The Hydrogen Atom

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In the previous post I kept going on and on about hydrogen hydrogen,

but the potential V was just written as ‘V’ and I moved on, but now this is really hydrogen.!~

So~ the feature of the hydrogen atom is!!!

It’s symmetric with respect to θ and Φ!!!

So that means “we just need to solve the equation for r!!!(Radial Equation)”

Aaaaah so what’s the goal of this post,

it’s plugging the electric potential of hydrogen into V in the equation for r !!!!

and finding the function of r!!!

Let’s find R(r) and also do the normalization cleanly!!!! That’s it!

Alright, let’s get into it.

The equation we got from the r-equation by substituting rR = u!! Writing it out again,

Now let’s plug the Potential of hydrogen into V here hehe

because of this, the Potential V that can be defined is

defined like this,

and that V in hydrogen is

(hydrogen has 1 proton and 1 electron, so the charge is e squared)

(E<0 means the electron is in a bound state!!!!- for details see general mechanics, Kepler’s laws section,)

Now let’s play around with that equation~~~

And now I’m gonna clean it up

I’ll finish the cleanup like this.

The reason I cleaned it up like this is,

pinky pinky that thing is just some random constant, so first I’ll

change it to this.

And for the variable substitution here, what I’ll do is,

because of that bluey bluey bastard

I’ll do this substitution.

Then we can get rid of the coefficient in front of the second derivative in that thing.!!

Cleaning it up nice and neat~~

It came out like this~~~~~~~~

But.. it’s another differential equation that’s hard to solve…T_T

Just like back when we did the algebraic solution for the harmonic oscillator!!!

Let’s try to find something like the ‘asymptotes’ of the equation’s solutions first….

Let’s look at each solution in these cases!​

So what do we do next!!!!

It is

we have to pray!lolololol

Go pray to whatever god you each happen to believe in

“Oh~ god~ u(ρ) was hard to find, but please please make v(ρ) … be findable…T_T I want to find it"​

Let’s plug the ‘solution to the differential equation’ that we hypothesized back into the diff eq.!!!!

So there’s something we have to find~~

Plug this into the differential equation above!!

We plugged the solution to the diff eq we got by thinking, into the diff eq!

Now all we need to do is find v(ρ) that satisfies that equation above!!! So how do we find it~~~~

This is when, just like with the algebraic solution of the harmonic oscillator, let’s do a power series expansion nice and smooth~~~~ and brute-force solve it!!

<since whatever the shape, it can be expressed as a polynomial!!!!!>

Okay so now we know the direction, let’s go run with it.

It’s expressed like this~~~

thus,

Now that we’ve prepared the ingredients, let’s stir-fry them,

Here too, ρ=0 can’t be a solution lololololololololol (trivial solution)​

Each of those coefficients being 0 must be the Solution!??!?!?!!!!

I’ll just color the coefficients once and then continue with the equation!!!!

The solution to the differential equation is

we got the condition on the coefficients of v(ρ) that makes it become this.

That condition is exactly the recurrence relation up there!!!!!

Let’s plug j=0 into that recurrence relation and look at a few examples, then move on

So the coefficient c-n of the n-th order term of v(ρ) keeps going like that,

and going on with respect to ρ is going on waaay~~~ with respect to r too, same thing,

so!!!!!!!!!!!!!!!!!!!!!c-n must become 0 from somewhere,

“the wave function must have an end.” - (because it has to be normalizable!!!!!!)

So look at the recurrence relation.

The n here is the principal quantum number that we know weeell~!!!!(principal quantum number)!!!!!!!

(cf. n-l, the minimum value n minus l can have is 1!!! Right!!!Since j-max can’t be negative,

so we get the condition n>l!!!! )

Yay it’s done.~~~~

Now we’ve found the whole Radial Equation~

Now let’s change all~~ of this to r.

Since u = rR,

And we should change ρ to r too righttt

ρ = Kr

Oho,,,,,,,,,,,K????????????

What was K again

Stop right here for a moment!!!!!!

The energy in the hydrogen atom with respect to n!!!!!

The energy when n=1?????The -13.6eV that almost everyone knows by common sense

It comes from right here!!!!

Anyway E is now hooked onto the principal quantum number n !!!

We’ve learned that the energy is ‘quantized’!

This isn’t this anymore either

it’s this.

Therefore,

That red guy that pops out at the very end here

The reciprocal of that guy

this guy, with subscript 0 added to a, is called the “Bohr radius.”

<The derivation of this guy is done easily easily in modern physics, so please refer to that.>

(This can be very easi~~~~~~~~ly proven using de Broglie’s matter waves lololol

But when Bohr published this, it was a whole 20 years before de Broglie published anything about matter waves.

Even before the concept of matter waves was introduced, Bohr already….just kind of intuitively knew it….

Bohr really…..seems worthy of being called the pope of modern physics lolololol)

<Just shooting the breeze>

(When quantum mechanics was juuuuust starting to come~~~~ into Korea, it was around the 1960s-70s.

That’s when Korean physics students finally got to learn this kind of science,,,but Bohr had passed away by then…..

Aa~~!!!!!! We want to show our respect to dear Niels Bohr,,,

Korean physics students who had no way to do so…………….apparently quite a few of them named their sons “Bohr” lolololololol

If you happen to know any Kim Bohrs, Park Bohrs, Lee Bohrs around you,,,,try asking them like this…“so whaddaya daddy do for a living, huh?” lolololololol….

But I don’t know any Kim Bohrs around me….hehehe

I heard this fun story in my professor’s class,,,,the professor said there are quite a few Bohrs around him lololol)

Anyway so

it’s this!!

We can finally fix this equation properly now

Just a litt~~le bit more,

The equation R(r) for r is a function that’s exactly determined only when n and l-ell are determined right?!?!?!!!!

So we can stylishly add n and ell as subscripts on R to emphasize that meaning.

The game of completely finding the equation for r is over!!!!

Now we have to talk about Normalization,

soo~~~~ at that time,

the normalization for the equation for r is

we just need to satisfy this,

ummm~~ now in R(r) there’s v(r) whose polynomial coefficients are defined by a recurrence relation!!!!

That function called v

people who do math call it the ‘(associated) Laguerre polynomial.’

We learned about the Laguerre polynomial too

and apparently the integration can be done by hand!

Now at last we can fully express the normalized wave function of the hydrogen atom, they say.

Ugh deriving the wave function once is gonna be tough………………………

When I have time later let’s find another book and try deriving this equation….hehehe

Anyway it’s outrageously complicated, but apparently for us earthlings it’s a really precious equation

There’s no other case besides hydrogen where it can be calculated this exactly T_T T_T T_T T_T phew~

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