Ladder Operators

If we summarize what we did before,

given mass m, and the potential’s

characteristic alone, the state function of the particle in there~~

at least the ground state, we can figure out!! That was it!

and

the energy at that time is

what was it that let us compute this far,

it was precisely by devising this thing called the ladder operator that we could get here.

And now, what other assumption did we have,

we assumed that the wave function would be independent of time, did variation on the wave function (separation of variables), and via the time-independent Schrödinger equation we got this far!

Ah, so now that we know the ground state,

using the raising operator among our ladder operators,

starting from the ground state and raising one by one by one,

all of them~~~

we can find in detail!

However!!! each time, we do need to normalize, right!?

So

raising one by one like this, step by step,

in general we can write it like this, got it!

The energy at each state — we already figured that out.

For the ground state, the energy first of all is

this, and

each time we apply it once, the energy goes up by

and

each time we apply it once, the energy goes down by

since we already did that before,

we can write it like this, something like that.

So now ultimately

each time the raising operator is applied, how do the constants

behave

and whether this can also be described in terms of n — that’s what we’re going to look into!!

&

here, if we find c_n and d_n, that’s what we need to do!!!!

(c_n and d_n are constants that have to be found through normalization at each state,

and whether that can be expressed in terms of the principal quantum number n or not! That’s the job of figuring out!!!!)

First, so we can see this,

Ahhh!!!!!

So the raising operator and the lowering operator are Hermitian conjugates!!!!

(Honestly, just by looking at the operators themselves, it’s easy to spot that they’re complex conjugates!!!!

Hermitian conjugate is a concept that hits home — BAM!!!! — when you look at it as a matrix,,,,

but for now let’s just say complex conjugate and move on. Assuming we’re doing quantum physics without linear algebra!!!)

The reason I showed you that property once more

is because we’re going to use that property to find c_n.

Let’s give it a go~

Wait!! Just keep this one thing in mind before we go

we’re going to do this, and since it’s easy, I’ll wrap it all up and write it in one go.

Then back to it, let’s gogogogo

By the same principle

for this guy too, gogogogo!!!!

Conclusion

the A_n in here can also be expressed in terms of n!!

by combining these two result equations, we can get it!!!

That conclusion!!!

When the potential is the harmonic oscillator potential, we solved the time-independent Schrödinger equation

with. New Operator (Mr. Ladder)

So then next time

the second method for solving the time-independent Schrödinger equation when the potential is the harmonic oscillator potential,

I’ll come see you with the method of plowing through it brute-force like an armored tank!!!

bb