th state has exactly $n$ nodes — positions where the probability of finding the particle drops to zero. More nodes means higher energy, the same pattern seen in the particle in a box (unit 12.04.01).

For large quantum numbers the probability distribution $|{\psi }_n(x){|}^2$ begins to resemble the classical probability density: the particle spends more time near the turning points and less time near the centre, where it moves fastest. This convergence of quantum behaviour to classical behaviour at large $n$ is an instance of the correspondence principle.

There is an algebraic shortcut that avoids solving any differential equation. Two operators — a lowering operator $a$ and a raising operator $a^{†}$ — step down or up the energy ladder. The Hamiltonian is built from them: $H=\hslash \omega (a^{†}a+1/2)$. The product $a^{†}a$ is the number operator $N$, whose eigenvalue on the $n$-th state is just $n$. Starting from the ground state and applying the raising operator repeatedly generates every eigenstate.

Why does this one model matter so much? Any system near a stable equilibrium oscillates harmonically to first approximation. A diatomic molecule vibrating along its bond, atoms in a crystal lattice rattling about their equilibrium positions, an electromagnetic field mode in a cavity — all are harmonic oscillators. In quantum field theory every free-field mode is an independent oscillator, and the photon number in that mode sits on the ladder $n=0,1,2,\dots$. Master this single system and you hold the key to molecular spectroscopy, solid-state physics, and quantum field theory.

Consider the carbon monoxide molecule. The C–O bond vibrates at $f\approx 6.4\times 10^{13}$ Hz, giving $\omega =2\pi f\approx 4.0\times 10^{14}$ rad/s. The spacing between vibrational levels is $\hslash \omega \approx 2.6\times 10^{-20}$ J, about 0.27 eV. The zero-point energy is half this, about 0.13 eV. At room temperature the thermal energy $k_{B}T\approx 0.026$ eV is an order of magnitude smaller than the level spacing, so nearly all CO molecules sit in the vibrational ground state.

A classical oscillator never ventures beyond its turning points — the positions where all its energy is potential and its velocity passes through zero. The quantum oscillator does. For the ground state the probability of finding the particle beyond the classical turning point is about 16%. The wave function does not cut off; it decays smoothly into the classically forbidden region, a hallmark of quantum tunnelling.

Visual [Beginner]

The picture shows a parabolic potential well with equally spaced energy levels drawn as horizontal lines inside it. Unlike the particle in a box, where the levels spread apart as $n^2$, the harmonic oscillator levels are uniformly spaced at intervals of $\hslash \omega$. The ground state ($n=0$) is a single Gaussian hump centred at the origin with no nodes. Each higher state adds one more node and one more peak.

The classical turning points for level $n$ sit where the horizontal energy line intersects the parabola, at $x=\pm \sqrt{(2n+1)\hslash /(m\omega )}$. Beyond these points the wave function decays but does not vanish — the quantum particle tunnels into the classically forbidden region.

Worked example [Beginner]

The CO molecule is modelled as a harmonic oscillator with reduced mass $\mu \approx 1.14\times 10^{-26}$ kg and vibrational angular frequency $\omega \approx 4.02\times 10^{14}$ rad/s.

Zero-point energy. The ground-state energy is

$$E_0=\frac{1}{2}\hslash \omega =\frac{1}{2}(1.055\times 10^{-34})(4.02\times 10^{14})\approx 2.12\times 10^{-20}\text{J}.$$

Converting to electron-volts ($1\text{eV}=1.602\times 10^{-19}$ J):

$$E_0\approx 2.12\times 10^{-20}/1.602\times 10^{-19}\approx 0.132\text{eV}.$$

Classical turning point. The ground state has $E_0=\frac{1}{2}\hslash \omega =\frac{1}{2}\mu {\omega }^2{A}^2$, so the classical amplitude is

$$A=\sqrt{\frac{\hslash }{\mu \omega }}=\sqrt{\frac{1.055\times 10^{-34}}{1.14\times 10^{-26}\times 4.02\times 10^{14}}}\approx 4.8\times 10^{-12}\text{m}=4.8\text{pm}.$$

The C–O bond length is about 113 pm, so the zero-point vibration amplitude is roughly 4% of the bond length — small but non-negligible.

Probability beyond the turning point. For the ground state, the probability of finding the bond stretched further than the classical amplitude $A$ is about 16% (the exact value involves the complementary error function and evaluates to $\text{erfc}(1)\approx 0.157$). Roughly one measurement in six would find the bond stretched beyond what a classical oscillator at the same energy could ever reach.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A particle of mass $m$ moves in one dimension under the quadratic potential

$$V(x)=\frac{1}{2}m{\omega }^2{x}^2,$$

where $\omega >0$ is the classical angular frequency of the oscillator. The Hilbert space is $\mathcal{H}=L^2(\mathbb{R})$, the space of square-integrable functions on the real line. The Hamiltonian is

$$H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2,$$

where $p=-i\hslash d/dx$ is the momentum operator 12.02.02 pending.

Dimensionless form. Introducing the oscillator length $\ell =\sqrt{\hslash /(m\omega )}$ and the dimensionless variable $\xi =x/\ell$, the time-independent Schrödinger equation $H\psi =E\psi$ becomes

$$\frac{d^2\psi }{d{\xi }^2}+(2\epsilon -{\xi }^2)\psi =0,\epsilon =\frac{E}{\hslash \omega }.$$

For large $|\xi |$ the ${\xi }^2$ term dominates and the asymptotic behaviour is $\psi \sim e^{-{\xi }^2/2}$. Writing $\psi (\xi )=h(\xi )e^{-{\xi }^2/2}$ leads to the Hermite equation

$$h^{\prime \prime }-2\xi h^{\prime }+(\lambda -1)h=0,\lambda =2\epsilon .$$

Square-integrability forces $\lambda -1=2n$ for a non-negative integer $n$, giving $\epsilon =n+1/2$. The polynomial solutions are the physicist's Hermite polynomials $H_n(\xi )$.

