Ciro Santilli @cirosantilli 37

In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.

Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.

This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table

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:

Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in $L^2$, where it always exists and converges pointwise to the function: Carleson's theorem.

Continuous version of the Fourier series.

Can be used to represent functions that are not periodic: math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.

Of course, every function defined on a finite line segment (i.e. a compact space).

Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.

As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.

Show up in the solution of the quantum harmonic oscillator after separation of variables leading into the time-independent Schrödinger equation, much like solving partial differential equations with the Fourier series.

I.e.: they are both:

Staring from a triangle wave, this explains why we always get the same musical notes:See also: solving partial differential equations with the Fourier series.

TODO: do higher overtones decay faster in time than the base ones?

Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.

This equation is a subcase of Equation "Schrödinger equation for a one dimensional particle" with $V(x)=x^2$.

We get the time-independent Schrödinger equation by substituting this $V$ into Equation "time-independent Schrödinger equation for a one dimensional particle":

Now, there are two ways to go about this.

The first is the stupid "here's a guess" + "hey this family of solutions forms a complete basis"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.

The second is the much celebrated ladder operator method.

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:

Describes perfect lossless waves on the surface of a string, or on a water surface.

Uniqueness: math.stackexchange.com/questions/1113622/uniqueness-of-solutions-to-the-wave-equation

As mentioned at: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from solving partial differential equations with the Fourier series citing courses.maths.ox.ac.uk/node/view_material/1720, analogously to the heat equation, the wave linear equation can be be solved nicely with separation of variables.