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Harmonic oscillator

A harmonic oscillator is a physical system in which a particle or body experiences a restoring force directly proportional to its displacement from an equilibrium position, leading to oscillatory motion described by a second-order linear differential equation.^[1] This proportionality, expressed as F = -kx where k is the force constant and x is the displacement, results in simple harmonic motion (SHM), characterized by sinusoidal position, velocity, and acceleration functions with constant frequency independent of amplitude.^[2] The simplest realization is a mass-spring system, where the spring provides the linear restoring force, and the motion is periodic with period T = 2\pi \sqrt{m/k}, with m as the mass.^[3] In classical mechanics, the harmonic oscillator approximates many real-world phenomena, such as the small-angle oscillations of a pendulum or the vibrations of elastic solids, due to the near-linearity of the restoring force near equilibrium.^[4] Damped and driven variants extend the model to include energy dissipation (e.g., friction) and external forces, explaining resonance in systems like musical instruments or electrical circuits.^[5] The harmonic oscillator's mathematical simplicity—solving to sinusoidal solutions—makes it foundational for understanding waves, as collections of coupled oscillators produce wave propagation.^[6] In quantum mechanics, the quantum harmonic oscillator treats the system as a quantum particle in a parabolic potential, yielding discrete energy levels E_n = \hbar \omega (n + 1/2), where \hbar is the reduced Planck's constant, \omega is the angular frequency, and n = 0, 1, 2, \dots.^[7] This model is crucial for describing atomic vibrations in molecules, phonons in solids, and the quantum theory of fields, including photons in the electromagnetic field.^[8] Applications span chemistry, where it models diatomic molecular vibrations, and beyond, underpinning technologies like lasers and quantum computing through concepts like coherent states.^[9]

Fundamental Principles

Definition and Characteristics

A harmonic oscillator is any physical system in which the restoring force or torque acting on it is directly proportional to the displacement from its equilibrium position and directed opposite to the displacement, resulting in periodic motion.^[10] This proportionality arises from the linear nature of the restoring mechanism, such as a spring or torsional fiber, and distinguishes harmonic oscillators from other oscillatory systems with nonlinear responses.^[8] Key characteristics of an ideal harmonic oscillator include its periodic motion, which traces a sinusoidal trajectory over time; conservation of mechanical energy between kinetic and potential forms without dissipation; and an oscillation frequency that remains independent of the amplitude of displacement.^[4] These properties make the harmonic oscillator a foundational model in physics for approximating small-amplitude vibrations in diverse systems, from mechanical devices to atomic bonds. The motion exemplifies simple harmonic motion, an idealized periodic trajectory.^[11] The early recognition of harmonic oscillator principles dates to the 17th century, with Christiaan Huygens analyzing pendulum motion in his 1673 treatise Horologium Oscillatorium, where he demonstrated harmonic behavior in cycloidal paths.^[12] Robert Hooke further advanced the concept in 1678 through his work De potentia restitutiva, establishing the law of elasticity that underpins linear restoring forces.^[13] In general, the restoring force for translational motion takes the form F = -kx, where k is the stiffness constant with units of newtons per meter (N/m), and x is the displacement. For rotational systems, the restoring torque is \tau = -\kappa \theta, where \kappa is the torsional stiffness constant (in N·m/rad) and \theta is the angular displacement.^[14] The natural angular frequency of oscillation, \omega = \sqrt{k/m} for a mass-spring system (where m is mass in kg), has units of radians per second (rad/s) and characterizes the system's intrinsic oscillation rate.^[4]

Simple Harmonic Motion

The motion of a harmonic oscillator arises from a restoring force proportional to the displacement from equilibrium, as described by Hooke's law, originally formulated by Robert Hooke in 1678./15%3A_Waves_and_Vibrations/15.2%3A_Hookes_Law) For a mass m attached to a spring with spring constant k, the restoring force is F = -kx, where x is the displacement. Applying Newton's second law, F = ma, yields the equation of motion m \frac{d^2x}{dt^2} = -kx, or rearranged, m \frac{d^2x}{dt^2} + kx = 0.^[15] Dividing through by m gives the standard form \frac{d^2x}{dt^2} + \frac{k}{m} x = 0. Defining the angular frequency \omega = \sqrt{k/m}, this simplifies to the second-order linear differential equation \frac{d^2x}{dt^2} + \omega^2 x = 0.^[4] The general solution to this equation is x(t) = A \cos(\omega t + \phi), where A is the amplitude and \phi is the phase constant.^[15] An equivalent form is x(t) = C \cos(\omega t) + D \sin(\omega t), where C and D are constants determined by initial conditions.^[4] To find these constants, consider initial conditions x(0) = x_0 and \frac{dx}{dt}(0) = v_0. Substituting into the cosine form gives x_0 = A \cos \phi and v_0 = -A \omega \sin \phi, allowing solution for A = \sqrt{x_0^2 + (v_0 / \omega)^2} and \phi = \tan^{-1} (-v_0 / (\omega x_0)).^[16] The velocity is then v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi).^[15] The period of oscillation T, the time for one complete cycle, is T = 2\pi / \omega = 2\pi \sqrt{m/k}, and the frequency f = 1/T = \omega / (2\pi).^[4] A key feature of simple harmonic motion is that T and f depend only on the system parameters m and k, remaining independent of the amplitude A or initial conditions.^[15] In phase space, plotting position x against velocity v, the trajectory forms a closed elliptical orbit given by \left( \frac{v}{\omega} \right)^2 + x^2 = A^2.^[17] The area enclosed by this ellipse is $2\pi E / \omega, where E is the total energy, highlighting the conserved nature of the motion.^[17] The total mechanical energy E of the oscillator is constant and equals \frac{1}{2} k A^2.^[18] This energy partitions between kinetic energy E_k = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t + \phi) and potential energy E_p = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega t + \phi), such that E = E_k + E_p at all times.^[18] At maximum displacement (x = \pm A, v = 0), all energy is potential; at equilibrium (x = 0, v = \pm A \omega), all is kinetic.^[19]