Eigenvalues and eigenfunctions. The energy spectrum is

$$\boxed{E_n=\hslash \omega \left(n+\frac{1}{2}\right),n=0,1,2,\dots }$$

The normalised eigenfunctions are

$${\psi }_n(x)={\left(\frac{m\omega }{\pi \hslash }\right)}^{1/4}\frac{1}{\sqrt{2^{n}n!}}{H}_n\left(\sqrt{\frac{m\omega }{\hslash }}x\right)\exp \left(-\frac{m\omega x^2}{2\hslash }\right).$$

The first few Hermite polynomials are $H_0=1$, $H_1=2\xi$, $H_2=4{\xi }^2-2$, $H_3=8{\xi }^3-12\xi$.

Ladder operators. Define the annihilation (lowering) and creation (raising) operators

$$a=\sqrt{\frac{m\omega }{2\hslash }}\left(x+\frac{ip}{m\omega }\right),a^{†}=\sqrt{\frac{m\omega }{2\hslash }}\left(x-\frac{ip}{m\omega }\right).$$

Their fundamental commutation relation is

$$[a,a^{†}]=1.$$

The position and momentum operators are recovered by inverting:

$$x=\sqrt{\frac{\hslash }{2m\omega }}(a+a^{†}),p=i\sqrt{\frac{m\hslash \omega }{2}}(a^{†}-a).$$

The Hamiltonian in ladder-operator form is

$$H=\hslash \omega \left(a^{†}a+\frac{1}{2}\right)=\hslash \omega \left(N+\frac{1}{2}\right),$$

where $N=a^{†}a$ is the number operator, with eigenvalues $n=0,1,2,\dots$

Orthonormality and completeness. The Hermite functions $\{{\psi }_n{\}}_{n=0}^{\infty }$ form a complete orthonormal basis of $L^2(\mathbb{R})$:

$$⟨{\psi }_m|{\psi }_n⟩={\delta }_{mn},\sum _{n=0}^{\infty }|{\psi }_n⟩⟨{\psi }_n|=1.$$

Completeness follows from the spectral theorem for the self-adjoint operator $H$, or from the classical result that the Hermite functions are dense in $L^2(\mathbb{R})$.

Time evolution. Each energy eigenstate picks up a phase at its own frequency:

$${\Psi }_n(x,t)={\psi }_n(x)e^{-i{E}_{n}t/\hslash }.$$

A general state is a superposition $\Psi (x,t)={\sum }_{n=0}^{\infty }{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$ with ${\sum }_n|c_n{|}^2=1$ and $c_n=⟨{\psi }_n|\Psi (0)⟩$.

Domain considerations

The QHO Hamiltonian is an unbounded operator on $L^2(\mathbb{R})$. Its domain is the dense subspace

$$D(H)=\{\psi \in L^2(\mathbb{R}):\psi \in H^2(\mathbb{R}),x^2\psi \in L^2(\mathbb{R})\}.$$

On this domain $H$ is essentially self-adjoint, guaranteeing a real spectrum and unitary time evolution via Stone's theorem. The Hermite functions lie in $D(H)$ and form a core for $H$.

Counterexamples to common slips

• The ladder operators $a$ and $a^{†}$ are not Hermitian — they are not observables. The number operator $N=a^{†}a$ is Hermitian; $a$ and $a^{†}$ are each other's adjoints.
• The ground state has quantum number $n=0$, not $n=1$ as in the particle in a box. The QHO quantum number starts at zero.
• The eigenfunctions extend over all of $\mathbb{R}$ — there are no hard walls. The Gaussian envelope $e^{-m\omega x^2/(2\hslash )}$ ensures normalisability despite the wave function being nonzero everywhere.
• The equal energy spacing $E_{n+1}-E_n=\hslash \omega$ is a special property of the quadratic potential. Changing the potential to, say, $V\propto |x|$ or $V\propto x^4$ destroys the equal spacing.

Key theorem with proof [Intermediate+]

Theorem (Ladder-operator diagonalisation of the harmonic oscillator). The operators

$$a=\sqrt{\frac{m\omega }{2\hslash }}\left(x+\frac{ip}{m\omega }\right),a^{†}=\sqrt{\frac{m\omega }{2\hslash }}\left(x-\frac{ip}{m\omega }\right)$$

satisfy $[a,a^{†}]=1$ and diagonalise $H$:

$$H=\hslash \omega \left(a^{†}a+\frac{1}{2}\right).$$

The spectrum of $H$ is $E_n=\hslash \omega (n+1/2)$ for $n=0,1,2,\dots$, with $|n⟩=(a^{†}{)}^n/\sqrt{n!}|0⟩$.

Proof.

Step 1: The commutation relation. Using the canonical commutation relation $[x,p]=i\hslash$ 12.02.02 pending:

$$[a,a^{†}]=\frac{m\omega }{2\hslash }\left[x+\frac{ip}{m\omega },x-\frac{ip}{m\omega }\right]=\frac{m\omega }{2\hslash }\cdot \frac{2\hslash }{m\omega }=1,$$

since the cross terms give $-i/(m\omega )[x,p]+i/(m\omega )[p,x]=2i\hslash \cdot i/(m\omega )\cdot (-1)\cdot m\omega /(2\hslash )$, and the $[x,x]$ and $[p,p]$ terms vanish.

