Ciro Santilli @cirosantilli 34 

One major application of this classification is that different boundary conditions are suitable for different types of partial differential equations as explained at: which boundary conditions lead to existence and uniqueness of a second order PDE.

Bibliography:

math.stackexchange.com/questions/1090299/why-are-elliptic-parabolic-hyperbolic-pdes-called-elliptic-parabolic-hyperb

Complete basis  Updated 2024-12-15  +Created 1970-01-01

Finding a complete basis such that each vector solves a given differential equation is the basic method of solving partial differential equation through separation of variables.

The first example of this you must see is solving partial differential equations with the Fourier series.

Notable examples:

Fourier series for the heat equation as shown at Fourier basis is complete for $L^{2}$ and solving partial differential equations with the Fourier series
Hermite functions for the quantum harmonic oscillator
Legendre polynomials for Laplace's equation in spherical coordinates
Bessel function for the 2D wave equation on a circular domain in polar coordinates

Control theory  Updated 2024-12-15  +Created 1970-01-01

This basically adds one more ingredient to partial differential equations: a function that we can select.

And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?

It's quite fun from a mathematics point of view!

Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.

Dirac equation  Updated 2024-12-15  +Created 1970-01-01

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

spontaneous emission coefficients.
fine structure, notably for example Dirac equation solution for the hydrogen atom
antimatter
particle creation and annihilation

Experiments not explained: those that quantum electrodynamics explains like:

Lamb shift
TODO: quantization of the electromagnetic field as photons?

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:

i \partial_{t} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} ψ_{3} ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤ = - i \partial_{x} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{4} ψ_{3} ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ + \partial_{y} ⎣ ⎢ ⎢ ⎢ ⎡ - ψ_{4} ψ_{3} - ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ - i \partial_{z} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{3} - ψ_{4} ψ_{1} - ψ_{2} ⎦ ⎥ ⎥ ⎥ ⎤ + m ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} - ψ_{3} - ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤

Expanded Dirac equation in Planck units

Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

Video 1.

Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)

Source.

Video 2.

PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)

Source.

Finite element method  Updated 2024-12-15  +Created 1970-01-01

Used to solve partial differential equation.

TODO understand, give intuition, justification of bounds and JavaScript demo.

Hamilton's equations  Updated 2024-12-15  +Created 1970-01-01

Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:

\frac{\partial H}{\partial q _{j}} = - \overset{p}{˙}_{j} \frac{\partial H}{\partial p _{j}} = \overset{q}{˙}_{j}

So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.

Hans Petter Langtangen  Updated 2024-12-15  +Created 1970-01-01

GitHub account: github.com/hplgit

It should be mentioned that when you start Googling for PDE stuff, you will reach Han's writings a lot under his GitHub Pages: hplgit.github.io/, and he is one of the main authors of the FEniCS Project.

Unfortunately he died of cancer in 2016, shame, he seemed like a good educator.

He also published to GitHub pages with his own crazy markdown-like multi-output markup language: github.com/hplgit/doconce.

Rest in peace, Hans.

History of quantum mechanics  Updated 2024-12-15  +Created 1970-01-01

The discovery of the photon was one of the major initiators of quantum mechanics.

Light was very well known to be a wave through diffraction experiments. So how could it also be a particle???

This was a key development for people to eventually notice that the electron is also a wave.

This process "started" in 1900 with Planck's law which was based on discrete energy packets being exchanged as exposed at On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).

This ideas was reinforced by Einstein's explanation of the photoelectric effect in 1905 in terms of photon.

In the next big development was the Bohr model in 1913, which supposed non-classical physics new quantization rules for the electron which explained the hydrogen emission spectrum. The quantization rule used made use of the Planck constant, and so served an initial link between the emerging quantized nature of light, and that of the electron.

The final phase started in 1923, when Louis de Broglie proposed that in analogy to photons, electrons might also be waves, a statement made more precise through the de Broglie relations.

This event opened the floodgates, and soon matrix mechanics was published in quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925), as the first coherent formulation of quantum mechanics.

It was followed by the Schrödinger equation in 1926, which proposed an equivalent partial differential equation formulation to matrix mechanics, a mathematical formulation that was more familiar to physicists than the matrix ideas of Heisenberg.

Inward Bound by Abraham Pais (1988) summarizes his views of the main developments of the subjectit:

Planck's on the discovery of the quantum theory (1900);
Einstein's on the light-quantum (1905);
Bohr's on the hydrogen atom (1913);
Bose's on what came to be called quantum statistics (1924);
Heisenberg's on what came to be known as matrix mechanics (1925);
and Schroedinger's on wave mechanics (1926).

Bibliography:

physics.stackexchange.com/questions/18632/good-book-on-the-history-of-quantum-mechanics on Physics Stack Exchange
www.youtube.com/watch?v=5hVmeOCJjOU A Brief History of Quantum Mechanics by Sean Carroll (2020) Given at the Royal Institution.