Damped and Forced Oscillations

Damped Harmonic Oscillator

The damped harmonic oscillator extends the simple harmonic oscillator model by including a dissipative force, typically viscous friction proportional to velocity, which causes the amplitude to decay over time. This system is described by Newton's second law applied to a mass m attached to a spring with constant k, opposed by a damping force -b \dot{x}, yielding the equation of motion

m \ddot{x} + b \dot{x} + k x = 0.

^[20] Dividing by m gives the standard form

\ddot{x} + 2\gamma \dot{x} + \omega_0^2 x = 0,

^[20] where \gamma = b/(2m) is the damping rate and \omega_0 = \sqrt{k/m} is the natural (undamped) angular frequency.^[20] To solve this linear homogeneous differential equation, assume a trial solution x(t) = e^{rt}, leading to the characteristic equation

r^2 + 2\gamma r + \omega_0^2 = 0.

^[20] The roots are

r = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}.

^[20] The nature of the roots, determined by the sign of the discriminant \gamma^2 - \omega_0^2, classifies the system's behavior into three regimes. In the overdamped regime (\gamma > \omega_0), the roots are real and negative, resulting in a non-oscillatory solution

x(t) = A e^{r_1 t} + B e^{r_2 t},

^[20] where r_1, r_2 < 0 are distinct, and the system approaches equilibrium via exponential decay without crossing the equilibrium position more than once.^[20] For critical damping (\gamma = \omega_0), the roots are equal (r = -\gamma), and the solution is

x(t) = (A + B t) e^{-\gamma t},

^[20] representing the fastest return to equilibrium without oscillation, as the linear t term allows a single overshoot but no subsequent cycles.^[20] In the underdamped regime (\gamma < \omega_0), the roots are complex conjugates, leading to oscillatory motion with decaying amplitude:

x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi),

^[20] where \omega_d = \sqrt{\omega_0^2 - \gamma^2} is the damped angular frequency, A is the initial amplitude, and \phi is the phase determined by initial conditions.^[20] The exponential factor e^{-\gamma t} ensures the oscillations diminish, approaching zero as t \to \infty. The quality factor Q = \omega_0 / (2\gamma) quantifies the damping level, representing the number of radians (or approximately Q / (2\pi) cycles) the oscillator completes before its energy decays to $1/e of its initial value; larger Q corresponds to weaker damping and more persistent oscillations. Energy dissipation in the damped oscillator arises from the damping force, which extracts power at an instantaneous rate proportional to the square of the velocity, P = -b \dot{x}^2, leading to an overall exponential decay of the mechanical energy E(t) \propto e^{-2\gamma t}.^[20]

Driven Harmonic Oscillator

The driven harmonic oscillator extends the damped model by incorporating an external time-dependent force F(t), which sustains oscillations against energy losses due to damping. The governing equation is the inhomogeneous second-order differential equation

m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + k x = F(t),

where m is the mass, b the damping coefficient, k the spring constant, and x(t) the displacement from equilibrium.^[21] This can be normalized by dividing through by m to yield

\frac{d^2 x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega_0^2 x = \frac{F(t)}{m},

with \gamma = b/(2m) the damping rate and \omega_0 = \sqrt{k/m} the natural angular frequency.^[22] The left-hand side represents the damped free oscillator, while the right-hand side introduces the driving term f(t) = F(t)/m.^[23] The general solution to this equation is the sum of the homogeneous solution x_h(t), which describes the transient response identical to the damped free oscillator and decays over time, and a particular solution x_p(t), which captures the steady-state response driven by F(t).^[24] Early in the evolution, the transient term dominates, leading to initial conditions-dependent behavior that fades as the system settles into the forced steady state. Detailed analysis of these components appears in subsequent sections on solutions.^[25] A key phenomenon in the driven case is resonance, where the amplitude of oscillation reaches a maximum under periodic driving F(t) = F_0 \cos(\omega t). For the undamped limit (\gamma \to 0), this occurs when the driving frequency \omega matches \omega_0. With damping present, the amplitude peaks at \omega = \sqrt{\omega_0^2 - 2\gamma^2} for \gamma < \omega_0 / \sqrt{2}, shifting the resonance below the natural frequency due to frictional effects.^[21] This condition amplifies the response significantly near \omega \approx \omega_0, enabling efficient energy transfer from the driver to the oscillator.^[26] Power absorption by the oscillator also exhibits resonance characteristics. The time-averaged power delivered by the driving force is P = \frac{1}{2} F_0 v_0 \cos \phi, where v_0 is the steady-state velocity amplitude and \phi the phase difference between force and velocity. This reaches its maximum at \omega = \omega_0, independent of damping, as the velocity aligns in phase with the force to optimize energy input.^[25] At this frequency, the oscillator extracts the most work from the driver, balancing dissipation.^[27] For a step input where the force suddenly becomes constant at F_0 for t > 0, the particular solution is a static shift to the new equilibrium x_p = F_0 / k, superimposed on the decaying transient oscillation from initial conditions. Over time, the response transitions from oscillatory transients to this steady displacement, illustrating how constant forcing repositions the equilibrium without sustained motion.^[28]

