Algebraic Solution to the Harmonic Oscillator

No preamble to introduce this, I’ll just pick up right where the previous post left off!!!!

Now I’m just going to solve this brute-force.

Solve what?

This differential equation !!!

But, they say it’s easier if you make the equation somewhat simpler

via a simple substitution, so I’ll follow along.

Using this, I’ll make it a bit simpler.

The time-independent Schrödinger equation ends up like this~~~~

And a thought that comes from looking at Large K….

Well, just that kind of thought came to me….haha

I’m not sure if this idea will be needed later….haha

OK then, now I’m going in with the assumption.

[CAUTION: assumption incoming]

Assumption: I’m going to look at the case where ξ is freakin’ huge.

Then the equation becomes this hahaha. Now solving this differential equation is easy-peasy super easy right?!

Its general solution is ~~~

It’s going to have this kind of shape.

Let’s check.

Oh ho~~~

But the fact that the wavefunction is zero at plus infinity and minus infinity in x is just a common-sense condition,

a common-sense constraint, so we can set B=0.

When xi is large, this is how it goes

This is the shape the general solution took.

(The shape of the B-containing term, even in the case where we throw away the assumption,

absolutely cannot exist, even in the case where xi is small!!!!!! It cannot exist!!!!!! It absolutely cannot exist)

Ah~~~then we think like this.

Now [CAUTION: imagination incoming]

I don’t know for sure, but the original equation

the general solution of this differential equation ~~~~

will be this combined with some other thing ~~~~ and that’s how the general solution’s shape got put together!!!

Ah~~~~~~ let’s think of it this way

The general solution of the original equation

will be that exponential term multiplied by some arbitrary function of xi!!!!!

Yup!~!@!!

Let’s wrap up the process of finding the general solution of the differential equation with this for now!!!

Alright!! Let’s go with this!!!! Then I’ll plug this general solution we drew up from imagination into the original equation!!!

Damn it -.- so what – how am I supposed to solve this!!!

From here, how are we going to solve it? “We’re going to Power series expand the function A(ξ).”

(Power series — yeah, that’s the one lol hehe)

When you learn power series in mathematical physics, “is there really something that can’t be power-series expanded~~~~? or is that power series unique~~~~~”

answers to these questions can be obtained from mathematical physics.

So trust it and let’s go!

I’m going to do a power series expansion like this. Of A(ξ)!

If you expand it this way

do this, and plug it into the equation above.

Let me write it out by same-degree terms, starting from the smallest.

All the way through~~~~~ all~~~ added together has to be 0, and

the case where xi is 0 is what, a Trivial solution? Since it’s a meaningless solution,

the coefficients all have to be 0~~~ that’s what it means, I guess

The coefficients have to be 0~~~ I’ll write it out.

Oh my!!!! A recurrence relation is born!!!!

A recurrence relation — you know how you lay out matches in a row drrrr~~ and light one end and drrrruk whoosh~ it all catches on fire, that’s why it’s called ignition right?

That means if you just know one thing! then drrrr~~~ruk! everything gets determined, can I put it that way?!

Oh! nice!

Ah damn but the recurrence jumps two at a time…T_T

So let’s split A(ξ) into two and think about it that way!!!

The ones with odd-degree xi have odd subscripts on the a coefficients inside,

and the ones with even-degree xi have even subscripts on the a coefficients inside!

When we split it this way, that it breaks into odd and even functions as the two big pieces is a bonus?!!

OK then, now let’s look at the wavefunction for juuust~~~ a moment again.

We had been predicting the wavefunction like this

Why did I say to look at this for a moment, it’s because I want to worry about normalization!!!!

For the square to have the meaning of a probability density, normalization has to be possible…..

What I want to say is, when we did the power series expansion,

A(ξ) is ‘absolutely not’ a never-ending infinite-degree series.

If A(ξ)’s terms kept going forever, normalization wouldn’t work!!!

That is, that recurrence relation, I mean

whether it ends up like this,

or like this, either way

it has to be something that determines A(ξ) to have an end.

(Not converging to 0 either, just plain 0 from somewhere onward!!!)

That is

in this recurrence relation, if from some j=n onward it’s 0

since the recurrence has this shape

it is possible!!!

From some j=n onward, forr~~~~ever 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …..

So that this function works, let’s install a device into the recurrence over there.

That device?

If you put in this kind of K, the device gets installed!!

That is,

If we modify the recurrence this way,

is this satisfied??????

No no no no no

The recurrence jumps two at a time,,,, so not yet..

Then let’s take a look at how this gets resolved, starting from n=0.

What if the max n of j is n=0?

If n=0 is the max number of j, then there’s no way around it, this

is what it’s saying, so,

is the only possibility, since after that every 2-jump is 0,

What if the max n of j is n=1?

By the same logic as above, if the max n of j is 1

it becomes like this so

the two survive?!?!?!

That’s No No

It becomes like this!!!!

If you plug in j=1, as the recurrence’s device dictates, 3, 5, 7, ~~ all become 0, that’s true!!

But if you plug in j=0

What this means is

if it’s not this, the even terms all survive on and on~~~~~~ so

under the condition that the max n of j is n=1

you have to fit in this!!

That is,

One more time?

What if the max n of j is n=2?

Here too,

Since here the non-zero ones are only

we get:

So depending on what the maximum value of j is, the wavefunction right now

gets determined this way, and

what I’m saying is pay attention to the red.

Looking at this, “Oh ho?!!!!! Looks like a Hermite polynomial?”

looking at it this way, actually the smart people have all tried this, (the process isn’t hard)

can’t we write this generally with respect to n ‘using Hermite polynomials’?!?!!

Using Hermite polynomials, it can be expressed!!!!!

(I’ll omit the process.)

And after that, I think there was some simple-check content about whether it follows the ‘correspondence principle’~ or not.

It’s simple simple content, so this posting I’ll wrap up here.

I’m going to change majors. I can’t do physics anymore, bye bye