How the Schrodinger Equation Predicts Real Life (and Why It's So Difficult) - Quantum Mech Parth G

Understanding the Schrodinger Equation theoretically is very useful... but the main aim of the equation is to predict what happens in real life!

In this video, we'll be looking at how the Schrodinger Equation can be used to predict the behaviour of a hydrogen atom. To do this, we'll first look at what the Hamiltonian is for a hydrogen atom, as well as how this is calculated. Then, we can substitute this into the Schrodinger Equation, and finally solve this to tell us the "allowed" wave functions for the system.

The Hamiltonian in the Schrodinger Equation is closely related to all the energy contributions in the system we happen to be studying. The easiest way to begin finding the Hamiltonian for the hydrogen atom is to think about the kinetic and potential energies in the system.

The kinetic energy of the hydrogen atom can be calculated using the reduced mass of the entire system (which is equivalent to finding the kinetic energy of the center of mass of the atom). This just makes life easier, rather than having to find the kinetic energy of the proton and electron separately.
https://en.wikipedia.org/wiki/Kinetic...

The potential energy of the hydrogen atom can be found by considering the electrostatic attraction between the positively charged proton, and negatively charged electron. We can directly import the classical expression for the potential energy for this system, which is equal to the product of the two charges divided by (4pi x epsilon 0 x the distance between the two particles). Epsilon 0 is a universal constant known as the "permittivity of free space". https://en.wikipedia.org/wiki/Electri...

Adding together the kinetic energy and potential energy for the system gives us a good first approximation for the Hamiltonian. We can then plug it into the Schrodinger equation and solve it to give us the allowed wave functions for the system. We find that the electron can be found in one of many discrete energy levels around the proton. The Schrodinger equation also accurately predicts the energies of these energy levels compared to what we measure experimentally.

But when we make more precise measurements, we find that these energy levels are further split up into very close energy levels. This is known as the fine structure of the atom. To predict this fine structure theoretically, we have to modify the Hamiltonian to account for other (smaller) energy contributions. There are three terms we need to add to fairly accurately predict the fine structure. These are the relativistic correction (since the electron can move pretty fast), a term relating to the spin of the electron (spin-orbit coupling), and an entirely quantum mechanical term known as the Darwin term. This deserves its own video.

Now finding the fine structure Hamiltonian is one thing, it is already an approximation as the relativistic correction is technically an infinite series of smaller and smaller terms. But solving the Schrodinger Equation with this new Hamiltonian is even harder. And this is just for a hydrogen atom. It only gets trickier when we consider larger atoms such as Helium, Lithium, Beryllium, and so on.

(Note: The methods used in this video are not always consistent with each other, they are mainly for illustrative purposes. For example, the reduced mass of the atom is sometimes used, but in other cases the proton is assumed to be stationary. Depends on the circumstance and the level of detail needed).

Timestamps:
0:00 - The Schrodinger Equation and the Hydrogen Atom
0:57 - The Hamiltonian as the Total Energy (Kinetic + Potential)
4:10 - Substituting the Hamiltonian into the Schrodinger Equation, and Solving
5:21 - The Fine Structure of the Hydrogen Atom
6:05 - The Three Extra Hamiltonian Terms
7:56 - Why Solving the Equation is Hard for Larger Atoms

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