Problem 1: Free Particle

Let’s solve the free particle

I guess

Newton’s Law F = ma

d^2 x / dt^2=0

Hence x = x_0 + vt

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

Whatever. Lagrangian version

L = T-V = \frac{1}{2}mv^2

Euler Lagrange Equations

\frac{d}{dt} \partial L / \partial \dot{q} = \partial_q L

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

S = \int L dt = \int \partial/\partial\dot{q} L \delta \dot{q} + \partial_q L \delta q

Nice.

H = \frac{p^2}{2m}

\dot{p}=-\partial_x H=0

p = Const

\dot{x}=\partial_p H=\frac{p}{m}

Okay. What about the quantum version?

Well p = \frac{\hbar}{i}\partial_x

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]=-\hbar/i

 

But what is

i \partial_t \psi = -\frac{\hbar^2}{2m}\nabla^2 \psi

E\psi =

Whatever. I’m bored.

Maybe I’ll do the path integral some other day

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