# Quantum harmonic oscillator The **quantum harmonic oscillator** is the quantum treatment of a particle in a parabolic potential, U = m·omega0²·x²/2 — the [[Simple_harmonic_motion|mass on a spring]] of classical mechanics, solved with the [[Schrödinger_equation|Schrödinger equation]]. It is the most-used exactly solvable model in physics, because every smooth potential looks parabolic near its minimum. Its [[Energy_level|energy levels]] are evenly spaced, E_n = hbar·omega0·(n + 1/2), n = 0, 1, 2, … and the lowest of them is not zero: even at absolute zero the oscillator retains hbar·omega0/2 of [[Zero-point_energy|zero-point energy]].[^qm-ch2] In the microsim below the reader climbs the ladder with a single control, n from 0 to 30. Each eigenfunction psi_n is drawn at the height of its own rung, with exactly n nodes; beside it |psi_n|² is drawn against the classical probability density 1/(pi·√(x_t² − x²)), whose turning points sit at ±x0·√(2·n + 1). The point of the sweep is a reversal. The ground state is most likely to be found at the centre, where a classical particle moves fastest and therefore spends the least time; by n = 30 the quantum density has migrated outward and its envelope peaks at the turning points, where the classical particle is slowest. The [[Correspondence_principle|correspondence principle]] is thus visible on one slider. A preset switches the potential to a box, where E_n = n²·h²/(8·m·L²), for the contrast: that ladder spreads out as n², while this one stays even. On the Physics flagship this article serves Part II — Core theories at the section *Quantization: the oscillator ladder* (row P36). Its microsim is a sibling of the wave-packet sim on the [[Schrödinger_equation|Schrödinger equation]] page: the same equation, a bound potential instead of a free particle, and stationary states instead of a spreading packet. ## One-dimensional harmonic oscillator The one-dimensional case contains almost all the physics, and everything in higher dimensions follows by adding independent copies of it. Two routes lead to the same answer: solving the differential equation directly, which produces the Hermite functions drawn in the microsim, and the operator algebra, which produces the spectrum without any functions at all. The sections below take them in that order and then set the scales, because without x0 and hbar·omega0 the pictures have no size. ### Hamiltonian and energy eigenstates The time-independent equation for the parabolic potential is −(hbar²/(2·m))·psi'' + (m·omega0²·x²/2)·psi = E·psi, with bounded, normalisable solutions only for E_n = hbar·omega0·(n + 1/2).[^qm-ch2] The eigenfunctions are a Gaussian times a polynomial: psi_n = [1/(2ⁿ·n!·√pi·x0)]^(1/2) · exp(−x²/(2·x0²)) · H_n(x/x0), where H_n is the nth Hermite polynomial — H₀ = 1, H₁ = 2·ξ, H₂ = 4·ξ² − 2, H₃ = 8·ξ³ − 12·ξ, H₄ = 16·ξ⁴ − 48·ξ² + 12 — and H_n has exactly n zeros, which are the n nodes drawn on screen.[^qm-ch2] The ground state is the bare Gaussian, psi₀ = (m·omega0/(pi·hbar))^(1/4)·exp(−m·omega0·x²/(2·hbar)).[^qm-ch2] This is what the microsim computes, with the HUD line `En = hbar*w0*(n + 1/2)`. One numerical point is worth stating because it is the reason the sim can reach n = 30 at all: evaluating H_n and the normalising factor 2ⁿ·n! separately overflows or loses all precision past about n = 20, so the sim uses the normalised recursion psi_(n+1) = √(2/(n+1))·ξ·psi_n − √(n/(n+1))·psi_(n−1), which never forms a factorial.[^manual04] The domain is ±10·x0 on 1,024 samples, which is enough because psi₃₀ places all thirty of its nodes inside ±7.81·x0 (derived). That recursion is not printed in the source text and is used here as an ILLUSTRATIVE numerical device; the eigenfunctions it generates are the book's.