Step 2: H in terms of a and $a^{†}$. Expanding:

$$a^{†}a=\frac{m\omega }{2\hslash }\left(x^2+\frac{p^2}{m^2{\omega }^2}+\frac{i}{m\omega }[x,p]\right)=\frac{m\omega }{2\hslash }{x}^2+\frac{1}{2m\hslash \omega }{p}^2-\frac{1}{2}.$$

Rearranging: $\hslash \omega (a^{†}a+1/2)=p^2/(2m)+m{\omega }^2{x}^2/2=H$.

Step 3: The lowering chain terminates. Let $|n⟩$ be an eigenstate of $N=a^{†}a$ with $N|n⟩=n|n⟩$. Then

$$N(a|n⟩)=a^{†}aa|n⟩=(a{a}^{†}-1)a|n⟩=a(N-1)|n⟩=(n-1)(a|n⟩).$$

So $a|n⟩$ is an eigenstate of $N$ with eigenvalue $n-1$, provided it is nonzero. Repeating lowers the eigenvalue by 1 each step. Since $N$ is positive-semidefinite ($⟨\psi |N|\psi ⟩=\Vert a|\psi ⟩{\Vert }^2\ge 0$), the eigenvalues cannot go below zero. The chain must terminate: there exists a state $|0⟩$ with $a|0⟩=0$.

Step 4: The ground state. From $a|0⟩=0$ and $H=\hslash \omega (N+1/2)$:

$$H|0⟩=\hslash \omega (0+\frac{1}{2})|0⟩=\frac{1}{2}\hslash \omega |0⟩.$$

So $E_0=\hslash \omega /2$.

Step 5: Normalisation and the raising chain. From the norm calculations $\Vert a|n⟩{\Vert }^2=⟨n|N|n⟩=n$ and $\Vert a^{†}|n⟩{\Vert }^2=⟨n|(N+1)|n⟩=n+1$, the normalised actions are

$$a|n⟩=\sqrt{n}|n-1⟩,a^{†}|n⟩=\sqrt{n+1}|n+1⟩.$$

Starting from $|0⟩$ and applying $a^{†}$ repeatedly:

$$|n⟩=\frac{(a^{†}{)}^n}{\sqrt{n!}}|0⟩,H|n⟩=\hslash \omega \left(n+\frac{1}{2}\right)|n⟩.\square$$

Corollary (Ground state in position space). The equation $a|0⟩=0$ becomes, in the position representation,

$$\frac{\hslash }{m\omega }\frac{d{\psi }_0}{dx}+x{\psi }_0=0,$$

with solution ${\psi }_0(x)=(m\omega /(\pi \hslash ){)}^{1/4}\exp (-m\omega x^2/(2\hslash ))$. This is the unique normalised ground state (up to phase), and all higher eigenstates are generated by applying $a^{†}$ and normalising.

Bridge. The ladder-operator diagonalisation builds toward 12.10.01 pending where the QHO propagator is derived by path-integral methods, recovering the same spectrum from a second, independent route. The result appears again in 14.05.01 pending as the vibrational Hamiltonian of molecular bonds, where the selection rule $\Delta n=\pm 1$ governs infrared absorption frequencies. The foundational reason the ladder operators succeed is that the canonical commutation relation $[x,p]=i\hslash$ decomposes into creation-annihilation pairs whose algebra is representation-independent, and this is exactly the structure that identifies the oscillator number basis $|n⟩$ with the Fock-space particle-number basis of quantum field theory. The bridge is between the algebraic structure of a single oscillator and the particle interpretation of quantised fields, where the same raising-chain argument constructs the many-particle Hilbert space from the vacuum.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The quantum harmonic oscillator presents a formalisation challenge beyond Mathlib's current scope. The key objects are:

• The Hilbert space $L^2(\mathbb{R})$ with its inner product and the dense subspace of Schwartz functions.
• The position and momentum operators $x$ and $p=-i\hslash d/dx$ as unbounded operators satisfying $[x,p]=i\hslash$.
• The ladder operators $a$, $a^{†}$ and the number operator $N=a^{†}a$.
• The spectral decomposition $H=\hslash \omega (N+1/2)$ with eigenvalues $\hslash \omega (n+1/2)$.
• The Hermite functions as a complete orthonormal basis.

Mathlib has Hermite polynomials (Polynomial.Hermite) and the spectral theorem for compact self-adjoint operators. It does not have: unbounded-operator commutator calculus on $L^2(\mathbb{R})$, the canonical commutation relation as a theorem about operators on a Hilbert space, or the identification of the Hermite-function spectrum with the QHO eigenvalues. A formalisation would proceed by proving $[a,a^{†}]=1$ in a suitable operator-algebra framework, then deriving the spectrum by the ladder argument. This unit ships without a lean_module.

Advanced results [Master]

Theorem (Displacement operator and coherent states). For $\alpha \in \mathbb{C}$, define the displacement operator $D(\alpha )=e^{\alpha a^{†}-{\alpha }^{\ast }a}$. The Baker-Campbell-Hausdorff formula for operators whose commutator is a scalar reduces to $e^{A+B}=e^{-[A,B]/2}{e}^A{e}^B$. Applying this with $A=\alpha a^{†}$, $B=-{\alpha }^{\ast }a$ (whose commutator is $[A,B]=|\alpha {|}^2$), gives $D(\alpha )=e^{-|\alpha {|}^2/2}{e}^{\alpha a^{†}}{e}^{-{\alpha }^{\ast }a}$. Since $e^{-{\alpha }^{\ast }a}|0⟩=|0⟩$, it follows that $D(\alpha )|0⟩=e^{-|\alpha {|}^2/2}{e}^{\alpha a^{†}}|0⟩=|\alpha ⟩$.

