Let’s solve the free particle

I guess

Newton’s Law F=ma

d2x/dt2=0

Hence x=x0+vt

At least that works. Not sure I derived it particularly. Or proved it unique.

Whatever. Lagrangian version

L=TV=12mv2

Euler Lagrange Equations

ddtL/q˙=qL

How do you get that? By varying the action with fixed endpoints it’s the one that minimizes the path.

S=Ldt=/q˙Lδq˙+qLδq

Nice.

H=p22m

p˙=xH=0

p=Const

x˙=pH=pm

Okay. What about the quantum version?

Well p=ix

How do I know that? In particular it’s hard to remember where the i goes. Well, I memorized it at some point. It follows that

[x,p]=/i

But what is

itψ=22m2ψ

Eψ=

Whatever. I’m bored.

Maybe I’ll do the path integral some other day