Formalism: Hilbert Space

So we’ve been strolling down this road called quantum mechanics for a while now.

Doo-doo da-da~~ look at me, just vibing down the road of quantum mechanics~!!!~~

“AH — you scared the hell out of me!!! Who are you?!?!?!”

(http://www.sejonmall.com/FrontStore/iGoodsView.phtml?iGoodsId=edg00092)

“Ahem~~ you there, you are walking the road of quantum mechanics, are you not~~~ Tell me — what is THE equation of classical mechanics~~~”

“Uh… F = ma?”

“Ohhh hoo~~ correct. And in quantum mechanics, it’s the Schrödinger Equation~~~~~ Remember it well~~~”

(poof — gone)

“Huh?? What?? Hey, where are you going! HEY!! HEY~!~!!!!”

Yeah. Stuff like this keeps happening to me.

OK so — from here on, this chapter is basically the one where we justify, mathematically, that all the stuff we’ve been saying actually makes sense.

How the field called quantum mechanics ended up where it is today, through what mathematical machinery —

and we’re only going to peek at the super super super easy baby-tier version of it.

If you want the story of how quantum mechanics was born, honestly, go read a book called Quantum Story — it’s great.

What I thought while reading it lol lol lol lol — the parts I’d already studied, oh man, those read like a thriller. So fun.

But the parts I hadn’t hit yet (the Zeeman Effect, for one) I understood absolutely zero of…..sigh.

Anyway — read it. I’m recommending it.

In the early days of quantum physics (post-Planck, so roughly after 1900), the field hit this stretch where quantum mechanics was being born and exploding out in every direction.

But Heisenberg, at some point, hit a wall. Like a massive wall. He had NO~~~~~ idea why!!!!! the math wouldn’t close, everything jammed up!!! argh!! totally stuck.

There was some experimental result at the time (there was something, OK, don’t ask me the specifics) and no matter how hard he thought about it — nothing. Nothing. Absolutely no idea.

He was getting super super deep in the frustration hole (Bohr and Heisenberg were a teacher-student pair, by the way) and eventually he was so fried he went to Bohr and was like,

“Hey professor, I’m cooked. I’m gonna go to the beach for a bit, clear my head, cool?”

Bohr was like yeah yeah go ahead, and Heisenberg went off to the seaside to chill. But you think he was actually going to stop thinking about it?! No chance.

Dude was seriously stuck.

And the story goes — at the beach, this thought hit him:

The reason errors kept piling up at the microscopic level is —

the physical system had completely changed, but physicists were still using the same old classical variables — $x$, $p$, $T$, $V$, all that — like nothing had changed.

Same old variables for a totally new world. No wonder it was all wrong~~

That’s apparently what clicked.

So right there, Heisenberg tried swapping out the variables entirely.

And the variable he picked as his shiny new modern-physics variable? None other than spectral intensity.

He took that as his variable, cranked through some formula step by step, and — boom. The numbers he wanted popped out, perfectly.

(Absolutely no idea what kind of “vacation” he was on lol)

He was so hyped he fired off a KakaoTalk message to Bohr right then and there!!!

(OK fine, KakaoTalk didn’t exist — a letter. He sent a letter.)

But Bohr got the letter and had no clue what Heisenberg was even talking about.

Then about 7 months later, Bohr is lying in bed trying to fall asleep —

and BAM!!!!!! a flash!! — that formula Heisenberg had sent him way back when,

that formula — “bro!!!! that was a MATRIX!!!!!!!!!!”

That’s apparently what came to him.

And that’s, you know, more or less how Heisenberg discovered what we now call matrix mechanics???

And from there linear algebra became like, ridiculously central and powerful in quantum mechanics?!?!?! (I mean, I don’t really know the full history~)

Oh — they say Heisenberg wrote this to his daughter in a letter around that time:

“Dear daughter… it seems your father has made a discovery as great as Newton’s……”

Damn. Damn. That is so freaking cool. Like, for real lol lol

OK OK why am I rambling about all this~~~~

The first thing we actually need to look at is Hilbert space.

So — the state functions $\psi$ we’ve been dealing with — these guys live in Hilbert space~~

Because $\psi$ has to be normalized for it to mean anything physical — we’re describing the world with it, after all —

that is,

$$1\quad =\quad \int _{a}^{b}{\left| \psi \right|^{2}dx}$$

we only deal with $\psi$ that satisfies this.

The set of functions that satisfy the above — the math people among us call it

$$L_{2}(a,b)$$

and that set is apparently what they call Hilbert space!!!

So: the wave function (state function) is a function (vector) that lives in Hilbert space!!!

Next up — brackets. There are a LOT of kinds of brackets out there.

We’ve got (), {}, [], <>, and so on.

We’re zooming in on the <> ones!!! Because they have a real mathematical definition!!!

Quick aside: you know how () is called “parentheses”, [] “square brackets”, {} “curly braces”?

And the <> guy — what do you even call it? “Angle brackets”? Yeah, angle brackets.

In physics these <> guys get called a “braket” — as in, bra + ket. Cute, right? You’ll see why in a sec.

OK so far so good~~~~??

Alright — math time. Linear algebra, specifically!!!

So, a vector in linear algebra can also be a function. (As long as you’ve got a proper basis set up.)

And the thing that takes a vector from one set and maps it to a vector in another set — that’s basically what we were taught to call a “function”, right? Probably?