Advanced Oscillator Types

Parametric Oscillators

A parametric oscillator is a harmonic oscillator in which one or more of the system's parameters, such as the stiffness or mass, varies periodically with time, leading to potential instability and energy amplification without an external additive force. The canonical mathematical description is given by the Mathieu equation,

\frac{d^2 x}{dt^2} + \omega_0^2 (1 + \epsilon \cos(\Omega t)) x = 0,

where \omega_0 is the natural frequency, \epsilon is the modulation amplitude, and \Omega is the driving frequency of the parameter variation. This form was introduced by Émile Mathieu in 1868 while studying vibrations in elliptical membranes.^[29] The underlying mechanism is parametric resonance, which occurs when the driving frequency \Omega satisfies \Omega \approx 2\omega_0 / n for integer n, particularly the primary resonance at n=1 where \Omega \approx 2\omega_0. In this regime, small perturbations to the equilibrium position grow exponentially due to the periodic modulation transferring energy to the oscillator, resulting in unbounded solutions even from infinitesimal initial conditions. This contrasts with directly forced oscillators, where energy input arises from an additive periodic term in the equation of motion; here, amplification stems from modulation of intrinsic parameters like the effective spring constant.^[30] A classic historical example of parametric resonance is the child on a swing, where the child periodically shifts their center of mass (effectively modulating the pendulum's length or moment of inertia) at approximately twice the natural swinging frequency, leading to amplified oscillations without direct tangential pushes.^[31] In applications, parametric oscillators are pivotal in nonlinear optics, notably optical parametric oscillators (OPOs) used in tunable lasers. In an OPO, a high-frequency pump laser beam interacts with a nonlinear crystal inside a resonator, where the pump photon's energy splits into lower-frequency signal and idler photons satisfying energy conservation (\omega_p = \omega_s + \omega_i) and phase-matching conditions, enabling broadband wavelength conversion for spectroscopy and sensing. The first demonstration of an OPO occurred in 1965 using lithium niobate.^[32] Stability analysis of parametric oscillators relies on Floquet theory, which posits that solutions to linear differential equations with periodic coefficients are of the form x(t) = e^{\mu t} p(t), where p(t) is periodic and \mu is the Floquet exponent determining growth (\Re(\mu) > 0) or decay (\Re(\mu) < 0). For the Mathieu equation, stable regions correspond to bounded oscillations, while unstable tongues—regions of exponential growth in the parameter space of \Omega/\omega_0 versus \epsilon—emerge, with the primary tongue bifurcating near \Omega = 2\omega_0. These instability tongues, analogous to Arnold tongues in nonlinear dynamics, delineate the boundaries between periodic bounded motion and parametric amplification.^[30]^[33]

Universal Oscillator Equation

The universal equation governing classical linear harmonic oscillators is a second-order linear ordinary differential equation of the form

\frac{d^2 x}{dt^2} + 2\zeta \omega_0 \frac{dx}{dt} + \omega_0^2 x = \frac{F(t)}{m},

where x(t) denotes the displacement from equilibrium, \omega_0 = \sqrt{k/m} is the undamped natural angular frequency, \zeta = b/(2\sqrt{km}) is the damping ratio (with b the viscous damping coefficient), m is the mass, k is the stiffness coefficient, and F(t) is any external driving force.^[34] This standard form arises from applying Newton's second law to a mass-spring system subject to linear restoring, dissipative, and external forces.^[25] It encompasses the undamped case (\zeta = 0, F(t) = 0), the purely damped case (\zeta > 0, F(t) = 0), and the driven case (F(t) \neq 0).^[34] In the conservative context without damping or driving, the equation derives from a quadratic potential energy V(x) = \frac{1}{2} k x^2, yielding a linear restoring force F_\text{restoring} = -\frac{dV}{dx} = -k x.^[35] The full form incorporates a velocity-proportional damping force -b \frac{dx}{dt} and the external force F(t), leading to m \frac{d^2 x}{dt^2} = -k x - b \frac{dx}{dt} + F(t), which divides by m to produce the standard equation.^[25] Many nonlinear systems approximate this linear harmonic form near stable equilibria through linearization of the equations of motion. For instance, the simple pendulum's nonlinear equation \frac{d^2 \theta}{dt^2} + \frac{g}{l} \sin \theta = 0 (with \theta the angular displacement, g gravity, and l length) reduces to the harmonic oscillator equation \frac{d^2 \theta}{dt^2} + \frac{g}{l} \theta = 0 under the small-angle approximation \sin \theta \approx \theta.^[36] For analytical convenience, the equation can be recast in non-dimensional variables, such as the scaled time \tau = \omega_0 t and a normalized displacement u(\tau) = x(t), transforming it to

\frac{d^2 u}{d\tau^2} + 2\zeta \frac{du}{d\tau} + u = f(\tau),

where f(\tau) = F(\omega_0^{-1} \tau)/(m \omega_0^2) represents the normalized forcing term.^[37] This form highlights the roles of the damping ratio \zeta and natural frequency \omega_0 independently of specific physical scales. An alternative approach to solving the equation involves the Laplace transform, which converts the time-domain differential equation into an algebraic one in the s-domain: s^2 X(s) + 2\gamma s X(s) + \omega_0^2 X(s) = s x(0) + x'(0) + 2\gamma x(0) + \mathcal{L}\{F(t)/m\}, where \gamma = \zeta \omega_0 = b/(2m) and \mathcal{L} denotes the Laplace transform operator; initial conditions x(0) and x'(0) appear explicitly on the right-hand side.^[38]