Theorem (Completeness of Hermite functions). The Hermite functions $\{{\psi }_n{\}}_{n=0}^{\infty }$ form a complete orthonormal basis of $L^2(\mathbb{R})$.

One proof route uses the generating function ${\sum }_n{H}_n(\xi )t^n/n!=e^{2\xi t-t^2}$ to show that finite linear combinations of Hermite functions are dense in the Schwartz space $\mathcal{S}(\mathbb{R})$, which is itself dense in $L^2(\mathbb{R})$. An equivalent argument uses the spectral theorem for the essentially self-adjoint operator $H$ on $D(H)$: since $H$ has pure point spectrum with no continuous part, its eigenprojections sum to the identity.

Theorem (Selection rule for dipole transitions). The position-operator matrix element $⟨m|x|n⟩$ vanishes unless $m=n\pm 1$: $$⟨n+1|x|n⟩=\sqrt{\frac{\hslash }{2m\omega }}\sqrt{n+1},⟨n-1|x|n⟩=\sqrt{\frac{\hslash }{2m\omega }}\sqrt{n}.$$

This follows from $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$: the operator $a$ lowers $n$ by 1 and $a^{†}$ raises $n$ by 1, with all other matrix elements vanishing by orthogonality of distinct number states. The selection rule $\Delta n=\pm 1$ means each normal mode absorbs and emits at exactly one frequency $\omega$, independent of $n$ — the spectroscopic signature of equally spaced levels. Anharmonic corrections to the potential introduce weakly allowed $\Delta n=\pm 2,\pm 3$ transitions (overtones) at reduced intensity.

Theorem (Mehler's formula). The QHO propagator kernel admits the closed form $$K(x^{\prime },x;t)=\sum _{n=0}^{\infty }{\psi }_n(x^{\prime }){\psi }_n^{\ast }(x)e^{-i\omega (n+1/2)t}=\sqrt{\frac{m\omega }{2\pi i\hslash \sin \omega t}}\exp \left(\frac{im\omega }{2\hslash \sin \omega t}[(x^{\prime 2}+x^2)\cos \omega t-2x^{\prime }x]\right).$$

The sum is evaluated by inserting the Hermite generating function and completing the square. The exponent is $i{S}_{\text{cl}}/\hslash$, where $S_{\text{cl}}$ is the classical action along the unique classical trajectory from $x$ to $x^{\prime }$ in time $t$. For the harmonic oscillator the semiclassical approximation is exact: the quadratic potential ensures that the path integral has no contributions beyond the Gaussian (quadratic) fluctuation determinant.

Theorem (Minimum-uncertainty states). The ground state $|0⟩$ and every coherent state $|\alpha ⟩$ saturate the Heisenberg uncertainty relation: $(\Delta x)(\Delta p)=\hslash /2$. For energy eigenstates $|n⟩$ with $n\ge 1$, the uncertainty product is $(\Delta x{)}_n(\Delta p{)}_n=\hslash (n+1/2)$.

The ground state is a Gaussian, which is the unique minimum-uncertainty wave function. Coherent states are displaced ground states related by the displacement operator $D(\alpha )$, which is unitary and generated by $x$ and $p$, preserving the uncertainty product.

Theorem (Ehrenfest theorem for the oscillator). For any state $|\psi (t)⟩$, the expectation values satisfy $$\frac{d}{dt}⟨x⟩=\frac{⟨p⟩}{m},\frac{d}{dt}⟨p⟩=-m{\omega }^2⟨x⟩.$$

These combine to $d^2⟨x⟩/d{t}^2+{\omega }^2⟨x⟩=0$: the quantum expectation value oscillates at the classical frequency $\omega$ regardless of the state. This exact correspondence is a special property of the quadratic potential; for anharmonic potentials the Ehrenfest equations are only approximately classical because $⟨V^{\prime }(x)⟩\ne V^{\prime }(⟨x⟩)$ when $V$ is not linear.

Theorem (Spectral zeta function and regularised vacuum energy). The zeta function of the QHO is ${\zeta }_H(s)=(\hslash \omega {)}^{-s}\zeta (-s,1/2)$ where $\zeta (z,q)$ is the Hurwitz zeta function. The zeta-regularised vacuum energy evaluates to $$E_{\text{vac}}=-\frac{1}{2}{\zeta }_H(-1)=-\frac{\hslash \omega }{24}.$$

This finite result, obtained by analytic continuation, governs the Casimir effect for conducting plates and the zero-point energy of bosonic string modes.

Synthesis. The results above illustrate the foundational reason the harmonic oscillator serves as the backbone of quantum physics: the equal spacing $E_{n+1}-E_n=\hslash \omega$ forces a linear ladder algebra whose matrix elements are exactly computable. This is exactly the structure that generalises to quantum field theory, where each field mode carries its own copy of the oscillator algebra, and the bridge is that creation and annihilation of field quanta are raising and lowering operations on a Fock space built from the vacuum. Putting these together, the selection rules, the virial theorem, the Mehler propagator, and the coherent-state minimum uncertainty all trace back to the single commutation relation $[a,a^{†}]=1$. The central insight is that the physics of the oscillator is entirely determined by its algebra, independent of the representation. The pattern recurs in molecular spectroscopy 14.05.01 pending, where vibrational transition intensities are set by the same matrix elements, and in solid-state physics 11.05.01 pending, where phonon occupation numbers follow the Bose-Einstein distribution derived from the oscillator partition function.