And we said our state function is a “vector” too — and its basis would be, you know, the basis that spans Hilbert space~?????

So now, the vector that represents that wave function!!! That’s also a vector, obviously!!!

And to write this vector down, a physicist named Dirac went —

bra-vector

ket-vector

— and proposed a notation.

“Hey guys~ writing it like this is way more convenient~” (Dirac notation)

In “braket”, bra is the front half — so among <, >, it refers to the <.

A bra-vector (for vector α) is defined as:

$$<\alpha |\quad =\quad (a_{1}^{*},a_{2}^{*},a_{3}^{*},\quad ...\quad ,\quad a_{n}^{*})$$

And a ket-vector is defined as

$$|\alpha >\quad =\quad \left( \begin{matrix}{a_{1}}\\{a_{2}}\\{a_{3}}\\{...}\\{a_{n}}\end{matrix} \right)$$

Yeah. That’s what he proposed.

Why write it like this, though?

Hmm,,,, this is all stuff from high school honestly,,,,

The inner product of a 3D vector is

$$\overrightarrow {A}\cdot \overrightarrow {B}\quad =\quad A_{1}B_{1}\quad +\quad A_{2}B_{2}\quad +\quad A_{3}B_{3}$$

obviously, and

cranking up the dimension, the inner product of an n-dimensional vector is

$$\quad A_{1}B_{1}\quad +\quad A_{2}B_{2}\quad +\quad A_{3}B_{3}\quad +\quad \cdot \cdot \cdot \cdot \quad +\quad A_{n}B_{n}$$

BUT!!! If your field is the complex numbers, the actual definition of the inner product is

$$\quad A_{1}^{*}B_{1}\quad +\quad A_{2}^{*}B_{2}\quad +\quad A_{3}^{*}B_{3}\quad +\quad \cdot \cdot \cdot \cdot \quad +\quad A_{n}^{*}B_{n}$$

(For real it should be an integral, but let me write it like this for now so the story flows.)

Any element (vector) that lives in Hilbert space can be seen as a linear combination of Hilbert space’s basis vectors,

so any arbitrary wave function

$\psi_{\alpha}$

can be written as a linear combination of the basis vectors of Hilbert space —

$\psi_{1},\psi_{2},\psi_{3},\psi_{4},....,\psi_{n}$

— these guys.

A “linear combination” is just a sum with some coefficients out front, right?!

$$\psi_{\alpha}\quad =\quad a_{1}\psi_{1}+a_{2}\psi_{2}+a_{3}\psi_{3}+\cdot \cdot \cdot \cdot +a_{n}\psi_{n}$$

And let me write down one more arbitrary wave function.

$$\psi_{\beta}\quad =\quad b_{1}\psi_{1}+b_{2}\psi_{2}+b\psi_{3}+\cdot \cdot \cdot \cdot +b_{n}\psi_{n}$$

The inner product of the two vectors is

$$\left( a_{1}^{*}\psi_{1}^{*}+a_{2}^{*}\psi_{2}^{*}+a_{3}^{*}\psi_{3}^{*}+\cdot \cdot \cdot \cdot +a_{n}^{*}\psi_{n}^{*} \right) \left( b_{1}\psi_{1}+b_{2}\psi_{2}+b\psi_{3}+\cdot \cdot \cdot \cdot +b_{n}\psi_{n} \right) \\ =\quad a_{1}^{*}b_{1}+\quad a_{2}^{*}b_{2}+a_{3}^{*}b_{3}+\cdot \cdot \cdot \cdot +a_{n}^{*}b_{n}$$

In matrix form:

$$\left( \begin{matrix}{a_{1}^{*}}&{a_{2}^{*}}&{a_{3}^{*}\cdot \cdot \cdot \cdot}&{a_{n}^{*}}\end{matrix} \right) \left( \begin{matrix}{b_{1}}\\{b_{2}}\\{b_{3}}\\{.}\\{\cdot}\\{b_{n}}\end{matrix} \right)$$

And in Dirac notation:

$$<\alpha |\beta >$$

From now on I’ll just be using Dirac notation,,,,

The thing I’ve been trying to get across this whole time is: yes, the entries of the bra-vector really are the complex conjugates!!

And what property does this definition give us? Well —

$$\left< \psi_{a}|\psi_{b} \right> \quad =\quad \left< \psi_{b}|\psi_{a} \right>^{*}$$

Even a property like this!!!

Stuff that now gets layered on top of Hilbert space…..

First — the basis vectors of Hilbert space have magnitude 1!!!!!!

Because these guys have to be normalized, first and foremost!!!!!!

$$\left< \psi_{1}|\psi_{1} \right> \quad =\quad \int _{-\infty}^{\infty}{\psi_{1}^{*}}\psi_{1}dx\quad =1$$

And as we said, the basis vectors of Hilbert space are orthogonal!!!!!

$$\left< \psi_{n}|\psi_{m} \right> \quad =\quad 0\quad (n\neq m)$$

(They’re basis vectors — kinda obvious they’d be perpendicular!!)

Magnitude 1 and orthogonal — mash those together and you get orthonormal!!!@!!@!@!@

(I just wanted an excuse to write the word “orthonormal”…..heh)

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Originally written in Korean on my Naver blog (2015-08). Translated to English for gdpark.blog.