quantum-mechanical calculation

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.