momentum

(noun)

The product of the mass and velocity of a particle in motion.

Related Terms

Examples of momentum in the following topics:

  • Wave Equation for the Hydrogen Atom

    • This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus.
    • Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, ℓ and mℓ (both are integers).
    • The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum.
    • In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wavefunctions must be found.
    • Due to angular momentum conservation, states of the same ℓ but different mℓ have the same energy.

  • Description of the Hydrogen Atom

    • This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus.
    • Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, ℓ and m (both are integers).
    • The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum.
    • The magnetic quantum number m = −, ..., +ℓ determines the projection of the angular momentum on the (arbitrarily chosen) z-axis.
    • In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found.

  • The Uncertainty Principle

    • Only partial knowledge of the momentum and position of a particle may be known at the same time.
    • The more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.
    • The reasoning was derived from considering the uncertainty in both the position and the momentum of an object.
    • The uncertainty in the momentum of the object follows from de Broglie's equation as h/λ.
    • A more formal inequality relating the standard deviation of position (${ \sigma }_{ x }$) and the standard deviation of momentum (${ \sigma }_{ \rho }$) was derived by Earle Hesse Kennard later that year (and independently by Hermann Weyl in 1928):

  • The de Broglie Wavelength

    • The de Broglie wavelength is inversely proportional to the momentum of a particle.
    • In his 1924 PhD thesis, de Broglie sought to expand this wave-particle duality to all material particles with linear momentum.
    • The de Broglie equations relate the wavelength (λ) to the momentum (p), and the frequency (f) to the kinetic energy (E) (excluding its rest energy and any potential energy) of a particle:
    • This was fortunately reminiscent of Bohr's observation about the angular momentum of an electron, which had already been established:
    • Use the de Broglie equations to determine the wavelength, momentum, frequency, or kinetic energy of particles

  • Quantum Numbers

    • This model describes electrons using four quantum numbers: energy (n), angular momentum (ℓ), magnetic moment (mℓ), and spin (ms).
    • For particles in a time-independent potential, per the Schrödinger equation, it also labels the nth eigenvalue of Hamiltonian (H) (i.e. the energy E with the contribution due to angular momentum, the term involving J2, left out).
    • The second quantum number, known as the angular or orbital quantum number, describes the subshell and gives the magnitude of the orbital angular momentum through the relation.
    • The magnetic quantum number describes the energy levels available within a subshell and yields the projection of the orbital angular momentum along a specified axis.
    • The fourth quantum number describes the spin (intrinsic angular momentum) of the electron within that orbital and gives the projection of the spin angular momentum (s) along the specified axis.

  • The Bohr Model

    • Although Rule 3 is not completely well defined for small orbits, Bohr could determine the energy spacing between levels using Rule 3 and come to an exactly correct quantum rule—the angular momentum L is restricted to be an integer multiple of a fixed unit:
    • Starting from the angular momentum quantum rule, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions.
    • The Bohr-Kramers-Slater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.

  • Indeterminacy and Probability Distribution Maps

    • Quantum indeterminacy can be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified.

  • Metallic Crystals

    • According to the theory of special relativity, increased mass of inner-shell electrons that have very high momentum causes orbitals to contract.

  • The Pauli Exclusion Principle

    • Half-integer spin means the intrinsic angular momentum value of fermions is $\hbar =\frac { h }{ 2\pi }$ (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.).

  • Mechanistic Background

    • Excited states may be classified as singlet or triplet based upon their electron spin angular momentum.