Maths

Physical symmetries assumptions

• Reality is knowable, it is then describable with a descriptor.

• Causality determinism is a must.

• Time is uniform, it is then presented with a propagator.

• Information cannot be lost, then every propagator has an inverse.

• Reality is continuous, then every derivative must exist.

Since every derivative must exist, linear differential equations are used, 'linear' means the order is growing progressively like in Taylor series. Note that the order of a differential equation is the order of the highest derivative in the equation. Operators in Quantum mechanics have to be linear to deal with linear differential equations describing superposition states. Other examples of linear series with an order progressively increasing: Legendre Polynomials, Magnus expansions, Fourier series, Floquet series, etc. Since developments may be extended to infinity, cutting the equation is equivalent to an approximation and is usually acceptable if high order terms have considerably small contribution. Approximations are set after confronting the experimental observations to the theoretical equations.

Exponential laws were found to faithfully describe all self-growing natural laws. Napier discovered the natural logarithm to transform multiplications in additions. Since, this function was extended do different bases. Exponential is found to be just a special basis for which Napier Logarithm is equal to 1. Exponential evolution (growth or decay) is widely used since it faithfully describes natural laws e.g. Boltzmann distribution, Arrhenius equation, etc. More importantly exponential functions are invariant with mathematical operations, this renders them very useful as eigenfunctions in quantum mechanics.

Exponential is a sum of Tailor series. Euler introduced the imaginary component (i) in the formula eiq = cosq + i.sinq, after exploring the series he developed for the exponential function while solving Bernoulli’s financial interest problem. The comparison of the series he got for ‘e’ with the series obtained earlier for (sin) and (cos) led to Euler equation. The addition of (sin) and (cos) gives (e) only if sin is multiplied by an imaginary number (i) for which the square is -1.

Euler formula is used in electronics, since capacitors exhibit a phase shift between the voltage input and the current output, they are not in phase. The output is the derivative of the input (cos to sin). When using exponential, calculations become easy since both the derivative and the integral of an exponential is an exponential, exponential being an eigenfunction. Note also that exponential functions are just multiplied by different orders of Legendre polynomials to represent the wavefunctions, and similarly, the NMR operators.   

Despite the imaginary part in the wavefunction expression, its square is described as electronic density and all observables in quantum physics are real. This imaginary component in Euler formulas is found in Quantum Mechanics since it could describe wavefunctions, which must always have a positive energy despite their eventual unphased evolution. This also explains why all Hamiltonians in Quantum Mechanics have to be Hermitian, named after Hermite i.e. operators equal to their conjugates; Hamiltonians have to deal with real and imaginary components equally, it should be Hermitian. Expectation values of Hermitian operators, associated with physical observables, must be real. Hamiltonians are scalar obeying the linearity obligation also. Linearity is fundamental in Q.M. It allows several versions of reality to simultaneously exist in the state vector and implies there is no interaction between them, these are known as superposition states. Wave functions represent a probability of existence for the electron and its square is nothing then electronic density. Note that the wavefunction squared has to be equal to 1 in the entire space, it is the integral of the electronic probability density function (pdf), a pdf is always equal to 1. 

Faraday, inspired by Coulomb, noticed the exchange between electricity and magnetism. Maxwell, inspired by Gauss, wrote the mathematical formalism considering Faraday observations, using 3D Gaussian probability distribution functions.  Note that Ampere proposed the interconversion of magnetic and electric currents earlier, as it was already noticed by Ampere. The equations of Maxwell explain why nuclei with a spin are magnetic dipoles and not monopoles. They are only electric monopoles as magnetic monopoles do not exist. 

Maxwell has proven mathematically that the speed of light (c) is equal to 1/µ0e0, µ0 and e0 being the permeability (magnetic) and permittivity (electric) of vacuum. Since, light is considered an electromagnetic wave. Photons (particles of light) were later suggested by Planck and Einstein to explain the interaction of light with electrons in matter. Electrons in matter were later seen as wave-particles by De Broglie. This led Schrodinger to suggest a wave function for electrons. Schrodinger built his waves on the circles of Bohr’s model using the equation of standing waves written by D’Alembert to describe the vibration of a violin cord associating the space with the time. Note that Huygens did interpret the light as a wave in the 1600s while Newton believed the light was made of particles called ether (photons, 300 years later). Because of D’Alembert equations for standing waves, the boundaries of the waves should be at zero, this allows only to some multiple frequencies to exist, and explains Planck’s quantification of energy states. Then, when waves superpose on circles starting and finishing at zero, they can be nothing than a multiple of Integer numbers.