Solutions and Analysis

Transient Solutions

The transient solutions of the harmonic oscillator describe the free, decaying motion that arises from the homogeneous part of the governing differential equation, particularly in systems subject to damping. For a damped oscillator, the homogeneous equation is given by

\frac{d^2 x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega_0^2 x = 0,

where \gamma is the damping coefficient, \omega_0 is the natural angular frequency, and x(t) is the displacement.^[39]^[40] This equation models the system's response without external forcing, capturing the evolution from initial conditions until the motion fades due to energy dissipation.^[41] The form of the transient solution x_h(t) depends on the damping regime, determined by the ratio of \gamma to \omega_0. In the underdamped case, where \gamma < \omega_0, the solution exhibits oscillatory decay:

x_h(t) = e^{-\gamma t} \left( A \cos(\omega_d t) + B \sin(\omega_d t) \right),

with the damped angular frequency \omega_d = \sqrt{\omega_0^2 - \gamma^2}.^[23] Here, A and B are constants determined by initial conditions. For the overdamped regime (\gamma > \omega_0), the solution is a sum of exponentials without oscillation:

x_h(t) = e^{-\gamma t} \left( C e^{\sqrt{\gamma^2 - \omega_0^2} \, t} + D e^{-\sqrt{\gamma^2 - \omega_0^2} \, t} \right),

where C and D are constants.^[39]^[41] At critical damping (\gamma = \omega_0), the solution takes the form

x_h(t) = (A + B t) e^{-\gamma t},

representing the fastest non-oscillatory return to equilibrium.^[40] The coefficients in these solutions are fixed by matching initial displacement x(0) and velocity \dot{x}(0), ensuring the transient response aligns with the system's starting state regardless of any external forcing.^[21] This independence from forcing terms highlights the transient's role as the "free" evolution superimposed on any driven behavior. All transient solutions share a common decay envelope e^{-\gamma t}, governed by the time constant \tau = 1/\gamma, which quantifies how quickly the oscillations or motion amplitude diminishes.^[41]^[23] In driven systems, the transient component x_h(t) dominates the short-term response, shaping the initial transients before the steady-state solution takes over as the homogeneous part decays to negligible levels.^[24]^[27]

Steady-State Solutions

In driven damped harmonic oscillators, the steady-state solution represents the long-term behavior after the initial transient response has decayed, resulting in sustained motion synchronized with the driving force. This particular solution is independent of initial conditions and persists indefinitely, as the driving force continuously supplies energy to counter damping losses.^[27] For a sinusoidal driving force F(t) = F_0 \cos(\omega t), the equation of motion is m \ddot{x} + b \dot{x} + k x = F(t), where m is mass, b is the damping coefficient, and k is the spring constant. To find the particular solution, represent the force in complex form as F(t) = \mathrm{Re}[F_0 e^{i \omega t}], and assume x_p(t) = \mathrm{Re}[A e^{i \omega t}]. Substituting yields A = \frac{F_0 / m}{\omega_0^2 - \omega^2 + 2 i \gamma \omega}, where \omega_0^2 = k/m is the natural frequency and \gamma = b/(2m) is the damping rate. Thus, the steady-state solution is x_p(t) = \mathrm{Re}\left[ \frac{F_0 / m}{\omega_0^2 - \omega^2 + 2 i \gamma \omega} e^{i \omega t} \right].^[27]^[21] The amplitude of this oscillation is |x_p| = \frac{F_0 / m}{\sqrt{ (\omega_0^2 - \omega^2)^2 + (2 \gamma \omega)^2 }}. The phase lag \delta between the driving force and the response satisfies \delta = \arctan\left( \frac{2 \gamma \omega}{\omega_0^2 - \omega^2} \right), indicating how the system's displacement trails the applied force due to inertia and damping.^[27]^[25] Resonance occurs when the driving frequency \omega maximizes the amplitude, at \omega = \sqrt{\omega_0^2 - 2 \gamma^2} for underdamped systems (\gamma < \omega_0 / \sqrt{2}); at this frequency, the response amplitude peaks, enhancing energy transfer from the driver. The phase lag reaches 90° precisely when \omega = \omega_0, where the driving force is perpendicular to the displacement in the phase plane.^[21]^[26] For a constant (step) driving force F(t) = F_0, the steady-state solution is a static displacement x_{ss} = F_0 / k, with no oscillation, as acceleration and velocity vanish in equilibrium.^[42] This mechanical system draws an analogy to electrical circuits via complex impedance, where the operator relating force to displacement is Z(\omega) = -\omega^2 m + i \omega b + k, such that x_p(\omega) = F(\omega) / Z(\omega); here, mass corresponds to inductance, damping to resistance, and the spring to inverse capacitance.^[25]^[43]

