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Joule-second

The joule-second (symbol: J s) is the coherent derived unit in the International System of Units (SI) for the physical quantities of action and angular momentum, defined as the product of one joule of energy and one second of time.^[1] Expressed in terms of SI base units, it equals kilogram meter squared per second (kg⋅m²⋅s⁻¹).^[1] In classical mechanics, action is a scalar quantity representing the integral over time of the Lagrangian of a physical system, where the Lagrangian is the difference between kinetic and potential energy; this integral has dimensions of energy multiplied by time and underpins the principle of least action, which states that the path taken by a system between two points in configuration space is the one that minimizes the action.^[2] Angular momentum, a vector quantity, quantifies the amount of rotation in a system and is given by the cross product of position and linear momentum vectors (L = r × p), with the same dimensional structure as action.^[1] The joule-second holds particular significance in quantum mechanics as the unit of the Planck constant h, fixed at exactly 6.626 070 15 × 10⁻³⁴ J s, which defines the scale at which quantum effects become prominent and relates the energy of a photon to its frequency through E = hν.^[3] The reduced Planck constant ħ = h / (2π), approximately 1.054 571 817 × 10⁻³⁴ J s,^[4] appears in the quantization of angular momentum, where orbital angular momentum levels are multiples of ħ.^[5] These constants link classical and quantum descriptions, influencing fields from atomic spectra to black hole physics.

Fundamentals

Definition

The joule-second (symbol: J s) is the coherent derived unit in the International System of Units (SI) formed by multiplying the joule (J), the unit of energy, by the second (s), the unit of time. It quantifies physical action, a scalar quantity that describes the evolution of a system over time, as well as angular momentum.^[6]^[7] In physics, action plays a central role as the unit of a fundamental quantity in Lagrangian mechanics, defined as the integral of the Lagrangian L—typically the kinetic energy minus the potential energy—over time along a system's trajectory:

S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt,

where q represents generalized coordinates and \dot{q} their time derivatives. This integral measures the "cost" of the path in phase space, and action has dimensions of energy times time. The principle of least action asserts that the true physical trajectory of a system is the one for which this action is stationary, meaning its first variation \delta S = 0, providing a variational foundation for deriving equations of motion.^[8]^[2] Action can also be expressed as the integral of momentum p with respect to displacement dq, S = \int p \, dq, which highlights its interpretation as a measure of phase space volume enclosed by a system's orbit in certain canonical formulations. One well-known physical constant expressed in joule-seconds is Planck's constant h.^[8] A concrete example of a quantity with units of joule-seconds is angular momentum \mathbf{L}, defined as the cross product of the position vector \mathbf{r} and linear momentum \mathbf{p}:

\mathbf{L} = \mathbf{r} \times \mathbf{p}.

The SI units of \mathbf{L} are kilogram meters squared per second (kg m²/s), equivalent to J s because one joule equals kg m²/s². This equivalence underscores the deep connection between action-like quantities and rotational dynamics in classical mechanics.^[7]

Dimensional Analysis

The joule-second, as a unit of action, possesses the dimensional formula [J s] = [M L^2 T^{-1}], where M denotes mass, L length, and T time.^[9] This structure arises from the multiplication of the dimensions of energy, which is [M L^2 T^{-2}] for the joule, by the dimension of time [T], yielding the combined form [M L^2 T^{-1}].^[10] Dimensional analysis reveals that the joule-second remains invariant under changes in the choice of units, as dimensions express the intrinsic scaling relationships between physical quantities independent of specific measurement systems.^[11] This invariance facilitates the construction of dimensionless quantities in physics, such as the fine-structure constant \alpha \approx 1/137, which incorporates the reduced Planck's constant \hbar (with units of joule-seconds) alongside other constants to form a unitless parameter characterizing electromagnetic interactions.^[12] In natural units where \hbar = [1](/page/1), the action becomes dimensionless, underscoring the joule-second's role as a fundamental scaling unit that aligns quantum mechanical principles with classical action formulations.^[13]

