Application of the Schrödinger Equation in Modeling the Hydrogen Atom

[Sandra Jane]

TL;DR

Hi! Can someone help me understand this topic?

So, I've just studied Schrödinger Equation and it's application in the hydrogen atom. At first, I understood it well but then, I kinda lost it and it ended up as "Quantum Numbers". Now I remember having them at high school, but I don't understand how the equation ended up as the quantum numbers such as n, l, and m. And I remember that there's something special about the "s" of the quantum numbers based on the equation. Please elaborate in words that may be understandable for a new engineering studs like me haha.

(I really need this for my upcoming final exam next week T_T)

 

[PeroK]

TL;DR: Hi! Can someone help me understand this topic?

So, I've just studied Schrödinger Equation and it's application in the hydrogen atom. At first, I understood it well but then, I kinda lost it and it ended up as "Quantum Numbers". Now I remember having them at high school, but I don't understand how the equation ended up as the quantum numbers such as n, l, and m. And I remember that there's something special about the "s" of the quantum numbers based on the equation. Please elaborate in words that may be understandable for a new engineering studs like me haha.

(I really need this for my upcoming final exam next week T_T)

How have you gone from not knowing what a differential equation is to Quantum Mechanics and the hydrogen atom in two or three days?!

 

[gentzen]

So, I've just studied Schrödinger Equation and it's application in the hydrogen atom. At first, I understood it well but then, I kinda lost it and it ended up as "Quantum Numbers".

Try to become familiar with differential equations first. Then try to become familiar with Eigenvalues and Eigenvectors of symmetric matrices, i.e. diagonalization. Next extent from symmetric matrices to Hermitian matrices, and from Hermitian matrices to self-adjoint operators. That last step is actually subtle and confusing, so maybe try to find some way for yourself to skip that step, or at least skip the subtle details.

Next you should probably focus on the Schrödinger equation for the harmonic oscillator, and look at visualizations of the lowest eigenmodes.
Next comes the topic of simultaneous diagonalization of commuting matrices/operators. That will explain in the end why you end-up with 4-quantum numbers for the hydrogen atom, instead of just one Eigenvalue like for the harmonic oscillator.

 

[Sandra Jane]

Try to become familiar with differential equations first. Then try to become familiar with Eigenvalues and Eigenvectors of symmetric matrices, i.e. diagonalization. Next extent from symmetric matrices to Hermitian matrices, and from Hermitian matrices to self-adjoint operators. That last step is actually subtle and confusing, so maybe try to find some way for yourself to skip that step, or at least skip the subtle details.

Next you should probably focus on the Schrödinger equation for the harmonic oscillator, and look at visualizations of the lowest eigenmodes.
Next comes the topic of simultaneous diagonalization of commuting matrices/operators. That will explain in the end why you end-up with 4-quantum numbers for the hydrogen atom, instead of just one Eigenvalue like for the harmonic oscillator.

Thanks so much! I'll try this guide.

 

[Sandra Jane]

How have you gone from not knowing what a differential equation is to Quantum Mechanics and the hydrogen atom in two or three days?!

so, I've been studying differential equation and modern physics at the same time for a whole semester as my major courses. Sure, I understand some of the modern physics but not differential equation (I mean not at all). Until I realize that I need to understand differential equation first to solve modern physics problems. T_T and my final exam is next week T_T

 

[Matterwave]

The "solution" for the hydrogen atom using Schrodinger's equation leads to quite a wide ranging discussion if we allow it to. It would probably help if you specified a bit clearer where you're coming from or what you are actually confused about.

Are you confused about the solution in spherical coordinates? What are Legendre polynomials, Laguerre polynomials, spherical harmonics or the like?

Are you confused about how to set up the problem algebraically? What's the maximal commuting set of observables?

Are you confused on terms we add to the hamiltonian as we get more and more in depth on relativistic correction? (Fine structure, Hyperfine structure)

Or you just don't know what n,l,m,s stand for? (These are quantum numbers that specify a particular eigenstate of the maximal commuting set of observables mentioned earlier. An actual hydrogen atom need not be in an eigen state.)

Give a bit of where you're at and I'm sure people can help. :)