Full General Solution

The full general solution to the driven damped harmonic oscillator equation, \ddot{x} + 2\gamma \dot{x} + \omega_0^2 x = f(t), where f(t) = F(t)/m is the normalized driving force, is given by x(t) = x_h(t) + x_p(t) for t > 0, with x_h(t) denoting the homogeneous (transient) solution and x_p(t) the particular solution that satisfies the nonhomogeneous equation.^[24]^[21] This superposition holds due to the linearity of the differential equation, and the solution must satisfy specified initial conditions x(0) and \dot{x}(0). The constants in the homogeneous solution, typically A and B (or amplitude and phase \phi) for the underdamped case where x_h(t) = e^{-\gamma t} (A \cos \omega_d t + B \sin \omega_d t) with \omega_d = \sqrt{\omega_0^2 - \gamma^2}, are determined by applying the initial conditions to the total solution: x(0) = x_h(0) + x_p(0) and \dot{x}(0) = \dot{x}_h(0) + \dot{x}_p(0).^[21] Solving this system of two equations yields the unique values for A and B (or \phi), ensuring the solution matches the system's starting state.^[24] For a sinusoidal driving force f(t) = f_0 \cos \omega t, the full solution takes the form

x(t) = e^{-\gamma t} \left( A \cos(\omega_d t + \phi) \right) + D \cos(\omega t - \delta),

where D = \frac{f_0}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2 \gamma \omega)^2}} is the amplitude of the steady-state response and \delta = \tan^{-1} \left( \frac{2 \gamma \omega}{\omega_0^2 - \omega^2} \right) is its phase shift relative to the drive.^[21] The constants A and \phi are then found from the initial conditions as described above, adjusting for the contribution of the particular solution at t=0.^[24] In general, for an arbitrary driving force f(t), the particular solution can be expressed using Duhamel's integral (also known as the convolution integral or Duhamel's principle):

x_p(t) = \int_0^t h(t - \tau) f(\tau) \, d\tau,

where h(t) is the impulse response function of the system, given by h(t) = \frac{1}{\omega_d} e^{-\gamma t} \sin \omega_d t for the underdamped case.^[44] This integral form arises from the variation of parameters method and represents the system's response to the accumulated effect of the force up to time t.^[44] The total solution is then x(t) = x_h(t) + x_p(t), with constants in x_h(t) adjusted via initial conditions. As t \to \infty, provided \gamma > 0, the transient term x_h(t) decays exponentially to zero due to damping, leaving only the steady-state particular solution x_p(t).^[24] This asymptotic behavior highlights the long-term dominance of the driven response over free oscillations. For complex or non-periodic driving forces f(t), direct evaluation of Duhamel's integral may be impractical, necessitating numerical methods such as Fourier transforms to decompose f(t) into frequency components or Laplace transforms to solve the differential equation in the s-domain before inverting back to time.^[21] These transform techniques efficiently handle the convolution inherent in the general solution, especially for broadband or transient excitations.

Physical Realizations and Applications

Mechanical Systems

The mass-spring system represents a fundamental mechanical realization of the harmonic oscillator, where a mass m attached to a spring with stiffness constant k undergoes simple harmonic motion when displaced from equilibrium. The restoring force follows Hooke's law, F = -k x, leading to the equation of motion m \ddot{x} + k x = 0, with natural angular frequency \omega_0 = \sqrt{k/m}.^[16] The total mechanical energy E of this undamped system is conserved and given by E = \frac{1}{2} m v^2 + \frac{1}{2} k x^2, where v is the velocity and x the displacement, oscillating between kinetic and potential forms without loss.^[16] A simple pendulum, consisting of a point mass m suspended from a massless string of length l, approximates harmonic motion for small angular displacements \theta. Under the small-angle approximation, where \sin \theta \approx \theta, the equation of motion simplifies to \ddot{\theta} + (g/l) \theta = 0, yielding natural frequency \omega_0 = \sqrt{g/l} and period T = 2\pi \sqrt{l/g}, independent of mass.^[45] This approximation holds for angles less than about 15 degrees, beyond which nonlinear effects distort the motion.^[45] For extended bodies, the physical pendulum generalizes the simple case, with motion determined by the torque from gravity about the pivot. The natural frequency is \omega_0 = \sqrt{m g d / I}, where d is the distance from the pivot to the center of mass and I is the moment of inertia about the pivot.^[46] This formula accounts for the distribution of mass, reducing to the simple pendulum result when I = m l^2 and d = l.^[46] Damping in mechanical oscillators arises from dissipative forces like air resistance or viscous drag, often modeled as proportional to velocity, F_d = -b v, where b is the damping coefficient. For a sphere of radius r moving slowly in a fluid of viscosity \eta, Stokes' law provides b = 6 \pi \eta r, enabling quantitative prediction of energy loss in oscillatory motion.^[47] In driven mechanical systems, an external periodic force F(t) = F_0 \cos(\omega t) is applied, as in a forced pendulum or vibration isolation setups, leading to resonance when the driving frequency \omega matches \omega_0, amplifying amplitude.^[48] The 1940 Tacoma Narrows Bridge collapse illustrates the dangers of resonant excitation, though driven by aeroelastic flutter rather than pure linear resonance, serving as a caution against unchecked oscillations in structures.^[48] For a damped mass-spring system, the total energy decays exponentially as E(t) = E_0 e^{-2 \gamma t}, where \gamma = b/(2m) is the damping rate, reflecting the quadratic dependence on the amplitude that diminishes as e^{-\gamma t}.^[41] This decay governs the transition from oscillatory to aperiodic motion as damping increases.^[41]