Mathematical formulation of quantum field theory  Updated 2024-12-15  +Created 1970-01-01

TODO holy crap, even this is hard to understand/find a clear definition of.

The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.

But what the heck is the mathematical model for a quantum field theory? TODO someone was saying it is equivalent to an infinite set of PDEs somehow. Investigate. Related:

The path integral formulation might actually be the most understandable formulation, as shown at Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

The formulation of QFT also appears to be a form of infinite-dimentional calculus.

Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:

The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.

Maxwell's equations  Updated 2024-12-15  +Created 1970-01-01

Unified all previous electro-magnetism theories into one equation.

Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.

The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.

The equations are a system of partial differential equation.

The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:

$E_{x} (t, x, y, z)$ , $E_{y} (t, x, y, z)$ , $E_{z} (t, x, y, z)$ : directions of the electric field $E : R^{4} \to R^{3}$
$B_{x} (t, x, y, z)$ , $B_{y} (t, x, y, z)$ , $B_{z} (t, x, y, z)$ : directions of the magnetic field $B : R^{4} \to R^{3}$

and two known input functions:

$ρ : R^{3} t o R$ : density of charges in space
$J : R^{3} \to R^{3}$ : current vector in space. This represents the strength of moving charges in space.

Due to the conservation of charge however, those input functions have the following restriction:

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0

Charge conservation

Also consider the following cases:

if a spherical charge is moving, then this of course means that $ρ$ is changing with time, and at the same time that a current exists
in an ideal infinite cylindrical wire however, we can have constant $ρ$ in the wire, but there can still be a current because those charges are moving
Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.

The goal of finding

E

and

B

is that those fields allow us to determine the force that gets applied to a charge via the Equation "Lorentz force", and then to find the force we just need to integrate over the entire body.

Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations:

d i v E = \frac{ρ}{ε _{0}}

(2)

Equation 2.

Gauss' law

d i v B = 0

(3)

Equation 3.

Gauss's law for magnetism

\nabla \times E = - \frac{\partial B}{\partial t}

(4)

Equation 4.

Faraday's law

\nabla \times B = μ_{0} (J + ε_{0} \frac{\partial E}{\partial t})

(5)

Equation 5.

Ampere's circuital law

You should also review the intuitive interpretation of divergence and curl.

Numerical analysis  Updated 2024-12-15  +Created 1970-01-01

Techniques to get numerical approximations to numeric mathematical problems.

The entire field comes down to estimating the true values with a known error bound, and creating algorithms that make those error bounds asymptotically smaller.

Not the most beautiful field of pure mathematics, but fundamentally useful since we can't solve almost any useful equation without computers!

The solution visualizations can also provide valuable intuition however.

Important numerical analysis problems include solving:

partial differential equations

Schrödinger equation  Updated 2024-12-15  +Created 1970-01-01

The partial differential equation of non-relativistic quantum mechanics.

Experiments explained:

via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

fine structure
spontaneous emission coefficients

To get some intuition on the equation on the consequences of the equation, have a look at:

The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.

Then, with that in mind, the general form of the Schrödinger equation is:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

Schrodinger equation

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$ , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The

x

argument of

ψ

could be anything, e.g.:

we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$ , we just add more arguments to it.
we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

Note however that there is always a single magical

t

time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.

The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.

Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation

Schrödinger equation for a one dimensional particle  Updated 2024-12-15  +Created 1970-01-01

We select for the general Equation "Schrodinger equation":

$x = x$ , the linear cartesian coordinate in the x direction
$\hat{H} = - \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)$ , which analogous to the sum of kinetic and potential energy in classical mechanics

giving the full explicit partial differential equation:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = [- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)] ψ (x, t)

Schrödinger equation for a one dimensional particle

The corresponding time-independent Schrödinger equation for this equation is:

[- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x)] ψ (x) = E ψ (x)

(2)

Equation 2.

time-independent Schrödinger equation for a one dimensional particle

Separation of variables  Updated 2024-12-15  +Created 1970-01-01

Technique to solve partial differential equations

Naturally leads to the Fourier series, see: solving partial differential equations with the Fourier series, and to other analogous expansions:

One notable application is the solution of the Schrödinger equation via the time-independent Schrödinger equation.

Bibliography:

math.libretexts.org/Bookshelves/Differential_Equations/Book%3A_Differential_Equations_for_Engineers_(Lebl)/4%3A_Fourier_series_and_PDEs/4.06%3A_PDEs_separation_of_variables_and_the_heat_equation on LibreTexts for the heat equation

System of partial differential equations  Updated 2024-12-15  +Created 1970-01-01

In many important applications, what you have to solve is not just a single partial differential equation, but multiple partial differential equations coupled to each other. This is the case for many key PDEs including: