The Schrodinger Equation is (Almost) Impossible to Solve.

Sure, the equation is easily solvable for perfect / idealized systems, but almost impossible for any real systems.

The Schrodinger equation is the governing equation of quantum mechanics, and determines the relationship between a system, its surroundings, and a system's wave function.

The wave function contains all the information we can know about a system, such as the probabilities of finding a particle within the system in different regions of space. Specifically, the square modulus of the wave function is related to any measurement probability.

The Schrodinger equation simplifies down to "kinetic energy + potential energy = total energy", but using the language and quantities defined by the theory of quantum mechanics.

We can set up the Schrodinger equation for any system that we are studying, simply by adding together all the kinetic energies and potential energies within the system. In this video, we see how to do that for a hydrogen atom and a helium atom. Then we "solve" the Schrodinger equation by finding the allowed wave functions.

With a hydrogen atom, we merely need to account for the potential energy that comes about due to the electron-proton interaction. Since they are both charged particles, they exert electrostatic forces on each other, and hence there is a potential energy between them.

In a helium atom (2 protons, 2 neutrons, 2 electrons), things become a bit more complicated. To make things simple, we make 3 assumptions: (1) the nucleus is stationary, since it's much more massive than the electrons, (2) the nucleus behaves as one single object in order to avoid accounting for the interactions between the particles making up the nucleus, and (3) the atom is isolated and does not interact with anything outside it.

These simplifications allow us to much more easily build the Schrodinger equation for our helium atom. We only need to account for the kinetic energies of the electrons since we assume the nucleus is stationary. We also only need to account for 3 sources of potential energy. Two of these are the interactions between the nucleus and the two electrons, and the third is the electron-electron interaction. We would have account for many more terms if we did not use the simplifications outlined above.

At this point, we have a differential equation that we can solve in order to find the allowed wave functions. But this equation is extremely difficult to solve analytically, and we don't have many techniques to do it. We instead have to resort to further simplifications or the use of computers to find "brute force" solutions. This last method is basically trial and error but with fairly educated guesses.

Thanks for watching, please do check out my socials here:
Instagram - @parthvlogs
Music Chanel - Parth G's Shenanigans

Here are some affiliate links for things I use! I make a small commission if you make a purchase through these links.

Cards linked in this video:

Timestamps:
0:00 - What Does the Schrodinger Equation Mean, and How Do We Solve It?
3:29 - Building the Schrodinger Equation for the Hydrogen Atom
5:05 - A Simplified Model of the Helium Atom
7:06 - Building the Schrodinger Equation for a Simplified Helium Atom
10:07 - Solving the Schrodinger Equation?

[theultimatereductionist]

I am a PhD Mathematician with a specialty in differential algebra: the study of how to solve all sorts of nonlinear FPDEs = functional PDEs. Recently, I tried to solve the S-equation for a Lennard-Jones potential. I then realized that the reason I cannot solve is that it is impossible to solve over an INFINITE spatial domain: -infinity < x < infinity. Psi would have to decay fast enough at both ends in order for the integral of Psi*Psi-bar over all real x to be FINITE let alone 1.

[mcwulf]

I remember our Quantum Physics lecturer solving this for Hydrogen. It took the whole hour.

[timbeaton]

Interesting. Even the simple model of the Helium atom you describe is essentially a sort of 3 body problem... 2 electrons, and one massive nucleus. And we know that that is difficult enough to solve even classically, let alone with the quantum complications.

[AmokBR]

Nice video! I did my PhD on solving this equation for multi-electron systems. You can’t solve them analytically, but fortunately they’re pretty easy (that’s very relative) to solve numerically.
There are some much scarier beasts out there, like non-linear equations or the Dirac equation, for which it’s even difficult to build an exact Hamiltonian if tou have more than one electron.

[_John_P]

For a moment I thought it was another channel on Physics.

[StatsScott]

This was a thoroughly informative and accessible walkthrough. Thanks!

[TheWyrdSmythe]

Cool video! I’ve heard for years about how we have a good solution for hydrogen but helium, let alone higher, was a whole other matter. Nice to have it confirmed and explained!

[zhelyo_physics]

Very much enjoyed this, thanks. The richness of the solutions of the Schrodinger equations and its difficulty in solving reminds of the Navier Stokes equations with only a handful of analytical solutions yet such great complex behaviour.

[coniferviveur]

You have a real knack for explaining difficult subject matter.

[darshanpatel]

Omg BRO
You are look like a PROFESSOR.😮👍🏻👍🏻

[_..---]

Man this channel is golden, everything is explained so clearly. It's so interesting that people have even managed to discover how to calculate things of this nature.

[Lukav]

Your videos have some of the best explaination about physics. They really are gold.

[eeshtarr]

I'd expect the |phi|^2 to be always smaller than a |phi| (if it were completely real for all values of x, like you show). It being normalized to 1, all values of it must be < 1, and hence the square of it would be smaller than itself, instead of larger.

[soumyajitroy]

new look is 🔥
also great vid as always

[ffhashimi]

So, how then it's useful if we almost can't solve it? Greate video I hope you continue simplifying the math of quantum physics " as physicists doing it" big thanks.

[benmcreynolds]

I was thinking about how quarks are attracted to each other but also repel in a way so they hold a formation in some manner. (It reminds me of a L2 Lagrange point of a orbiting set of multiple body's in space. )

[SocratesAlexander]

If it is not possible to solve the Schrodinger Eqn. of complex atoms, how do we construct the modern model of the atom? (I mean, those s and p's and d orbitals and their quantum numbers etc.)

[thomaskoppen]

Thanks for the video, well made and very interesting!

[top-secret]

This channel kicks ass! Instead of making my brain cells die like 85% of everything in my feed, this makes me smarter!

[ayushdhingra]

Schrodinger be like
I know it tough but it works