Electrical and Optical Systems

In electrical systems, the harmonic oscillator manifests through LC circuits, where an inductor (L) and capacitor (C) are connected in series, leading to undamped oscillations of charge and current. The governing differential equation for the charge q on the capacitor is derived from Kirchhoff's voltage law: L \frac{d^2 q}{dt^2} + \frac{1}{C} q = 0.^[49] The natural angular frequency is \omega_0 = \frac{1}{\sqrt{LC}}, and the solution for the charge is q(t) = Q \cos(\omega_0 t + \phi), where Q is the amplitude and φ is the phase constant determined by initial conditions.^[50] Energy oscillates between the magnetic field in the inductor and the electric field in the capacitor without loss in the ideal case.^[51] Introducing a resistor (R) in series forms an RLC circuit, modeling damped harmonic motion. The equation becomes L \frac{d^2 q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = 0, with damping coefficient \gamma = \frac{R}{2L}.^[52] For underdamped conditions (R < 2\sqrt{L/C}), the charge decays exponentially while oscillating at \omega = \sqrt{\omega_0^2 - \gamma^2}. Resonance occurs when the driving frequency matches \omega_0, maximizing current amplitude, as exploited in radio tuning circuits to select specific frequencies.^[53] The electrical harmonic oscillator is analogous to the mechanical spring-mass system, with mappings: mass m ↔ inductance L, damping coefficient b ↔ resistance R, spring constant k ↔ reciprocal capacitance 1/C, displacement x ↔ charge q, and applied force F ↔ applied voltage V.^[54] This equivalence facilitates analysis using familiar mechanical intuition for circuit design. In optical systems, Fabry-Pérot cavities serve as resonators analogous to harmonic oscillators, consisting of two parallel mirrors separated by distance L, supporting standing electromagnetic waves. The resonant angular frequencies for modes are \omega_m = m \pi c / (L n), where m is the mode number, c is the speed of light, and n is the refractive index of the medium. These cavities exhibit high finesse, enabling narrow linewidth resonances for precise frequency control. Driven optical oscillators include lasers, where a gain medium (e.g., doped crystal or gas) provides amplification to sustain oscillations against cavity losses, analogous to an external force driving a mechanical oscillator. Parametric down-conversion in nonlinear crystals, such as beta-barium borate, generates signal and idler photons from a pump beam, enabling tunable optical parametric oscillators when placed in a cavity.^[55] Applications in electronics leverage LC and RLC circuits as bandpass filters and oscillators for signal processing, while in optics, mode-locking techniques in laser cavities synchronize multiple longitudinal modes to produce ultrashort pulses, typically femtoseconds long, vital for time-resolved spectroscopy and microscopy.^[56]

Quantum Harmonic Oscillator

The quantum harmonic oscillator extends the classical model into quantum mechanics by treating the position and momentum as operators satisfying the commutation relation [x, p] = i \hbar, leading to discrete energy states and wavefunctions that describe the probability distribution of the particle's position. The time-independent Schrödinger equation for this system is

-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + \frac{1}{2} m \omega_0^2 x^2 \psi = E \psi,

where m is the mass, \omega_0 is the angular frequency, \hbar is the reduced Planck's constant, and E is the energy eigenvalue. This equation, derived as part of the eigenvalue problem for quantization, yields exact solutions that reveal the quantized nature of the oscillator's energy, fundamentally differing from the continuous classical spectrum.^[57] The energy eigenvalues are given by E_n = \hbar \omega_0 \left(n + \frac{1}{2}\right), where n = 0, 1, 2, \dots is the quantum number, introducing a zero-point energy of \frac{1}{2} \hbar \omega_0 even in the ground state, which prevents the particle from being at rest and reflects quantum fluctuations.^[57] The corresponding eigenfunctions are

\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{m \omega_0}{\pi \hbar} \right)^{1/4} e^{-m \omega_0 x^2 / 2 \hbar} H_n \left( \sqrt{\frac{m \omega_0}{\hbar}} x \right),

with H_n denoting the Hermite polynomials, which ensure orthogonality and completeness of the basis. These wavefunctions oscillate with increasing nodes as n increases, providing a quantum description of the particle's localization.^[57] To simplify the algebra, creation (a^\dagger) and annihilation (a) operators are introduced, satisfying the bosonic commutation relation [a, a^\dagger] = 1, with position and momentum expressed as x = \sqrt{\frac{\hbar}{2 m \omega_0}} (a + a^\dagger) and p = i \sqrt{\frac{m \omega_0 \hbar}{2}} (a^\dagger - a). These operators raise or lower the energy by \hbar \omega_0, facilitating the construction of states from the ground state.^[58] The ground state wavefunction \psi_0(x) is a Gaussian, \psi_0(x) = \left( \frac{m \omega_0}{\pi \hbar} \right)^{1/4} e^{-m \omega_0 x^2 / 2 \hbar}, achieving the minimum uncertainty \Delta x \Delta p = \frac{\hbar}{2} as per the Heisenberg uncertainty principle, illustrating the balance between position and momentum spreads in quantum mechanics.^[57] This model finds broad applications in physics, serving as an approximation for molecular vibrations where diatomic bonds are treated as quantized oscillators, explaining infrared spectra through transitions between levels. In solid-state physics, it underlies the description of phonons as collective vibrations in crystal lattices, quantized modes that govern thermal and electrical properties. In quantum field theory, fields are expanded in terms of infinite sets of non-interacting harmonic oscillators, providing the foundation for particle creation and annihilation processes.^[58]