[^manual04] ### Ladder operator method The algebraic solution avoids the differential equation entirely. Define the lowering operator a = √(m·omega0/(2·hbar))·(x + i·p/(m·omega0)) and its adjoint a†. Their commutator is [a, a†] = 1, and the Hamiltonian becomes H = hbar·omega0·(a†·a + 1/2). Since a†·a has non-negative eigenvalues, there must be a state annihilated by a — the ground state — and a† applied to any eigenstate produces the next one up: a†|n⟩ = √(n + 1)·|n + 1⟩ and a|n⟩ = √n·|n − 1⟩. The spectrum follows without solving anything. The recursion the microsim uses is the position-space image of exactly this: applying a† in the ξ variable is what turns psi_n into psi_(n+1). The same algebra reappears whenever a system is built from independent modes, which is why it is the backbone of quantum field theory as well as of this problem: a mode of the electromagnetic field is an oscillator, a† creates a [[Photon|photon]] in it, and the ladder index becomes a particle number. The operators also give the matrix elements of position in one line — x has non-zero entries only between neighbouring rungs, proportional to √n — which is the selection rule behind the vibrational spectra discussed below. ### Natural length and energy scales Two combinations of m, omega0 and hbar fix every scale in the problem: x0 = (hbar/(m·omega0))^(1/2), E0 = hbar·omega0/2.[^qm-ch2] Measuring position in units of x0 makes the equation dimensionless, which is why the sim's horizontal axis is ξ = x/x0 rather than metres. The source gives the sizes: x0 is around 0.3 nm for an electron bound in an atom and 0.01–0.1 nm for atoms vibrating in a solid, so the quantum spread is a respectable fraction of an interatomic distance in both cases.[^qm-ch2] At the other extreme, a micromechanical membrane of mass about 2×10⁻¹⁴ kg with omega0 ≈ 3×10¹⁰ s⁻¹ has x0 ≈ 4.19×10⁻¹⁶ m, smaller than a nucleus (derived; the source gives about 4×10⁻¹⁶ m).[^qm-ch2] Its zero-point energy is 1.58×10⁻²⁴ J, or 9.9 μeV, and k_B·T equals hbar·omega0 at 0.23 K (derived).[^codata] That last number is the whole engineering problem of mechanical quantum devices: below a quarter of a kelvin the ladder is resolvable, and above it thermal occupation buries it. ### Coherent states The eigenstates are stationary and do not move, which is not what a pendulum does. The states that do are the coherent states, the eigenstates of the lowering operator a. A coherent state is a Gaussian of exactly the ground-state width whose centre traces the classical orbit at omega0 and which never spreads — the closest a quantum state comes to a classical oscillation, and the reason it saturates the uncertainty relation at all times rather than only at one instant. Its occupation numbers are Poisson-distributed with mean equal to the squared amplitude, which is the photon statistics of an ideal [[Laser|laser]] field. ### Highly excited states The classical particle spends time in inverse proportion to its speed, giving the probability density 1/(pi·√(x_t² − x²)) with the amplitude x_t = x0·√(2·n + 1) — a curve that is lowest at the centre and diverges at the turning points, the exact opposite of the quantum ground state.[^qm-ch2] Climbing the sim's control resolves the contradiction: |psi_n|² develops n nodes, its envelope flattens in the middle and rises at the edges, and by n = 30 the envelope tracks the classical curve closely, the remaining difference being the oscillation between nodes and a short exponential tail beyond the turning points where no classical particle can go. The agreement is exact in one respect at every n, not only at large n. The quantum expectation ⟨x²⟩ in state n is x0²·(n + 1/2), and the time average of x² for a classical orbit of amplitude x_t = x0·√(2·n + 1) is x_t²/2 = x0²·(n + 1/2) — the same number (derived). The second moment matches from the ground state upward; what converges slowly is the shape. ### Phase space solutions In phase space the eigenstates are rings. The Wigner function of the ground state is a single Gaussian mound of area of order hbar, and the excited states are concentric ridges with regions of negative value between them, a feature no classical probability density can have and a standard marker of non-classicality. The semiclassical reading of the ladder is the area rule ∮p·dx = 2·pi·hbar·(n + 1/2), which assigns each state a phase-space area of one quantum plus a half; for this potential it is exact at every n, although as an approximation it is generally justified only for n ≫ 1.[^qm-ch2] ### Two-dimensional harmonic oscillators Two independent oscillators give E = hbar·(omega_x·(n_x + 1/2) + omega_y·(n_y + 1/2)). When the frequencies are equal the energy depends only on n = n_x + n_y and the level is (n + 1)-fold degenerate; that degeneracy reflects a symmetry larger than the obvious rotational one, and it is broken as soon as the two frequencies differ. Anisotropic ratios reproduce the Lissajous families of the classical problem in the pattern of the nodal lines. ## <i>N</i>-dimensional isotropic harmonic oscillator In N dimensions with a single frequency the Hamiltonian separates into N identical one-dimensional problems, so E = hbar·omega0·(n + N/2), n = n₁ + n₂ + … + n_N, and the zero-point energy is N halves rather than one. The degeneracy of level n is the number of ways of writing n as an ordered sum of N non-negative integers. The isotropic problem is also separable in spherical coordinates, giving the same spectrum labelled by a radial quantum number and an angular momentum instead, which is the form used when the oscillator stands in for a central potential. ### Example: 3D isotropic harmonic oscillator For N = 3 the levels are E_n = hbar·omega0·(n + 3/2), with degeneracies (n + 1)·(n + 2)/2, that is 1, 3, 6, 10, 15 for n = 0 to 4 (derived). The three-dimensional oscillator is the starting potential of the nuclear shell model, where a parabolic well plus a strong spin–orbit term reproduces the observed stability of certain nucleon numbers in the [[Atomic_nucleus|atomic nucleus]]; it also serves as the confining potential of an isotropic [[Quantum_dot|quantum dot]], where the same degeneracies appear as shell structure in the addition spectrum. ## Applications The oscillator is less a physical system than a normal form: wherever a system sits near a stable equilibrium, its small excitations are quanta of this ladder. The applications below run from the largest scale to the smallest — a whole crystal, a single bond, the general Taylor argument that licenses both — and then invert the potential, which turns the same algebra into a barrier problem, and make it relativistic. ### Harmonic oscillators lattice: phonons A crystal is a lattice of masses coupled by springs. Changing coordinates to normal modes turns it into a collection of independent harmonic oscillators, one per mode, and quantising each gives a ladder whose quanta are [[Phonon|phonons]]. Heat capacity then follows from how many modes have hbar·omega below k_B·T: high temperatures excite all of them and recover the classical [[Equipartition_theorem|equipartition]] result, while low temperatures freeze out the stiff modes, which is the content of the [[Debye_model|Debye model]] and the reason [[Heat_capacity|heat capacities]] fall toward zero as T → 0. [[Thermal_conduction|Heat conduction]] in an insulator is phonon transport. ### Molecular vibrations A chemical bond near its equilibrium length is a harmonic oscillator, and the ladder is what [[Infrared_spectroscopy|infrared spectroscopy]] measures: absorption at hbar·omega0 and, through anharmonicity, weaker overtones. Real bonds are not parabolic — the potential flattens at large separation and the molecule dissociates, which a true parabola never permits; for H₂ that dissociation costs about 435 kJ/mol, roughly 4.5 eV per molecule.[^qm-ch2] The ladder spacing is visible in thermodynamics as well. The measured [[Heat_capacity|heat capacity]] of H₂ behaves as three degrees of freedom below about 60 K, five between roughly 300 and 600 K, and seven only above about 3,000 K.[^up2-ch2] The last step is the vibrational mode switching on, and it puts hbar·omega0 for H₂ at the scale of k_B × 3,000 K = 0.26 eV (derived): the [[Kinetic_theory_of_gases|kinetic theory]] of a diatomic gas cannot be got right without this ladder. Below that temperature the vibrational rung is simply out of reach of a collision, so the mode contributes nothing to the heat capacity — a quantised ladder showing up in a thermodynamic measurement made a century before anyone could see the levels directly. ### Hooke's law The reason for the model's reach is a Taylor expansion. Near a minimum of any smooth potential, U(x) ≈ U(x₀) + (1/2)·U''(x₀)·(x − x₀)², because the linear term vanishes by definition of a minimum. The quadratic coefficient is the spring constant of [[Hooke's_law|Hooke's law]], and omega0 = √(U''/m). Every stable equilibrium in nature is therefore harmonic for small enough displacements, and its quantum excitations are this ladder — which is why the same spectrum describes a bond, a lattice [[Vibration|vibration]], a trapped ion and a mode of the electromagnetic field. ### The inverted harmonic oscillator Replacing omega0 by i·omega0 turns the well upside down into a parabolic barrier, and the bound ladder becomes a transmission problem. The exact result is the Kemble formula, T = 1/(1 + exp(2·pi·(U_max − E)/(hbar·omega0))), which gives T = 1/2 exactly at the top of the barrier, where the semiclassical [[WKB_approximation|WKB]] estimate gives 1.[^qm-ch2] Below the top the two converge: at (E − U_max)/(hbar·omega0) = −1 the Kemble value is 0.001864 against 0.001867 from WKB, while at −0.5 they are 0.0414 and 0.0432 (derived).[^manual04] The inverted oscillator is therefore the standard smooth-barrier model for [[Quantum_tunnelling|tunnelling]] in chemical reaction rates and in field emission, and it is also the linearised description of any unstable equilibrium. ### The Dirac oscillator The Dirac oscillator is a relativistic wave equation obtained by adding a term linear in position to the momentum in the Dirac equation, so that the equation stays first order in both. Its non-relativistic limit is the ordinary harmonic oscillator plus a strong spin–orbit coupling, and its spectrum is exactly solvable, which makes it a rare relativistic problem that can be written down in closed form. It was introduced under that name by Moshinsky and Szczepaniak in 1989 and is used as a test case for relativistic shell models and, more recently, for analogue experiments in trapped-ion systems.[^moshinsky1989] ## See also - [[Particle_in_a_box]] - [[Energy_level]] - [[Zero-point_energy]] - [[Correspondence_principle]] - [[Schrödinger_equation]] - [[Phonon]] - [[Hooke's_law]] - [[WKB_approximation]] ## Notes The classical density drawn beside |psi_n|² in the microsim is the time-average density of a classical orbit at the same energy, normalised over the interval between the turning points; it diverges at those points, so the sim clips it for display. That clipping is cosmetic and is the only quantity on the panel that is not computed from the stated formulas. The horizontal axis is ξ = x/x0, so the picture is the same for an electron in an atom and for a micromechanical membrane; only the labels change. Footnote definitions are under References. ## References [^qm-ch2]: Likharev, Konstantin (2013). *Part QM: Quantum Mechanics* (Essential Graduate Physics). Chapter 2, "1D wave mechanics" (the oscillator equation and the ladder E_n = ħω₀(n + ½) at pp. ~96–99; the variational Gaussian returning ħω₀/2 exactly at p. ~97; x₀ = (ħ/mω₀)^½, the ground-state wave function and the 0.3 nm, 0.01–0.1 nm and 4×10⁻¹⁶ m scales at p. ~98; the Hermite polynomials, the normalised ψ_n and its n zeros at pp. ~99–101; the Wilson–Sommerfeld area rule ∮p dx = 2πħ(n + ½) and its validity for n ≫ 1 at pp. 53–54; WKB at p. 55 and the exact Kemble result for a parabolic barrier, with T = ½ at the top, at pp. 55–56; the H₂ bond at 435 kJ/mol ≈ 4.5 eV per molecule at p. ~69). The page labels in this subsection of the extract are inconsistent, so the oscillator pages are given as approximate and remain pages to pin. Portal Book 047. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^manual04]: Portal Books research note, sub-manual 04 "Atomic, Quantum, Statistical and Electromagnetic Physics", §3.7 and §3.2: the normalised recursion ψ_(n+1) = √(2/(n+1))·ξ·ψ_n − √(n/(n+1))·ψ_(n−1) is not in the extract and is supplied to avoid the overflow and cancellation that separate evaluation of H_n and 2ⁿn! produces beyond n ≈ 20; the domain ±10x₀ on 1,024 samples and the ψ₃₀ node span are recorded there; the derived Kemble-against-WKB pairs 0.001864/0.001867 at (E − U_max)/ħω₀ = −1 and 0.0414/0.0432 at −0.5 were computed for that manual and are not printed in the book. §A.1 of the same note records the page-label inconsistency in Portal Book 047 §2.9 (pages to pin). [^up2-ch2]: Sanny, Jeff; Ling, Samuel; et al. (2016). *University Physics Volume 2*. OpenStax. Chapter 2, "The Kinetic Theory of Gases", p. 97 (the hydrogen heat-capacity plateaus at d = 3 below about 60 K, d = 5 near 300–600 K and d = 7 above about 3,000 K); the chapter runs pp. 75–114. Portal Book 078. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^up3-ch7]: Sanny, Ling et al. (2016). *University Physics Volume 3*. OpenStax. Chapter 7, "Quantum Mechanics", pp. 295–346 (the quantum harmonic oscillator at introductory level, alongside the particle in a box and barrier tunnelling; page to pin). Portal Book 079. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^codata]: National Institute of Standards and Technology. "CODATA Internationally recommended values of the Fundamental Physical Constants" (reduced Planck constant, Boltzmann constant, electronvolt). NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cuu/Constants/ [^schrodinger1926]: Schrödinger, Erwin (1926). "Quantisierung als Eigenwertproblem (Erste Mitteilung)." *Annalen der Physik*, 384 (4): 361–376. https://doi.org/10.1002/andp.19263840404 [^heisenberg1925]: Heisenberg, Werner (1925). "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen." *Zeitschrift für Physik*, 33: 879–893. The paper's worked example is the anharmonic oscillator, which is where the matrix elements of x for this ladder first appear. [^dirac1927]: Dirac, Paul A. M. (1927). "The quantum theory of the emission and absorption of radiation." *Proceedings of the Royal Society A*, 114 (767): 243–265. The first use of the ladder operators of this problem to quantise a field. [^moshinsky1989]: Moshinsky, Marcos; Szczepaniak, Adam (1989). "The Dirac oscillator." *Journal of Physics A: Mathematical and General*, volume 22. ## Bibliography - Likharev, Konstantin (2013). *Part QM: Quantum Mechanics*. Chapter 2, pp. 31–106. Portal Book 047 — the page-cited source for the ladder, the eigenfunctions, the natural scales and the parabolic-barrier result.[^qm-ch2] - Sanny, Ling et al. (2016). *University Physics Volume 3*. Chapter 7, pp. 295–346. Portal Book 079 — the same material at introductory level.[^up3-ch7] - Sanny, Ling et al. (2016). *University Physics Volume 2*. Chapter 2, pp. 75–114. Portal Book 078 — kinetic theory and the heat-capacity plateaus that date the vibrational ladder.[^up2-ch2] - Heisenberg (1925), Dirac (1927), Schrödinger (1926) and Moshinsky and Szczepaniak (1989), cited above.[^heisenberg1925][^dirac1927][^schrodinger1926] ## External links - [Part QM: Quantum Mechanics](https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics), Konstantin Likharev, Open Textbook Library record — Portal Book 047 - [University Physics Volume 3](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3), OpenStax, Open Textbook Library record — Portal Book 079 - [Fundamental Physical Constants](https://physics.nist.gov/cuu/Constants/), NIST <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Quantum_harmonic_oscillator.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Quantum harmonic oscillator* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Quantum_harmonic_oscillator.html" data-title="Quantum harmonic oscillator"></div> *Built from `MICROSIM_GUIDE/specs/sims/Quantum_harmonic_oscillator.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator) : [Wikitube](https://en.wikitube.io/wiki/Quantum_harmonic_oscillator) · pinned revision [1374308768](https://en.wikipedia.org/w/index.php?oldid=1374308768) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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