Units and Equivalents

Expression in SI Base Units

The joule-second (J s), as a derived SI unit for action and angular momentum, is expressed in terms of the base units of mass, length, and time as $1 \, \mathrm{J \cdot s} = 1 \, \mathrm{kg \cdot m^2 \cdot s^{-1}}.^[14] This composition involves only the kilogram (kg) for mass, the metre (m) for length, and the second (s) for time, reflecting the purely mechanical nature of the unit without dependence on electrical, thermal, or other base quantities such as the ampere, kelvin, mole, or candela.^[14] The derivation follows from the definition of the joule itself, which is the SI unit of energy equivalent to one newton-metre (N m). The newton is defined as $1 \, \mathrm{[N](/page/N+)} = 1 \, \mathrm{kg \cdot m \cdot s^{-2}}, so the joule becomes $1 \, \mathrm{J} = 1 \, \mathrm{N \cdot m} = 1 \, \mathrm{kg \cdot m^2 \cdot s^{-2}}.^[14] Multiplying by one second yields the joule-second: $1 \, \mathrm{J \cdot s} = 1 \, \mathrm{kg \cdot m^2 \cdot s^{-2}} \times 1 \, \mathrm{s} = 1 \, \mathrm{kg \cdot m^2 \cdot s^{-1}}.^[14] This explicit expression aligns directly with the definitions provided in the SI Brochure for coherent derived units.^[14]

Conversions to Other Systems

The joule-second, with dimensions of kg m² s⁻¹ in SI base units, converts to the centimeter-gram-second (CGS) system as 1 J s = 10⁷ erg s, where the erg is the CGS unit of energy equivalent to g cm² s⁻².^[15] In Gaussian units, a CGS-based system prevalent in electromagnetic contexts, the unit of action remains erg s, dimensionally g cm² s⁻¹, so 1 J s = 10⁷ g cm² s⁻¹.^[16] In atomic units, the standard unit of action is the reduced Planck's constant ħ, defined as exactly 1 atomic unit of action and equal to 1.054 571 817 × 10⁻³⁴ J s in SI units.^[17] This provides the conversion factor: a quantity of action in J s divided by this value yields the measure in atomic units. The atomic unit of time, ħ / E_h where E_h is the hartree energy of 4.359 744 722 × 10⁻¹⁸ J, is 2.418 884 327 × 10⁻¹⁷ s, linking action to temporal scales in atomic physics.^[18]^[19] A practical conversion example is Planck's constant h = 6.626 070 15 × 10⁻³⁴ J s, which equals 4.135 667 696 × 10⁻¹⁵ eV s when expressed in electronvolt-seconds, a form often used in quantum optics and particle physics to align with energy scales in eV.^[20]^[21] Similarly, the reduced constant ħ ≈ 6.582 119 569 × 10⁻¹⁶ eV s facilitates comparisons across energy-frequency relations.^[22]

Physical Applications

In Classical Mechanics

In classical mechanics, the joule-second (J s) functions as the standard unit for angular momentum, a vector quantity that describes the rotational analog of linear momentum. For a rigid body rotating about a fixed axis, the angular momentum \vec{L} is expressed as \vec{L} = I \vec{\omega}, where I is the moment of inertia (with units kg m²) and \vec{\omega} is the angular velocity (with units rad/s).^[23] This yields units for L of kg m²/s, equivalent to J s, since 1 J = 1 kg m²/s².^[24] Angular momentum is conserved in isolated systems without external torques, playing a key role in analyzing rotational dynamics such as planetary motion or spinning objects.^[23] The joule-second also appears as the unit for the physical action in Lagrangian and Hamiltonian formulations of mechanics. The action S is defined through the integral S = \int_{t_1}^{t_2} \mathcal{L} \, dt, where \mathcal{L} = T - V is the Lagrangian, with T as kinetic energy and V as potential energy, both in joules.^[25] Integrating energy over time gives S dimensions of J s, reflecting action as a measure of the system's dynamical path.^[25] This quantity underpins variational principles, enabling the derivation of equations of motion from optimization criteria rather than direct force balances. A specific application arises in central force problems, such as orbital motion under gravity, where the orbital angular momentum magnitude is L = m v r for velocity \vec{v} perpendicular to the position vector \vec{r} from the central body.^[26] Here, m is the orbiting mass (kg), v is the tangential speed (m/s), and r is the orbital radius (m), yielding units kg m²/s or J s.^[24] Conservation of this angular momentum dictates that v \propto 1/r for circular orbits, stabilizing trajectories in systems like planetary systems or satellite paths.^[26] Hamilton's principle further illustrates the role of the joule-second by stating that the actual trajectory of a mechanical system between two times renders the action stationary, meaning \delta S = 0 for variations around the true path.^[25] Deviations from this path increase or decrease S (measured in J s), quantifying how closely a proposed trajectory aligns with the equations of motion derived from Newton's laws.^[25] This principle unifies diverse mechanical phenomena, from particle motion to rigid body rotations, by treating dynamics as a minimization of action.^[25]