References

[1]
15.1 Simple Harmonic Motion – University Physics Volume 1
A very common type of periodic motion is called simple harmonic motion (SHM). A system that oscillates with SHM is called a simple harmonic oscillator.
[2]
21 The Harmonic Oscillator - Feynman Lectures - Caltech
The simplest mechanical system whose motion follows a linear differential equation with constant coefficients is a mass on a spring.Missing: authoritative | Show results with:authoritative
[3]
Simple Harmonic Motion and Resonance | Middle Tennessee State ...
SHM occurs around an equilibrium position when a mass is subject to a linear restoring force. A linear restoring force is one that gets progressively larger ...
[4]
[PDF] Chapter 1 - Harmonic Oscillation - MIT OpenCourseWare
Oscillators are the basic building blocks of waves. We begin by discussing the harmonic oscillator. We will identify the general principles that make the ...Missing: authoritative | Show results with:authoritative
[5]
[PDF] 5 The Harmonic Oscillator Fall 2003 - andrew.cmu.ed
In many systems the restoring force is approximately proportional to the displacement ... harmonic oscillator. An important special case is a driving force ...
[6]
[PDF] Lecture 1: Simple Harmonic Oscillators
This course studies those oscillations. When many oscillators are put together, you get waves. Almost all physical processes can be explained by breaking them ...<|separator|>
[7]
[PDF] Harmonic oscillator Notes on Quantum Mechanics
Nov 30, 2006 · harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of. the particle. In classical physics ...
[8]
[PDF] Harmonic Oscillators and Coherent States
Harmonic oscillators are ubiquitous in physics. For example, the small vibrations of most me- chanical systems near the bottom of a potential well can be ...
[9]
[PDF] The Quantum Harmonic Oscillator - Georgia Institute of Technology
The harmonic oscillator is extremely useful in chemistry as a model for the vibrational motion in a diatomic molecule. Polyatomic molecules can be modeled ...
[10]
Simple Harmonic Oscillator - Galileo
The simple harmonic oscillator, a nonrelativistic particle in a potential 12kx2, is a system with wide application in both classical and quantum physics.
[11]
Who first solved the classical harmonic oscillator?
Feb 17, 2020 · Huygens considered the motion of pendula, and for simple cases knew the "law of the conservation of living force" (mechanical energy), as ...Who really discovered/invented the Hooke's law?Who solved the quantum harmonic oscillator?More results from hsm.stackexchange.com
[12]
Springs - The Physics Hypertextbook
He was successful in this regard and in 1678 Hooke made the solution to the anagram, and the true theory of springiness that now bears his name, public ...
[13]
Torsion Harmonic Oscillator | Wolfram Formula Repository
A torsion harmonic oscillator is a twisting system that, when displaced from its equilibrium position, experiences a restoring force proportional to the ...
[14]
15.1 Simple Harmonic Motion – General Physics Using Calculus I
x ( t ) = A cos ( ω t + φ ) . This is the generalized equation for SHM where t is the time measured in seconds, is the angular frequency with units of inverse ...
[15]
[PDF] Chapter 23 Simple Harmonic Motion
Jul 23, 2013 · The accuracy of clocks was increased and the size reduced by the discovery of the oscillatory properties of springs by Robert Hooke. By the ...
[16]
13. Adiabatic Invariants and Action-Angle Variables
The phase space elliptical orbit has semi-axes with lengths √2mE, √2E/mω2, so the area enclosed is πab=2πE/ω. The bottom line is that as we gradually change ...
[17]
15.2 Energy in Simple Harmonic Motion - UCF Pressbooks
Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant: E Total = 1 2 m v 2 + 1 ...
[18]
[PDF] Simple Harmonic Oscillator, Classical Pendulum, and General ...
[Conservation of Energy]. (57) dt. Our simple harmonic oscillator system satisfies conservation of energy by virtue of Eq.(20). We can show by di erentiating ...
[19]
[PDF] Lecture 2: Driven oscillators
Driven oscillators. 1 Introduction. We started last time to analyze the equation describing the motion of a damped-driven oscil- lator: d2x dt2 + γ dx dt. + ω0.
[20]
[PDF] 2.6 The driven oscillator
That is, we want to solve the equation. M d2x(t) dt2. + γ dx(t) dt. + κx(t) ... The problem we want to solve is the damped harmonic oscillator driven by a ...
[21]
[PDF] 2 Damped and Driven Harmonic Oscillation - Xie Chen
then xD(t)+x0(t) is also a solution of the inhomogeneous equation, describing the driven, damped oscillation. While xD(t) describes the periodic driven ...
[22]
Driven Oscillators - HyperPhysics
If a damped oscillator is driven by an external force, the solution to the motion equation has two parts, a transient part and a steady-state part.
[23]
Driven Damped Harmonic Oscillation
Driven damped harmonic oscillation is a steady oscillation maintained by continuously feeding energy to offset frictional losses in a damped oscillator.
[24]
[PDF] RES.8-009 (Summer 2017), Lecture 5: Driven Oscillations
One of the main features of resonance is that the amplitude of oscillation is at a value higher than it would be if the system were not in resonance. For the ...
[25]
Damped Driven Oscillator - Galileo
The phase lag of the oscillations behind the driver, θ=tan−1(bω/(k−mω2)), is completely determined by the frequency together with the physical constants of the ...
[26]
[PDF] Transients and Oscillations in RLC Circuits - Course Websites
• Commonly discuss impulse response and step response. Physics 401. 3 ... • A good reference LTI system is a driven damped harmonic oscillator. m. d.
[27]
Computation and Applications of Mathieu Functions: A Historical ...