Full proof set [Master]

Proposition (Parity of eigenstates). The $n$-th energy eigenstate ${\psi }_n(x)$ has parity $(-1{)}^n$: ${\psi }_n(-x)=(-1{)}^n{\psi }_n(x)$.

Proof. The Hamiltonian $H=p^2/(2m)+m{\omega }^2{x}^2/2$ is invariant under $x\to -x$. The parity operator $\Pi :(\Pi \psi )(x)=\psi (-x)$ commutes with $H$. Since the eigenvalues of $H$ are non-degenerate, each eigenstate is a simultaneous eigenstate of $\Pi$ with eigenvalue $\pm 1$.

The ground state ${\psi }_0(x)\propto e^{-m\omega x^2/(2\hslash )}$ is even: $\Pi {\psi }_0=+{\psi }_0$. The creation operator transforms as $\Pi a^{†}\Pi =-a^{†}$ because $x$ is odd and $p=-i\hslash d/dx$ is even under $x\to -x$. Each application of $a^{†}$ therefore flips parity. Starting from the even ground state, $|n⟩=(a^{†}{)}^n/\sqrt{n!}|0⟩$ has parity $(-1{)}^n$. $\square$

Proposition (Position-space variance and uncertainty product). For the energy eigenstate $|n⟩$, $$(\Delta x{)}_n^2=\frac{\hslash }{2m\omega }(2n+1),(\Delta p{)}_n^2=\frac{m\hslash \omega }{2}(2n+1),(\Delta x{)}_n(\Delta p{)}_n=\hslash \left(n+\frac{1}{2}\right).$$

Proof. Since $⟨x{⟩}_n=0$ and $⟨p{⟩}_n=0$ (Exercise 3), the variance reduces to $⟨x^2{⟩}_n$ and $⟨p^2{⟩}_n$. Using $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$: $$x^2=\frac{\hslash }{2m\omega }(a^2+(a^{†}{)}^2+a{a}^{†}+a^{†}a).$$

The terms $a^2$ and $(a^{†}{)}^2$ have vanishing expectation value in $|n⟩$ (they produce states orthogonal to $|n⟩$). The surviving terms give $⟨n|a{a}^{†}+a^{†}a|n⟩=(n+1)+n=2n+1$. Therefore $(\Delta x{)}_n^2=\hslash (2n+1)/(2m\omega )$.

An identical calculation with $p=i\sqrt{m\hslash \omega /2}(a^{†}-a)$ gives $(\Delta p{)}_n^2=m\hslash \omega (2n+1)/2$. The product $(\Delta x{)}_n(\Delta p{)}_n=\hslash (n+1/2)\ge \hslash /2$, with equality at $n=0$. $\square$

Proposition (Algebraic proof of the virial theorem). For every energy eigenstate $|n⟩$, $⟨T{⟩}_n=⟨V{⟩}_n=E_n/2$.

Proof. The potential energy is $V=\frac{1}{2}m{\omega }^2{x}^2=\hslash \omega (a+a^{†}{)}^2/4$. Expanding: $$⟨V{⟩}_n=\frac{\hslash \omega }{4}⟨n|(a^2+(a^{†}{)}^2+a{a}^{†}+a^{†}a)|n⟩=\frac{\hslash \omega }{4}(0+0+(n+1)+n)=\frac{\hslash \omega (2n+1)}{4}=\frac{E_n}{2}.$$

Since $E_n=⟨T{⟩}_n+⟨V{⟩}_n$, it follows that $⟨T{⟩}_n=E_n/2$. The kinetic and potential energies share the total energy equally in every eigenstate, a consequence of the quadratic potential's symmetry under the dilation $x\to \lambda x$, $p\to p/\lambda$. $\square$

Coherent states [Master]

The eigenstates of the annihilation operator $a$ are called coherent states. They satisfy $a|\alpha ⟩=\alpha |\alpha ⟩$ for $\alpha \in \mathbb{C}$. Expanding in the number basis:

$$|\alpha ⟩=e^{-|\alpha {|}^2/2}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}|n⟩.$$

The probability of measuring $n$ quanta is the Poisson distribution $P(n)=e^{-|\alpha {|}^2}|\alpha {|}^{2n}/n!$ with mean $⟨N⟩=|\alpha {|}^2$ and variance $\Delta N=|\alpha |$.

The position-space wave function of $|\alpha ⟩$ is a Gaussian with the same width as the ground state but displaced from the origin. Under time evolution, $|\alpha (t)⟩=|\alpha e^{-i\omega t}⟩$: the Gaussian oscillates back and forth like a classical particle, maintaining its shape. This is the closest quantum analogue of a classical oscillator trajectory — a wave packet that does not spread.

Coherent states saturate the Heisenberg uncertainty relation: $\Delta x\cdot \Delta p=\hslash /2$. They are minimum-uncertainty states with equal uncertainties in $x$ and $p$ (up to the scale set by $\omega$ and $m$). The ground state $|0⟩$ is the coherent state with $\alpha =0$.

In quantum optics, coherent states describe the output of an ideal laser. The Poisson photon-number distribution is the signature of a coherent state, distinguishing it from thermal light (Bose-Einstein distribution) or single-photon sources (delta-function distribution). The 2005 Nobel Prize in Physics (Glauber) was awarded for the theory of optical coherence built on these states.

