quantum superposition principle

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1: 18.39 Applications in the Physical Sciences

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§18.39(i) Quantum Mechanics

… ►which is the quantum superposition principle. … … ►

The Quantum Coulomb Problem

… ►where $n$ is now the Bohr Principle Quantum Number. …

2: 17.17 Physical Applications

… ►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys. ►Quantum groups also apply $q$-series extensively. Quantum groups are really not groups at all but certain Hopf algebras. They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. … ►A substantial literature on $q$-deformed quantum-mechanical Schrödinger equations has developed recently. …

3: 25.17 Physical Applications

§25.17 Physical Applications

►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. …See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). …

4: 32.16 Physical Applications

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Statistical Physics

►Statistical physics, especially classical and quantum spin models, has proved to be a major area for research problems in the modern theory of Painlevé transcendents. … ►For the Ising model see Barouch et al. (1973), Wu et al. (1976), and McCoy et al. (1977). ►For applications in 2D quantum gravity and related aspects of the enumerative topology see Di Francesco et al. (1995). …

5: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

6: 28.33 Physical Applications

… ►The general solution of the problem is a superposition of the separated solutions. … ►

Meixner and Schäfke (1954, §§4.1, 4.2, and 4.7) for quantum mechanical problems and rotation of molecules.

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Fukui and Horiguchi (1992) for quantum theory.

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7: 24.18 Physical Applications

… ►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

8: 36.14 Other Physical Applications

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§36.14(iii) Quantum Mechanics

►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …

9: 8.24 Physical Applications

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§8.24(i) Incomplete Gamma Functions

►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). … ►

§8.24(iii) Generalized Exponential Integral

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

10: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …