The Time-Independent Schrödinger Equation [Quantum Mechanics I Studied #4]

Before we dive into Chapter 2!!!

Quick fun detour — I want to talk about how the Schrödinger equation, the F=ma of quantum mechanics, actually came to be.

OK “the F=ma of quantum mechanics” sounds way too grand. Let’s just call it a fun little story about quantum mechanics?????

We’ll get into the real details when I post the modern physics stuff, but for now —

In 1900, uncle Max Planck drops the idea of energy quantization,

$$E=nh\nu$$

Then uncle Einstein, perfect timing, announces his experimental write-up of the photoelectric effect, and basically goes: light is a bunch of little light particles. So the energy of light — which, remember, is supposed to be an electromagnetic wave — is

$$E = n\text{ pieces} \times h\nu$$

i.e. one little light particle carries energy

$$h\nu$$

(Anyway. Geniuses.) That same year, 1905, this guy also drops special relativity,,, and out of that pops the rest mass

$$m_0 : \text{mass when } v=0$$

and the relation

$$\text{Total } E = \sqrt{p^2c^2 + m_0c^2}$$

A photon has rest mass zero, so for light specifically

$$E = pc$$

But the energy of one photon is

$$h\nu$$

right????

$$pc = h\nu \\ \therefore \quad p = \frac{h\nu}{c} = \frac{h}{\lambda}$$

And boom — Einstein gets on base with a clean single.

So now: Planck on second, Einstein on first.

Einstein wrapped up the particle-side of “waves” in 1905. Next batter steps up — de Broglie, 1924!!!!!!

de Broglie’s job: wrap up the wave-side of “particles”. Simple. Just, ya know, haha

$$\lambda = \frac{h}{p} \quad , \quad \nu = \frac{E}{h}$$

And now — last batter, cleanup hitter, 4th in the order — Schrödinger walks up to the plate in 1926.

Bases are LOADED. Loaded!! Planck on third, Einstein on second, de Broglie on first.

Schrödinger’s thinking: “OK, so what form of wave do electrons in the microscopic world actually take???”

But — as you know from Fourier series — any wave function, any wave function whatsoever, can be written as a linear combo of sines and cosines.

So to keep it simple, let’s just say it’s a plain old sine.

$$\psi(x) = A\sin kx$$

Now this thing satisfies

$$\frac{d^2\psi(x)}{dx^2} = -k^2\psi(x)$$

(differentiate twice, you get yourself back!)

And $k$ is $2\pi / \lambda$. Plug in the de Broglie matter-wave wavelength for $\lambda$:

$$k = \frac{p}{h}2\pi = \frac{p}{\hbar}$$

Express momentum in terms of energy:

$$\frac{p^2}{2m} = K = E-V \\ \therefore p = \sqrt{2m(E-V)}$$$$\therefore k = \frac{1}{\hbar}\sqrt{2m(E-V)}$$

Slide it back in:

$$\frac{d^2\psi(x)}{dx^2} = -k^2\psi(x) = -\frac{2m}{\hbar^2}(E-V)\psi(x)$$$$\psi(x)(E-V) = -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}$$$$E\psi(x) = -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V\psi(x)$$

And there it is. Welp. I’ve basically derived the time-independent Schrödinger equation right there.

Here’s the thing I actually want to say:

Just like that!!!! The Schrödinger equation is not some mystical thing that fell out of the sky.

Historically — a whole bunch of people farmed this thing into existence together?!?!?! That kind of vibe.

Same way F=ma wasn’t derived from this-and-that expression — it was put together by stitching a bunch of principles into one clean line.

The Schrödinger equation is the same. Not derived from a tidy chain of equations,,,

And honestly, I’ve heard it’s better for your mental health to just file it under “this is how the world works”??? ,,,,heh heh heh heh heh

heh heh heh heh heh heh

Anyway — cleanup hitter Schrödinger swings, hits a foul, gets caught out!! Because

the Schrödinger equation, at the end of the day, is just a complex number…. it doesn’t mean anything by itself,….(and to be fair, he didn’t know what it meant either)

Was there nobody who could send one out of the park….(sob)~~~

Our physics team is sitting in despair. But then a guy named Max Born, batting fifth, steps up to the plate,,, 1926 …

And announces that the square of the — the wave equation has probabilistic meaning. GRAND SLAM. Clears the bases~~~~~~~~~~~~~~

But — and here’s the thing — as Einstein, our 2nd batter, is jogging home to score, he goes:

“God does not play dice.”

Which is the irony, right? He himself just scored (= he literally put a run on the board for quantum mechanics)

but his face was glum the whole way around (= he never bought into quantum mechanics….) haha haha haha haha haha

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