Squeezed states generalise coherent states by allowing $\Delta x$ and $\Delta p$ to differ while still saturating the uncertainty bound. A squeezed state has reduced noise in one quadrature (say $x$) below $\sqrt{\hslash /(2m\omega )}$, at the cost of increased noise in the conjugate quadrature. Squeezed light is used in gravitational-wave detectors (LIGO) to beat the standard quantum limit on displacement sensitivity.

The Wigner function of a coherent state is a Gaussian in phase space centred at $(\text{Re}\alpha ,\text{Im}\alpha )$ with circular cross-section determined by $\hslash$. For a number state $|n⟩$ the Wigner function has $n$ concentric rings of alternating sign — a highly non-classical phase-space distribution. The transition from the ring structure of $|n⟩$ to the single Gaussian peak of $|\alpha ⟩$ as $n\to \infty$ with $|\alpha {|}^2\approx n$ is another manifestation of the correspondence principle. Negative values of the Wigner function signal non-classicality; coherent-state Wigner functions are everywhere non-negative, consistent with their status as the most classical quantum states.

Path integral for the harmonic oscillator [Master]

The QHO propagator — the matrix element $K(x^{\prime },x;T)=⟨x^{\prime }|e^{-iHT/\hslash }|x⟩$ — is one of the few propagators known in closed form. By the Feynman path integral (unit 12.10.01):

$$K(x^{\prime },x;T)=\sqrt{\frac{m\omega }{2\pi i\hslash \sin \omega T}}\exp \left(\frac{im\omega }{2\hslash \sin \omega T}[(x^{\prime 2}+x^2)\cos \omega T-2x^{\prime }x]\right).$$

The exponent is $i{S}_{\mathrm{cl}}/\hslash$ where $S_{\mathrm{cl}}$ is the classical action along the unique classical path from $x$ to $x^{\prime }$ in time $T$. For the harmonic oscillator the semiclassical approximation is exact: the path integral receives contributions from the classical trajectory and its quadratic fluctuations, and the latter are summed by the Gaussian integration with no higher-order corrections.

The prefactor $\sqrt{m\omega /(2\pi i\hslash \sin \omega T)}$ is the fluctuation determinant — the ratio of determinants of the operator $-{\partial }_t^2-{\omega }^2$ on the space of paths with fixed endpoints. This determinant is computed by the Gelfand-Yaglom method and produces the $\sqrt{1/\sin \omega T}$ factor.

The propagator has poles at $T=n\pi /\omega$ (the classical period), where $\sin \omega T=0$. These poles correspond to the classical periodicity: at half-integer multiples of the period the particle returns to its starting point with reversed momentum, and at integer multiples with the same momentum. Near each pole the propagator reduces to a sum over energy eigenstates via the spectral representation $K={\sum }_n{\psi }_n(x^{\prime }){\psi }_n^{\ast }(x)e^{-i{E}_{n}T/\hslash }$.

The QHO propagator is the free propagator from which all perturbative quantum-field-theory amplitudes are built: every free-field mode is an independent oscillator.

The propagator at imaginary time $T=-i\tau$ (with $\tau >0$) becomes the thermal density-matrix element ${\rho }_{\beta }(x^{\prime },x)=⟨x^{\prime }|e^{-\beta H}|x⟩$ with $\beta =\tau /\hslash$. This analytic continuation from real to imaginary time is the Matsubara formalism in thermal field theory, where the Euclidean propagator encodes finite-temperature quantum statistics. The poles of the propagator at real $T=n\pi /\omega$ become the Matsubara frequencies ${\omega }_n=n\pi /\beta$ after continuation, connecting the oscillator spectrum to the thermal Green functions of many-body physics 11.05.01 pending.

Thermal states and the partition function [Master]

At temperature $T$, the QHO is described by the canonical density matrix

$$\rho =\frac{1}{Z}{e}^{-\beta H},Z=\mathrm{tr}{e}^{-\beta H}=\frac{1}{2\sinh (\beta \hslash \omega /2)},$$

where $\beta =1/(k_{B}T)$. Evaluating the geometric series $Z=e^{-\beta \hslash \omega /2}/(1-e^{-\beta \hslash \omega })$ gives the stated form. The mean energy is

$$⟨E⟩=-\frac{\partial \ln Z}{\partial \beta }=\frac{\hslash \omega }{2}\coth \left(\frac{\beta \hslash \omega }{2}\right)=\hslash \omega \left(\frac{1}{2}+\bar{n}\right),$$

where $\bar{n}=(e^{\beta \hslash \omega }-1{)}^{-1}$ is the Bose-Einstein occupation number. At high temperature ($k_{B}T\gg \hslash \omega$): $⟨E⟩\to k_{B}T$, recovering the classical equipartition result (each quadratic degree of freedom contributes $k_{B}T/2$ for kinetic and $k_{B}T/2$ for potential energy). At low temperature ($k_{B}T\ll \hslash \omega$): $⟨E⟩\to \hslash \omega /2$, the zero-point energy.

The specific heat $C=\partial ⟨E⟩/\partial T$ exhibits activation behaviour: $C\approx k_B(\beta \hslash \omega {)}^2{e}^{-\beta \hslash \omega }$ at low $T$, vanishing exponentially as the thermal energy falls below the level spacing. This is the Einstein model of specific heat (1907), the first quantum theory of solids. The Debye model (unit 11.05.01) replaces the single frequency $\omega$ by a continuum of phonon frequencies with a cutoff and recovers the $T^3$ law at low temperature.

The thermal density matrix in position representation is obtained by evaluating the Mehler kernel at imaginary time $t=-i\beta \hslash$: $${\rho }_{\beta }(x^{\prime },x)=\frac{1}{Z}\sqrt{\frac{m\omega }{2\pi \hslash \sinh \beta \hslash \omega }}\exp \left(-\frac{m\omega }{2\hslash \sinh \beta \hslash \omega }[(x^{\prime 2}+x^2)\cosh \beta \hslash \omega -2x^{\prime }x]\right).$$

The diagonal ${\rho }_{\beta }(x,x)$ gives the thermal position probability density. At low temperature it collapses to $|{\psi }_0(x){|}^2$, the ground-state distribution. At high temperature it broadens into the classical Boltzmann distribution $\propto e^{-\beta m{\omega }^2{x}^2/2}$, a Gaussian with variance $k_{B}T/(m{\omega }^2)$ set by equipartition.

Multidimensional oscillators and normal modes [Master]

For a particle in $d$ dimensions with a quadratic potential $V(\mathbf{x})=\frac{1}{2}{\mathbf{x}}^{T}K\mathbf{x}$ where $K$ is a symmetric positive-definite matrix, the system decouples into $d$ independent one-dimensional oscillators. Diagonalise $K$ by an orthogonal transformation: $K=O^T\mathrm{diag}({\omega }_1^2,\dots ,{\omega }_d^2)O/m$. The total Hamiltonian separates as $H={\sum }_{i=1}^d{H}_i$ where each $H_i$ is a 1D oscillator with frequency ${\omega }_i$, and the total energy is

$$E_{n_1,\dots ,n_d}=\sum _{i=1}^d\hslash {\omega }_i\left(n_i+\frac{1}{2}\right).$$

For a molecule with $N$ atoms, there are $3N$ degrees of freedom. Subtracting 3 translational and 3 rotational modes (2 for a linear molecule), the remaining vibrational normal modes are the eigenvectors of the mass-weighted Hessian matrix. The CO molecule has one vibrational mode; water has three; a protein with thousands of atoms has thousands of modes. The normal-mode decomposition — diagonalising the Hessian — is the bridge between the abstract one-dimensional oscillator and the vibrational spectroscopy of real molecules (unit 14.05.01).

Consider carbon dioxide, a linear triatomic molecule O=C=O. It has $N=3$ atoms and $3N=9$ degrees of freedom. Subtracting 3 translational and 2 rotational modes (linear molecule) leaves 4 vibrational modes, but the symmetric stretch appears only once and the bending mode is doubly degenerate (bending in the plane and out of the plane). The three distinct vibrational frequencies of CO${}_2$ — symmetric stretch at $\sim 1388$ cm${}^{-1}$, bending at $\sim 667$ cm${}^{-1}$, and asymmetric stretch at $\sim 2349$ cm${}^{-1}$ — are each independent harmonic oscillators to leading approximation. The infrared absorption at 667 cm${}^{-1}$ is the strongest fundamental band and is the primary mechanism by which CO${}_2$ absorbs terrestrial infrared radiation, underpinning the greenhouse effect.

When all $d$ frequencies are equal (${\omega }_i=\omega$), the system has a $U(d)$ dynamical symmetry group that is larger than the geometric symmetry group. The resulting degeneracies — the number of ways to distribute $n$ quanta among $d$ modes is $\left(\genfrac{}{}{0ex}{}{n+d-1}{d-1}\right)$ — are the molecular-physics analogue of the hydrogen-atom degeneracy that arises from the $SO(4)$ symmetry.

The harmonic oscillator as the foundation of quantum field theory [Master]

In quantum field theory, a free scalar field $\varphi (\mathbf{x},t)$ with mass $m$ in a box of volume $V$ is decomposed into Fourier modes:

$$\varphi (\mathbf{x},t)=\sum _{\mathbf{k}}\frac{1}{\sqrt{2{\omega }_{\mathbf{k}}V}}\left(a_{\mathbf{k}}{e}^{i(\mathbf{k}\cdot \mathbf{x}-{\omega }_{\mathbf{k}}t)}+a_{\mathbf{k}}^{†}{e}^{-i(\mathbf{k}\cdot \mathbf{x}-{\omega }_{\mathbf{k}}t)}\right),$$

where ${\omega }_{\mathbf{k}}=\sqrt{|\mathbf{k}{|}^2+m^2}$. Each mode $\mathbf{k}$ is an independent harmonic oscillator. The operators $a_{\mathbf{k}}$ and $a_{\mathbf{k}}^{†}$ annihilate and create a quantum of the field (a particle) with momentum $\mathbf{k}$. The commutation relations $[a_{\mathbf{k}},a_{{\mathbf{k}}^{\prime }}^{†}]={\delta }_{\mathbf{k}{\mathbf{k}}^{\prime }}$ are precisely the oscillator algebra, one copy per mode.

The vacuum $|0⟩$ satisfies $a_{\mathbf{k}}|0⟩=0$ for every $\mathbf{k}$ — it is the tensor product of all individual ground states. A state with $n_{\mathbf{k}}$ particles in mode $\mathbf{k}$ has total energy $E={\sum }_{\mathbf{k}}{\omega }_{\mathbf{k}}(n_{\mathbf{k}}+1/2)$. The vacuum energy ${\sum }_{\mathbf{k}}{\omega }_{\mathbf{k}}/2$ is divergent — the sum over all modes of the zero-point energy — and gives rise to the cosmological constant problem discussed in the Historical and philosophical context section. The Casimir effect provides a physical manifestation: two conducting plates in vacuum restrict the available field modes, altering the zero-point sum and producing a measurable attractive force proportional to $\hslash c{\pi }^2/(240d^4)$ where $d$ is the plate separation. The force has been measured to within 1% of the theoretical prediction.

