quantum-mechanical calculation

(noun)

Branch of physics which studies matter and energy at the level of atoms and other elementary particles, and substitutes probabilistic mechanisms for classical Newtonian ones.

Related Terms

Examples of quantum-mechanical calculation in the following topics:

  • Indeterminacy and Probability Distribution Maps

    • Quantum mechanics provides a recipe for calculating this probability distribution.
    • An adequate account of quantum indeterminacy requires a theory of measurement.
    • Many theories have been proposed since the beginning of quantum mechanics, and quantum measurement continues to be an active research area in both theoretical and experimental physics.
    • In quantum mechanical formalism, it is impossible that, for a given quantum state, each one of these measurable properties (observables) has a determinate (sharp) value.
    • In the world of quantum phenomena, this is not the case.

  • Linear Combination of Atomic Orbitals (LCAO)

    • An LCAO approximation is a quantum superposition of atomic orbitals, used to calculate molecular orbitals in quantum chemistry.
    • This function can be used to calculate the probability of finding any electron in any specific region around an atom's nucleus.
    • An orbital may also refer to the physical region where the electron can be calculated to exist, as defined by the orbital's particular mathematical form.
    • A linear combination of atomic orbitals, or LCAO, is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry.
    • In quantum mechanics, electron configurations of atoms are described as wave functions.

  • Description of the Hydrogen Atom

    • The hydrogen atom (consisting of one proton and one electron, not the diatomic form H2) has special significance in quantum mechanics and quantum field theory as a simple two-body problem physical system that has yielded many simple analytical solutions in closed-form.
    • Bohr's results for the frequencies and underlying energy values were confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925–1926.
    • From this, the hydrogen energy levels and thus the frequencies of the hydrogen spectral lines can be calculated.
    • The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function (orbital) for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.
    • Identify the unique features of the hydrogen atom that make it important for calculations in quantum mechanics

  • Bonding in Coordination Compounds: Valence Bond Theory

    • In 1927, physicist Walter Heitler and collaborator Fritz London developed the Heitler-London theory, which enabled the calculation of bonding properties of the covalently bonded diatomic hydrogen molecule (H2) based on quantum mechanical considerations.
    • Calculate the theoretical hybridization of a metal in a coordination complex based on valence bond theory

  • The de Broglie Wavelength

    • He related this to the principal quantum number n through the equation:
    • Recent experiments even confirm the de Broglie relations for molecules and macromolecules, which are normally considered too large to undergo quantum mechanical effects.
    • The researchers calculated a de Broglie wavelength of the most probable C60 velocity as 2.5 pm.
    • More recent experiments prove the quantum nature of molecules with a mass up to 6910 amu.
    • Even macroscopic objects like tennis balls have a calculable de Broglie wavelength; however, they would be much too small to observe experimentally, and their wave-like nature is not intuitive to common experience.

  • Planck's Quantum Theory

    • As a result of these observations, physicists articulated a set of theories now known as quantum mechanics.
    • In some ways, quantum mechanics completely changed the way physicists viewed the universe, and it also marked the end of the idea of a clockwork universe (the idea that universe was predictable).
    • The wavelength or frequency of any specific occurrence of EM radiation determine its position on the electromagnetic spectrum and can be calculated from the following equation:
    • One photon of light carries exactly one quantum of energy.
    • Planck is considered the father of the Quantum Theory.

  • The Bohr Model

    • Due to its simplicity and correct results for selected systems, the Bohr model is still commonly taught to introduce students to quantum mechanics.
    • The quantum theory from the period between Planck's discovery of the quantum (1900) and the advent of a full-blown quantum mechanics (1925) is often referred to as the old quantum theory.
    • Bohr's model is significant because the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule.
    • 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.
    • This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.

  • The Uncertainty Principle

    • In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position (x) and momentum (p), can be known simultaneously.
    • Heisenberg offered such an observer effect at the quantum level as a physical explanation of quantum uncertainty.
    • It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems and that it arises in quantum mechanics simply due to the matter-wave nature of all quantum objects.
    • Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it.
    • These include, for example, tests of number-phase uncertainty relations in superconducting or quantum optics systems.

  • Wave Equation for the Hydrogen Atom

    • The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum.
    • This leads to a third quantum number, the principal quantum number n = 1, 2, 3, ....
    • The principal quantum number in hydrogen is related to the atom's total energy.
    • Note that the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i.e. ℓ = 0, 1, ..., n − 1.
    • According to the usual rules of quantum mechanics, the actual state of the electron may be any superposition of these states.

  • The Pauli Exclusion Principle

    • The Pauli exclusion principle, formulated by Austrian physicist Wolfgang Pauli in 1925, states that no two fermions of the same kind may simultaneously occupy the same quantum state.
    • In the theory of quantum mechanics, fermions are described by antisymmetric states.
    • In contrast, particles with integer spin (bosons) have symmetric wave functions; unlike fermions, bosons may share the same quantum states.
    • Electrons, being fermions, cannot occupy the same quantum state, so electrons have to "stack" within an atom—they have different spins while at the same place.
    • As spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli exclusion principle.