Ecumenism

 

Hello folks, creatives and science-centric people, today with this post I will embrace a new line of articles with the aim of not to explain what our target project will be all about – A Python-Centric Pseudo-Code Web-Portfolio – but to explain the science behind some of the most inventive spirits – works of other people as well as founding our own in the future…!

 

Nature is not classical but quantum-mechanical. People’s minds try to model nature and the environment around them so that they can understand them better. The language of the universe, some people name it “the code of nature” relies on fluid probabilities. The math of quanta, have been profound enough to make it clear that quantum physics can explain ‘almost’ everything. But what does that mean?

 

The discovery of wave-particle duality of matter and the wave-particle duality of light made it clear once and for all that if the particle-carrier of the electromagnetic power is the photon, there must be corresponding particles-carriers that can address the rest of the fundamental forces of nature. Do these particles truly exist? Almost all. The strong nuclear force is being represented by gluons, the weak nuclear force by the W and Z bossons, as well as the force of gravity that is being represented by baryons, even though… not yet invented. It is worth mentioning that the weak and strong nuclear forces due to the weak ranking have not yet the power to be manifested as classical interactions but only as quantum fields. For gravity, the verification of the wave-particle duality principle is going to wait for too long, even though nobody seems to doubt this result…

 

It seems that quantum physics can still shed some light for the grand unification of the 4 forces of nature, meaning, combined in one, and that The School of Copenhagen is going to exemplify to the world the ecumenical impact of the math of quanta.

 

Some of the most peculiar arguments on the interpretation of quantum physics rise from the double slit experiment conducted by Thomas Yang. Separation of the electron in two, branching universe, the effect of consciousness, wave-driver and many more, as well as the probable manifestation of God into nature, derive one of the most “divide and rule” phenomenon in the scientific community of Physicists that makes it clear what it’s all about: Jonsson Vs. Jonsson. That is, the version of the experiment by the German Physicist Clauss Jonsson.

 

As a project that will be continuously upgraded in the future by elements of QFT, RQFT, QED, QCD, these are the acronyms for Quantum Field Theory, Relativistic Quantum Field Theory, Quantum Electrodynamics, Quantum Chromodynamics, it’s worth noting that quantum physics can have groundbreaking impact in many other disciplines, including the above theories. To understand the ecumenical impact of the math of quanta, that is one of the building blocks of creativity, means first of all we have to do the math. Simply put: define the problem, define the boundary conditions, transform the problem in math, that is define the form Schrodinger’s equation is going to take and then try and play with the equation as it is obvious that in quantum physics we don’t always come up with differential equations’ solutions but just learning how to use them. It simply means that if there’s a problem with solid math then these are things we can put into a computer, ‘almost’ all of the times and come up with explanations, perhaps with comparative analyses of formulas as well, of complex problems. When can’t we put them into it? When we make a measurement in physics this lies on the principle of filtering which means we can only measure unique eigen values and only afterwards confirm Schrodinger. Schrodinger’s equation can only be solved precisely on the Hydrogen atom, at least to the limit where all other interactions between the electron and the proton are ignored, except the electrostatic.

 

Scientific programming though isn’t merely coming up with measurements. One scientist embarks on a journey to measure the world and that demands multi-faceted expertise as well as multi-disciplinary knowledge. As a beginning, the building block of quantum physics offers a unique and competitive stand to explain all about it, that is, the basics of the world of quanta not only for the dream of grand unification, but the act of using global functions to come up with innovative explanations of scientific as well as artistic phenomena. So, here’s what to expect from the journey into the fascinating world of quanta:

 

Utility Foundation of Quantum Mechanics:

 

Light, particles, the 2-slit experiment, the photoelectric effect, wave-particle dualism, De Broglie equation, elements of electromagnetism and the status quo before Quantum Mechanics, Heisenberg uncertainty, “Schrodinger's cat” thought experiment, basic equations. 

 

Required math:

Complex numbers, complex calculus and polar form of complex numbers, proof of Euler's identity for complex numbers, 2-dimensional derivatives of functions, partial derivatives, vector fields, Del operator, divergence, curl, Laplacian, operations with r, derivatives in 3 dimensions, Integrals (double, triple), coordinate systems, Cartesian coordinates, polar coordinates, cylindrical coordinates, spherical coordinates and their utility, geometric interpretations thereof, volume of a sphere.

 

The Schrodinger equation:

 

The classical wave equation and problems in quantum mechanics, the Schrodinger equation and the wave function of the universe, discussion and interpretation of the Schrodinger equation, superposition and evolution coefficients, normalization, probability density in quantum mechanics, continuity equation and probability current density, quantum mechanical operators and expectation values. 

 

Solving the stationary Schrodinger equation with examples:

 

Time-independent potential and product ansatz (separation of time dependence and spatial dependence), stationary Schrodinger equation, infinite potential well, finite potential well, eigenvector equations, energy levels, eigensystems and eigenvalues, potential staircase, general potentials, particles in a ring, wavepackets and uncertainty minimization, Fourier transform, quantum tunneling.

 

Scientific Programming in Python: 

 

Anaconda, Jupyter Notebook and Python programming of the stationary Schrodinger equation.

 

Theoretical extra background:

 

Kronecker symbols, delta and delta distribution, different representations of wave functions and quantum mechanical operators, Dirac notation or Bra-ket notation, Hermitian operators, adjoint operator, Commutators and proof through them of Heisenberg uncertainty, time evolution and Ehrenfest theorem. 

 

Second quantization and harmonic oscillator: 

 

Presentation of simple classical oscillators, solving the quantum mechanical harmonic oscillator problem in the general form with Hermite polynomials, Ladder Operators and their physical interpretation, comparison and illustration of classical and quantum mechanical probability. 

 

The hydrogen atom:

 

Solving the general Schrodinger equation of hydrogen by splitting the original equation into subproblems. Separation of the radial dependence from the angular dependences – polar, azimuthal angle – and analytical solution of the equations in general forms. Legendre polynomials and other functions. Spherical Harmonics. Eigenvectors and eigenenergies in the hydrogen atom.

Relativistic quantum mechanics and electron spin:

 

Introduction to special relativity, frames of reference, Lorenz Transformations – length contraction, time dilation – mass energy equivalence, relativistic energy dispersion with Taylor series. Klein-Gordon equation, muons, mionic hydrogen. Pauli equation, introduction to electronic spin and also the Dirac equation. The Dirac equation is the relativistic counterpart of the Schrodinger equation and includes it as well as the Pauli equation as special cases. Stern-Gerlach experiment and physical realization of spin through magnetic moment. Expressing spin-up, spin-down wavefunctions, Pauli matrices, spin calculations and related equations. Relativistic 1/c^2 corrections and Hamiltonian expressions with new terms – Zeeman, Darwin, Spin-orbit coupling.

 

Quantum Computing:

 

Introduction to quantum computing and formalism, physical realization of qubits, spin, tensor product and related equations. Quantum gates, Hadamard gate, CNOT gate and SWAP gate. Single-qubit, double qubit functions, expectation values, propagator, Larmor frequency and related equations. Entanglement and quantum teleportation. “Spooky action at a distance”. 

 

Epilogue: 

 

Historical interpretations of the theory of quantum physics based on mathematical formalism, alternative probabilism of parallel universes and hidden variables: The Copenhagen interpretation, Many-Worlds theory, Bohm theory. 

 

Time after time these mean we’ll have to always get deeper and deeper into the principles that started it all, right? But that’s what science is all about…! Have a great time folks!

 

Photo Credits: Menelaos Gkikas