The operator product $a_{\mathbf{k}}{a}_{{\mathbf{k}}^{\prime }}^{†}$ can be decomposed into a normal-ordered part (all creation operators to the left of annihilation operators) plus a c-number using $[a,a^{†}]=1$. This decomposition is Wick's theorem, and the c-number terms are the contractions — the propagators that appear as internal lines in Feynman diagrams. In this sense every perturbative QFT calculation is a combinatorial exercise in oscillator algebra.

This construction — the Fock space built from oscillator ladder operators — is the entire mathematical framework of free quantum field theory. Interacting theories are treated perturbatively around this free-field foundation. Every Feynman diagram is a bookkeeping device for combining oscillator matrix elements: vertices come from the interaction Hamiltonian, propagators from the free oscillator Green function, and the combinatorics of contractions from Wick's theorem applied to the oscillator ladder algebra.

Connections [Master]

The harmonic oscillator is the linchpin connecting quantum mechanics to its applications across physics and chemistry.

In molecular physics (unit 14.05.01), every chemical bond vibrates, and the small-amplitude approximation makes each normal mode a harmonic oscillator. Infrared spectroscopy measures transitions $n\to n+1$ between adjacent levels; the selection rule $\Delta n=\pm 1$ (derived from the non-vanishing matrix element $⟨n+1|x|n⟩$) means each mode absorbs at exactly one frequency $\omega$. Anharmonic corrections shift and split these lines, but the harmonic approximation is the zeroth-order description from which perturbation theory departs.

In solid-state physics, the lattice vibrations of a crystal are quantised as phonons — collective oscillation modes, each an independent QHO. The phonon specific heat at low temperature follows the Debye model (unit 11.05.01), which treats a continuum of harmonic oscillators with a cutoff frequency. Bose-Einstein condensation of phonon modes underlies superfluidity in liquid helium.

In quantum optics, the electromagnetic field in a cavity is a collection of harmonic oscillators, one per mode. Photon number states $|n⟩$ sit on the oscillator ladder; coherent states describe laser light; squeezed states describe noise reduction below the vacuum level. The entire field of continuous-variable quantum information is built on oscillator phase space.

In quantum field theory, the free-field Hamiltonian is a sum of oscillator Hamiltonians, one per momentum mode. The perturbative treatment of interactions (Feynman diagrams) expands around this oscillator basis. The QHO propagator (unit 12.10.01) is the free propagator from which all perturbative amplitudes are constructed.

The mathematical structure — the Fock space generated by $a^{†}$ acting on $|0⟩$ — appears identically in quantum optics, solid-state physics, nuclear physics (giant resonances), and particle physics. No other single model has this reach.

Historical and philosophical context [Master]

The harmonic oscillator holds a unique position in the history of quantum mechanics: it was the first nontrivial system solved by matrix mechanics, and it provided the earliest evidence that Heisenberg's new theory was correct.

In July 1925, Heisenberg published his "Umdeutung" paper, introducing matrix mechanics. Born and Jordan, recognising the matrix structure, computed the energy levels of the one-dimensional oscillator by diagonalising an infinite matrix. Their result — equally spaced levels $E_n=\hslash \omega (n+1/2)$ — agreed with the known half-integer quantisation rules and with the empirical spacing of molecular vibrational spectra. This was the first concrete success of the new quantum mechanics.

Schrödinger, in the first of his 1926 papers on wave mechanics, solved the oscillator by a different route: he wrote down the differential equation with the quadratic potential and found the Hermite-polynomial solutions. The energy levels matched Heisenberg's matrix-mechanics result — the first evidence that matrix mechanics and wave mechanics were equivalent. Schrodinger himself proved this equivalence later that year, and Dirac gave an independent proof in 1926.

Dirac, in 1927, introduced the ladder-operator method. His insight was that the algebra of $a$ and $a^{†}$ — independent of any representation — captures the essential physics. The same algebra works in the matrix representation (where $a$ is a sub-diagonal matrix), in the position representation (where $a$ is a first-order differential operator), and in the abstract bra-ket formalism Dirac was developing. The ladder operators are the natural language of quantum field theory: every creation and annihilation operator in QFT is an oscillator ladder operator in disguise.

Glauber, in 1963, introduced the modern theory of optical coherence by identifying laser light with coherent states of the electromagnetic field. His work showed that the photon-counting statistics of an ideal laser follow a Poisson distribution — the signature of a coherent state — and that higher-order correlation functions factorise for coherent light but not for thermal or chaotic sources. The 2005 Nobel Prize in Physics was awarded to Glauber "for his contribution to the quantum theory of optical coherence," cementing the coherent-state framework as the foundation of quantum optics. The squeezed-state extension, developed theoretically by Stoler (1970) and experimentally by Slusher's group (1985), opened the path to precision measurements below the standard quantum limit.

Philosophically, the zero-point energy $\hslash \omega /2$ raises a deep question. Every mode of the electromagnetic field contributes $\hslash \omega /2$ of vacuum energy. Summing over all modes gives a divergent result — the vacuum catastrophe. In non-gravitational physics this infinite constant is unobservable and can be subtracted by normal-ordering. In general relativity, vacuum energy gravitates and contributes to the cosmological constant. The observed cosmological constant is smaller than the naive sum of zero-point energies by roughly 120 orders of magnitude — the cosmological constant problem, one of the deepest unsolved problems in theoretical physics.

Bibliography [Master]

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• Shankar, R. Principles of Quantum Mechanics, 2nd ed. Springer, 1994. Ch. 7.
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• Schrodinger, E. "Quantisierung als Eigenwertproblem." Ann. Phys. 79, 361–376 (1926).
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