Wave is a self-sustaining disturbance in a continuous medium, that can move through space without transporting the medium. A wave function relates space and time, D’Alembert. It is written as linear partial differential equation, they obey superposition principle. Complex notations are commonly used (e = cos + i.sin) describing eventual phase shift. Euler generalized D’Alembert equation for 3 D waves. Standing waves are equivalent to harmonic motions (same frequency, same phase). It is a superposition of two travelling waves of same frequency and amplitude in opposite directions. Normal modes are the solutions at defined n levels, n = 1, 2, 3, etc. the wave equation could be decomposed with a linear combination of harmonic functions (Fourier series). For a travelling wave, Fourier transformation is used instead of Fourier series. A wave vector (k) defines the direction of wave propagation. Plane waves expansion expresses a plane wave as a linear combination of spherical waves made of Legendre polynomials multiplied by Bessel functions, wave vectors and position vectors. Bessel function is a generalization of the sine function. It can be interpreted as the vibration of a string with variable properties (thickness, tension, medium etc.). they should be the ones used to describe the ultra-soft potential in CASTEP using plane waves approach where the wave vector depends on the symmetry of the system studied. Once defined in a Brillouin zone, it decreases the weight of calculations.

Lagrange and Hamilton transformed the discussion into Energy and set the last math equations for Energy conservation. Leibniz and Newton did not use the term ‘Energy’ when they established their laws. Conservation laws are established thanks to Noether’s theorem found in 1915. According to Noether’s theorem, the conservation laws in physics derive from the symmetry character of a physical system. Then, for every system presenting a ‘continuous symmetry’ in the mathematical transformation of the coordinate system (which keeps invariant the integral of action ‘the Lagrangian’), there are corresponding quantities whose values are conserved. This is related to the principle of least action established by Lagrange. Then, rotational symmetry leads to the conservation of angular momentum while time translation symmetry leads to the conservation of Energy.

Potential is a point, field (change of potential) is a vector, field gradient (change of field) is an ellipsoid. The location of any point in space may be defined using different coordinate bases: (i) cartesian coordinates (x, y, z) or (ii) spherical coordinates (r, theta, phi) called radial, azimuthal and polar successively (these terms are also used when using a compass). Spherical coordinates are often used for mathematical simplification of the equations. Polar coordinates are a particular case of spherical coordinates. 

Spherical harmonics are used to describe orbitals for electrons but also the tensors and tensor operators for the NMR interactions. These spherical harmonics are nothing else then Legendre polynomials growing in order. S (0) (Isotropic) or Anisotropic (0th order), P (1) Anisotropic (1st order), D (2) anisotropic (2nd order). The first order anisotropic interactions are described with 2nd rank tensors (+ isotropic part if the tensor has a trace). Second order anisotropic interactions are represented by 0th, 2nd and 4th rank tensors, note that 4th rank tensors result from a multiplication of two 2nd order tensors giving an additional 0th rank tensor (e.g. the quadrupolar induced shift QIS for the 2nd order quadrupolar interaction.). Spherical harmonics associated with Legendre polynomials were introduced by Laplace in 1782.

The wave functions are nothing else than exponentials multiplied by the normal modes of stationary states with different orders and different n. The n is interpreted as principal quantum number. n and other quantum numbers (l and m) are introduced by Somerfield and his students. Fermi suggested the last quantum number for electrons, the ‘spin’ as a hidden angular momentum. The wavefunction of an electron is described with 4 quantum numbers, n, l, ml and ms. The total angular momentum being the sum of orbital and spin terms. Clebsch-Gordan series expansion answers the following question: How do wave functions associated with l, ml, s, ms combine to form wave functions associated with j and mj knowing that j = l+s, it is the total angular momentum. G-C coefficients are non-zero only when J= l+s. These coefficients are used in MQMAS. We only use the z components of these terms, as we do not have access to other components.

Pauli principle comes from indistinguishability of electrons. While their energy eigenvalues are the same, their wavefunctions are different. They are symmetric and asymmetric (plus or minus) fermions. Particles with only symmetric wave functions are bosons.

Two operators are said to commute if their commutator vanishes [X , Y] = 0. A commutator in quantum mechanics tells us if we can measure two 'observables' at the same time. If the commutator of two 'observables' is zero (if they commute), then they can be measured at the same time, otherwise, an uncertainty relationship occurs between the two, e.g. position and momentum do not commute, time and frequency do not commute, Heisenberg. In NMR, the same logic applies for interactions, when two interactions are recoupled simultaneously, we can determine each of them separately, precisely, only if they commute. They won’t have effects on each other when one of them is selected despite their simultaneous presence since they commute.