In Quantum Mechanics

In quantum mechanics, the joule-second serves as the unit of action, most prominently embodied in Planck's constant h, which quantifies the scale at which quantum effects become significant. Following the 2019 revision of the International System of Units (SI), h is defined exactly as $6.626\,070\,15 \times 10^{-34} J s, anchoring the definitions of several base units including the kilogram.^[27] This constant introduces discreteness into physical processes, contrasting with the continuous trajectories of classical mechanics. The reduced Planck's constant, denoted \hbar = h / 2\pi, has the value $1.054\,571\,817 \times 10^{-34} J s and plays a central role in the mathematical formalism of quantum theory, appearing in commutation relations and wave functions.^[4] For instance, the energy E of a photon is given by E = h f, where f is the frequency, as proposed by Einstein in his explanation of the photoelectric effect. Similarly, the de Broglie relation connects a particle's momentum p to its associated wavelength \lambda via p = h / \lambda, establishing wave-particle duality for matter. The time-energy form of the Heisenberg uncertainty principle, \Delta E \, \Delta t \geq \hbar / 2, limits the precision with which energy and time can be simultaneously known, reflecting the intrinsic probabilistic nature of quantum systems. Furthermore, angular momentum is quantized in integer multiples of \hbar, as derived from the solutions to the Schrödinger equation for central potentials, forming the basis for discrete energy levels in atoms and molecules.

Distinctions and Confusions

Difference from Joules per Second

The joule per second (J/s) is the SI derived unit of power, equivalent to the watt (W), which quantifies the rate of energy transfer or conversion.^[28]^[29] In contrast, the joule-second (J s) serves as the unit of action, representing the product of energy and time rather than a rate.^[30]^[31] A key distinction arises in their dimensional formulations: the joule-second has dimensions of [M L^2 T^{-1}], derived from energy ([M L^2 T^{-2}]) multiplied by time ([T]), while joules per second possess dimensions of [M L^2 T^{-3}], reflecting energy divided by time.^[32] This fundamental difference underscores that action accumulates over time in a manner independent of instantaneous rates, whereas power describes instantaneous flow. A common misconception confuses these units in technical specifications, such as laser outputs or electrical systems, where "joules per second" correctly denotes power rather than cumulative action.^[33] For instance, in laser applications, device power is rated in watts to indicate energy delivery rate, avoiding misinterpretation as the action integral used in quantum contexts. Similarly, in electrical work, applying power (J/s) over a duration yields total energy (J), but computing action requires multiplying that energy by time in a distinct physical framework, such as phase space integrals in mechanics.^[29]^[34]

Relation to Other Action Units

The joule-second, as the SI unit of action, connects to other units employed in alternative measurement systems and specialized physical domains. In the centimetre-gram-second (CGS) system, the equivalent unit is the erg-second, reflecting the historical use of CGS in classical mechanics and early 20th-century physics literature. Specifically, 1 J s = $10^7 erg s, since the erg is defined as exactly $10^{-7} J. In particle and quantum physics, the electronvolt-second (eV s) serves as a convenient alternative, particularly for quantities involving fundamental constants like Planck's constant. The value of h is $4.135\,667\,696 \times 10^{-15} eV s, facilitating calculations in high-energy contexts where energies are expressed in electronvolts.^[20] Although the joule-second shares dimensional similarities with certain mechanical quantities, it must be distinguished from the newton-second (N s), which denotes impulse or change in linear momentum with dimensions kg m/s. In contrast, the joule-second has dimensions kg m²/s, corresponding to action or angular momentum, highlighting their distinct physical interpretations despite superficial unit resemblances in some derivations.^[35]

Historical Context

Origin of the Concept

The concept of action in physics originated in the 18th century with Pierre Louis Maupertuis, who in 1744 proposed the principle of least action as a foundational law governing the motion of physical systems. Maupertuis defined action as the integral of momentum with respect to position along the path, expressed as \int p \, dq, where p is momentum and dq is an infinitesimal displacement in configuration space. This formulation posited that nature selects the path minimizing this quantity, providing a variational approach to mechanics that unified diverse phenomena under a single principle.^[8] In the 1830s, William Rowan Hamilton advanced this idea through his principle of stationary action, formalizing the action S as the time integral of the Lagrangian function, S = \int L \, dt, where L = T - V with T kinetic energy and V potential energy. This Hamiltonian action explicitly carries dimensions of energy multiplied by time, establishing action as a fundamental quantity in classical mechanics with consistent units across systems. Hamilton's work, detailed in his 1834 and 1835 essays, extended Maupertuis' abbreviated action to more general cases, influencing subsequent developments in dynamics.^[36] Prior to the formal definition of the joule in 1889, action was quantified using pre-SI systems such as the foot-pound-second (FPS) framework prevalent in British and American engineering. In this system, energy was measured in foot-pounds (force), yielding action in foot-pound-seconds, equivalent to the product of mechanical work and duration. This unit reflected the practical applications in 19th-century mechanics before the international adoption of metric standards.^[37] A pivotal implicit application of action units occurred in Max Planck's 1900 derivation of the blackbody radiation law, where he introduced the quantum of action h to resolve the ultraviolet catastrophe in classical theory. Planck's constant h, with dimensions of energy times time, discretized energy exchanges in oscillators, marking the birth of quantum mechanics and directly tying the action concept to atomic-scale phenomena.^[38]

Standardization in SI

The joule, the base unit underlying the joule-second, was formally adopted as a practical unit of work at the International Electrical Congress in Paris in 1889, defined as $10^7 erg (cgs units of work), with early electrical realizations relying on the Clark cell to standardize voltage measurements for energy calculations.^[39] This electrical definition was later refined and aligned with mechanical equivalents, establishing the joule as equivalent to one kilogram meter squared per second squared (\mathrm{kg \cdot m^2 \cdot s^{-2}}) through absolute measurements by the International Committee for Weights and Measures (CIPM) in 1946 and ratification by the 9th General Conference on Weights and Measures (CGPM) in 1948.^[14] The joule-second (J s) was incorporated into the International System of Units (SI) upon its establishment at the 11th CGPM in 1960 as a coherent derived unit for action (and angular momentum), expressed in base units as \mathrm{kg \cdot m^2 \cdot s^{-1}}, without a special name to distinguish it from the joule (J) and second (s), which retain their own names.^[14] The International Bureau of Weights and Measures (BIPM) plays a central role in upholding this coherence by coordinating global metrology standards, disseminating SI definitions through CGPM resolutions, and ensuring traceability for measurements involving the joule-second in scientific and industrial applications.^[40] This standardization evolved significantly with the 2019 SI redefinition, approved by the 26th CGPM in 2018 and effective from May 20, 2019, which fixed the Planck constant h exactly at $6.626\,070\,15 \times 10^{-34} J s, shifting the joule-second from reliance on physical artifacts (such as the international prototype kilogram) to invariant fundamental constants for enhanced precision and universality.^[14] Prior to this, definitions depended on absolute realizations of base units like the kilogram and second; post-2019, the joule-second's scale is directly anchored to h, confirming its status as a stable derived unit while maintaining compatibility with pre-existing measurements.

References

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- **Planck's Constant (h):**
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