Mathieu functions of period 𝜋 or 2 ⁢ 𝜋 , also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified ...
[28]
[PDF] Mathieu's Equation and Its Generalizations - Cornell Mathematics
1 Introduction. Mathieu's equation is one of the archetypical equations of non- linear vibrations theory [1].
[29]
[PDF] PARAMETRIC RESONANCE - Eugene Butikov
The most familiar example of paramet- ric resonance is a child swinging on a swing. The swing is a physical pendulum whose reduced length changes periodi ...
[30]
Tunable Coherent Parametric Oscillation in LiNb O 3 at Optical ...
Tunable Coherent Parametric Oscillation in LiNb O 3 at Optical Frequencies. J. A. Giordmaine and Robert C. Miller. Bell Telephone Laboratories, Murray Hill ...Missing: paper | Show results with:paper
[31]
Parametric resonances: from the Mathieu equation to QASER
Jun 6, 2016 · We use the Floquet theory and develop a projection method that can properly capture gains near the primary resonance and subharmonic frequencies ...
[32]
[PDF] The Impact Damped Harmonic Oscillator in Free Decay
Between two successive impacts the equations of motion are simply those of a damped harmonic oscillator and of a free particle. Well separated bounces are.
[33]
[PDF] Chapter 8 The Simple Harmonic Oscillator
The functional form of a simple harmonic oscillator from classical mechanics is V (x) = 1 2 kx2 . Its graph is a parabola as seen in the figure on the left.
[34]
Simple Pendulum
This is known as the small-angle approximation. Making use of this ... Equation (1.52) is the simple harmonic oscillator equation. Hence, we can ...
[35]
[PDF] controlling the nature of bifurcation in friction-induced vibrations with ...
Sep 12, 2013 · Figure 1: Damped harmonic oscillator on a moving belt ... where primes denote derivative with respect to the non-dimensional time τ = ω0 t.
[36]
[PDF] The LaPlace Transform: 1 s
Take inverse Laplace Transform to get X(t) oscillatory behavior. Page 4. Damped Harmonic Oscillator. Initial Conditions: Laplace Transform Example: s2x(s) ...
[37]
[PDF] The Harmonic Oscillator
May 23, 2008 · The harmonic oscillator is a common model used in physics because of the wide range of problems it can be applied to.
[38]
[PDF] 2-9 The damped harmonic oscillator
This form of the equation is often used in discussing mechanical oscilla- tions. The total energy of the oscillator is (2-144) -27e-** + 7²te-rt. (2-146) a (C1 ...Missing: standard | Show results with:standard
[39]
Damped Harmonic Oscillator - HyperPhysics
A damped harmonic oscillator has a damping force linearly dependent on velocity, causing exponential decay. Underdamped cases oscillate about zero, with a ...
[40]
Driven Oscillators - HyperPhysics Concepts
If a constant force is applied to a damped oscillator, it will stretch out to a final position x = F/k determined by its spring constant. However, depending ...Missing: formula | Show results with:formula<|control11|><|separator|>
[41]
http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html
[42]
[PDF] 18.03 Supplementary Notes - MIT Mathematics
Many differential equations textbooks present beats as a system response when a harmonic oscillator is driven ... Edwards and Penney call it. “Duhamel's principle ...
[43]
Oscillation of a Simple Pendulum - Graduate Program in Acoustics
The period of this sytem (time for one oscillation) is T = 2 π ω = 2 π L g . Small Angle Assumption and Simple Harmonic Motion. animation showing three ...
[44]
https://math.mit.edu/~hrm/papers/sup.pdf
[45]
5.2 Drag Forces – College Physics - UCF Pressbooks
... drag force is given by Stokes' law,. F s = 6 π η r v ,. where r is the radius of the object, η is the fluid viscosity, and v is the object's velocity.
[46]
November 7, 1940: Collapse of the Tacoma Narrows Bridge
The collapse of the Tacoma Narrows Bridge was driven by wind-generated vortices that reinforced the twisting motion of the bridge deck until it failed. Join ...
[47]
LC Circuit
The oscillations of an LC circuit can, thus, be understood as a cyclic interchange between electric energy stored in the capacitor, and magnetic energy stored ...
[48]
[PDF] 31. LC oscillator and mechanical analogue. Electric/magnetic ...
Nov 30, 2020 · magnetic energy: UB = 1. 2. LI2. (stored on inductor). • electric energy: UE = Q2. 2C. (stored on capacitor). • total energy: E = UB + UE = ...
[49]
Resonant RLC Circuits - HyperPhysics
An example of the application of resonant circuits is the selection of AM radio stations by the radio receiver. The selectivity of the tuning must be high ...
[50]
Analogous Electrical and Mechanical Systems - Swarthmore College
To apply this analogy, every node in the electrical circuit becomes a point in the mechanical system. Ground becomes a fixed location, resistor become friction ...<|control11|><|separator|>
[51]
Optical Parametric Oscillators – OPO, nonlinear ... - RP Photonics
Optical parametric oscillators are coherent light sources based on parametric amplification in a resonator, in some ways similar to lasers.
[52]
Mode Locking – laser pulse generation, active, passive, ultrashort ...
Mode locking denotes a group of techniques for generating ultrashort pulses with lasers. Various active and passive mode locking techniques are discussed.What is Mode Locking? · Active Mode Locking · Passive Mode Locking
[53]
Quantisierung als Eigenwertproblem - Wiley Online Library
First published: 1926. https://doi.org/10.1002/andp.19263840404. Citations ... Download PDF. back. Additional links. About Wiley Online Library. Privacy ...
[54]
The quantum theory of the emission and absorption of radiation
The quantum theory of the emission and absorption of radiation. Paul Adrien Maurice Dirac.