Fourier Transform (FT) splits any function in scaled sinusoidal functions with different frequencies and phases. More importantly, from the Fourier equations, it is clear that time and frequency do not commute: for a short (exact) time, the frequency is wide (not exact) while for a long (uncertain) time, the frequency is sharp (exact). This is similar to Heisenberg uncertainty, for position and momentum. Position for momentum is like time for energy (or frequency). Position and momentum operators do not commute. their commutator is not zero. Similarly, in FT-NMR spectroscopy, a short pulse in time irradiates a wide frequency area in frequency domain. A FT of a square pulse is a sinc (sinus cardinal). Note that the only function that stays the same after FT is the Gaussian. This may explain the interest is Gaussian pulses in some conditions, Gauss pulse gives Gauss irradiation.

Hilbert infinite-dimensional vector space allows the methods of linear algebra and calculus to be generalized from Euclidean finite-dimensional vector space. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric spa A Hilbert space is a special case of a Banach space. Wavefunctions are seen as vectors in the Hilbert space (a complex space) obeying all the obligations of QM. That’s why it is used to describe them. Wavefunctions are eigenvectors or eigenfunctions of the energy operator (Hamiltonian).

In Hilbert space, eigenvalues scale eigenfunctions or eigenvectors. Eigenvectors are those that do not change their direction after undergoing an operation that changes their coordinate system, they are only scaled by eigenvalues. This explains why Euler exponentials are usually describing the wave functions. Exponential has the particularity to be only scaled by constants after integration or derivation. Wave functions are seen as vectors in the Hilbert space (a complex space) obeying all the obligations of Quantum Mechanics. Wave functions are eigenvectors or eigenfunctions that are scaled by Energy (eigenvalue) after the application of the energy operators (Hamiltonian) on them.

While gravity and electric forces are conservative, magnetic force is not. Magnetic force is not conservative because it does a work, see Lagrange and Noether. Forces from magnetic field change direction of charges velocity vector but not its speed. Electric and magnetic fields transform into each other when moving between different inertial frames. Actually, magnetism is nothing than a relativistic effect of electricity. Magnetism is needed to make the physical behavior of electricity look the same in all the different reference frames.

Even though the developed mathematical concepts allowed us to establish complicated equations explaining the mechanics of the universe going from elementary particles to planets, a considerable gap still exist between the microscopic and the macroscopic physics, namely thermodynamics and quantum physics. Quantum thermodynamics are under development... The paradox of the time arrow in both is a real challenge.

Chronology of Maths concepts

• 1200  Fibonacci Bigollo          Fibonacci sequence.

• 1494  Luca Pacioli                      Father of accounting

• 1545 Gerolamo Cardano complex numbers and probability.

• 1591 Franciscus Viète      new notations in algebra.

• 1614 John Napier            logarithms.

• 1637  René Descartes             Cartesian coordinate system.

• 1637  Pierre de Fermat          Fermat’s Last Theorem.

• 1665  Blaise Pascal                   Pascal’s Triangle.

• 1675 Leibniz-Newton      infinitesimal calculus.

• 1738  Jacob Bernoulli              Bernoulli principle.

• 1740 Leonhard Euler         Euler equations.

• 1747 Jean D'Alembert     D'Alembert equation.

• 1782 Legendre                Legendre polynomials.

• 1788 Joseph-Louis Lagrange Lagrange equation.

• 1795 Carl Friedrich Gauss number theory, geometry, and probability.

• 1807 Joseph Fourier        Fourier series.

• 1814 Pierre Simon Laplace  Laplace equation.

• 1833  William Hamilton         Hamiltonian Mechanics.

• 1850 charles Hermite e is transcendantale.

• 1850  George Boole                 Boolean logic.

• 1877  Georg Cantor                 Set theory.

• 1890  David Hilbert                  Hilbert’s basis theory.

• 1904  Henri Poincaré              Poincaré conjecture.

• 1913  Srinivasa Ramanujan  Landau-Ramanujan constant.

• 1931 Eugene Wigner       formalism for quantum mechanics.

• 1932  John von Neumann    Operator theory and quantum mechanics.

• 1942  Alan Turing                      Father of computer science.

• 1950  John Forbes Nash        Nash embedding theorem.

• 1993  Andrew Wiles                Proving “Fermat’s Last Theorem”.