The Math of Quantum Mechanics

As was the case with relativity theory, there is a structure to the mathematics of quantum mechanics.

A good starting point is the Schrodinger Wave Equation. A wavefunction gives information about the "state"of a particle. The Schrodinger equation describes particle behavior over time.

Because in particle physics the place and velocity of a particle cannot be simultaneously determined (due to the basic wave-particle duality of fundamental entities), it is the probability of finding particle in a particular space that is determined by the wave function.

Particles interact as waves, but arrive probabilistically as particles in space and time.

Another way of arriving at a picture of quantum processes is to arrange observables in a matrix.

Perhaps the most intriguing insight of quantum mechanics is that at the most fundmental level, reality is connected. This means the interactions, the events, the processes are what is real, not the entities.

This one insight responds to centuries of metaphysical speculation about the nature of reality. Combined with the ideas of entanglement and nonlocality, this takes us to a place where metaphysics and ethics may intersect.

Is it possible that being responsible for our environment and fellow citizens is hardwired into our nature? That when we diverge from this ideal we go against the grain of our being and cause injury to ourselves?

Admittedly this a lot to digest if you are not familiar with quantum physics. Basically what particle physicists have found is that at the most fundamental level, nature is neither mass nor energy, but a "combination" of both.

When we talk of quarks, electrons, strings, and other subatomic entities there is no mental picture possible. Only mathematical formalism which leads to abstract metaphors.

The names associated with the developments of quantum mechanics include Schrodinger, Heisenberg, Planck, Bohr, Dirac, deBroglie, Pauli, and Einstein.


They all developed significant parts of the theory and performed amazing mathematical wizardry in the process.

Schrodinger's wave function was accompanied by Heisenberg's matrix, Bohr's atom and deBroglie's duality. Heisenberg Uncertainty, Pauli Exclusion, Dirac's Sea, and Bohr's Complementarity were critical perspectives.

The fact that the extension of the wavelength described by the Fourier Transform cannot be made arbitrarily small is another way of viewing Heisenberg Uncertainty.

Uncertainty, probability, is built into the very structure of nature as we see it. Wave intensity is linked to probability and wavelength is linked to momentum. The very nature of particle position and movement is ethereal and fuzzy.

Before going too deeply into particle physics, it is instructive to visit aspects of classical mechanics.

We should consider how Newton's laws bleed into quantum mechanics.

Newton's Second Law of particle motion relates force, mass and acceleration. Classical particle dynamics rests on this law. The Euler-Lagrange Equation generalizes the law for particle motion. Hamilton's Principle provides for the simplification of the math of particle trajectory.

Hamiltonians and Lagrangians are key elements of the math structure of classical mechanics and they are key to quantum mechanics as well. They provide for robust evaluation of particle dynamics and field theory.

The Lagrangian determindes the amplitude of a particle's wave and therefore the probability of tha associated particle's existence.

Feynman diagrams allow us to visualize the results of particle interactions.

A system of n particles is described by 3n differential equations. Hamiltonians simplify multi-dimensional calculations.

Finding the probability that a particle is in a region of space involves integrating the wave function. The time indepenent Schrodinger equation has an eigenvalue form.

The Hamiltonian can be used to determine the possible results of the measurement of the energy of a wavefunction. The eigenvalues of the Hamiltonian are the energies of the system.

The trajectory of an electron is determined by position, speed, and force. Interactions between matter and light involve em fields and photons.

Operators are critical components of the math of quantum mechanics. Operators work on wavefunctions and assist in the measurement of values. Observable events, those involving position and momentum are represented by operators.

Eigenvalues of a matrix which represents an operator give measurement results. Eigenvalues of matrices representing Hamiltonians are possible energies of wavefunctions. Eigenvectors of a Hermitian matrix can be utilized in developing simlarity transformations.

A particle's motion can be represented in either position or momentum form. Fourier transforms relate these two forms. In quantum mechanics much of the activity occurs in mathematical spaces like Hilbert Space.

In state space, functions take the role of vectors. Spaces of any number of dimension are mathematically constructable.

For a vector space, the number of vectors in a basis gives the dimension. To make basis vectors orthonormal, the Gram -Schmidt procedure is employed.

Eigenvalues and Eigenfunctions of Hermitian Operators acting on functions of continuous variables provide additional imformation on particle behavior.

Wave mechanics can give a complete theory of events in the absense of relativity. Dirac equations allow for relativistic and field effects. Fourier analysis gives particles freedom by releasing confinements inposed by wave equations.

Studying atoms and molecules involves working in three dimensions.

Again, this all sounds a little abstract if we don't keep our eye on the prize...that all this math ultimately tells us what is going on deep in the heart of nature, ourselves, and in others. That knowledge can lead us to a deeper understanding of the mysteries of existence which have puzzled philosophers for millenia.

Lee